Finite-size and asymptotic behaviors of the gyration radius of knotted cylindrical polygons

a r X i v :c o n d -m at/0002192v 1 14 F e b 2000Heat conduction in one dimensional nonintegrable systemsBambi Hu 1,2,Baowen Li 1,3∗,and Hong Zhao 1,41Department of Physics and Centre for Nonlinear Studies,Hong Kong Baptist University,China2Department of Physics,University of Houston,Houston TX 77204-55063Department of Physics,National University of Singapore,119260Singapore 4Department of Physics,Lanzhou University,730000Lanzhou,China(Received 17June 1999)Two classes of 1D nonintegrable systems represented by the Fermi-Pasta-Ulam (FPU)model and the discrete φ4model are studied to seek a generic mechanism of energy transport in microscopic level sustaining macroscopic behaviors.The results enable us to understand why the class represented by the φ4model has a normal thermal conductivity and the class represented by the FPU model does not even though the temperature gradient can be established.PACS numbers:44.10.+i,05.70.Ln,05.45.-a,66.70.+fHeat conduction in one-dimensional (1D)noninte-grable Hamiltonian systems is a vivid example for study-ing microscopic origin of the macroscopic irreversibility in terms of deterministic chaos.It is one of the oldest but a rather fundamental problem in nonequilibrium statisti-cal mechanics [1].Intended to understand the underlying mechanism of the Fourier heat conduction law,the study of heat conduction has attracted increasing attention in recent years [2–11].Based on previous studies,we can classify the 1D lat-tices into three categories.The first one consists of inte-grable systems such as the harmonic chain.It was rig-orously shown [12]that,in this category no tempera-ture gradient can be formed,and the thermal conductiv-ity is divergent.The second category includes a num-ber of nonintegrable systems such as the Lorentz gas model [2,10],the ding-a-ling and alike models [3],and the Frenkel-Kontorova (FK)model [5]and etc..In this category,the heat current is proportional to N −1and the temperature gradient dT/dx ∼N −1,thus the ther-mal conductivity κis a constant independent of system size N .The Fourier heat conduction law (J =−κdT/dx )is justified.The third category also includes some non-integrable systems such as the FPU [4,6]chain,the di-atomic Toda chain [7],the (mass)disorder chain [8],and the Heisenberg spin chain [9]and so on.In this cate-gory,although the temperature gradient can be set up with dT/dx ∼N −1,the heat current is proportional to N α−1with α∼0.43,the thermal conductivity κ∼N αwhich is divergent as one goes to the thermodynamic limit N →∞.These facts suggest that the nonintegrability is neces-sary to have a temperature gradient,but it is not suffi-cient to guarantee the normal thermal conductivity in a 1D lattice.This picture brings us to ask two questionsof fundamental importance:(i)Why do some noninte-grable systems have normal thermal conductivity,while the others fail?(ii)How can the temperature gradient be established in those nonintegrable systems having di-vergent thermal conductivity?The reason for the divergent thermal conductivity in integrable system is that the energy transports freely along the chain without any loss so that no temperature gradient can be established.The set up of temperature gradient in nonintegrable systems implies the existence of scattering.However,the different heat conduction be-havior in the two categories of nonintegrable systems in-dicates that the underlying mechanism must be different.To get the point,let’s write the Hamiltonian of a generic 1D lattice as H =iH i ,H i =p 2i∗To whom correspondence should be addressed.Email:bwli@.hk1models are the simplest anharmonic approximation of a monoatomic solid.In theφ4model,V takes the harmonic form,and the external potential U(x)=mx2/2+βx4/4, with mfixed to be zero in this paper.In the FPU model,U vanishes and V takes the anharmonic form of (x i−x i−1)2/2+β(x i−x i−1)4/4,andβ=1throughout this paper.In the case ofβ=0,the FPU model reduces to the harmonic chain.In our numerical simulations the Nos´e-Hoover ther-mostates[14]are put on thefirst and the last particles, keeping them at temperature T+and T−,respectively. The motion of these two particles are governed by¨x1=−ζ+˙x1+f1−f2,˙ζ+=˙x21/T+−1¨x N=−ζ−˙x N+f N−f N+1,˙ζ−=˙x21/T−−1.(2) where f i=−(V′+U′)is the force acting on the i’th particle.The equation of motion of other particle is ¨x i=f i−f i+1.The eighth-order Runge-Kutta algorithm was used.All computations are carried out in double ually the stationary state set in after107 time units.We should point out that we have performed computations by using other types of thermostate,and no qualitative difference has been found.thethehar-ofthe Fig.1(a)shows temperature profiles.In all noninte-grable systems,the temperature scales as T=T(i/N). However,in the FPU case there is a singular behavior near the two ends,which is a typical character of1D nonlinear lattices having divergent thermal conductivity. In the samefigure we also show the temperature pro-files for two integrable lattices:the harmonic and the monoatomic Toda models.In these two cases no temper-ature gradient could be set up and the stationary state corresponds to T=(T++T−)/2,which is consistent with the rigorous result[12].In Fig.1(b),we plot the quantity J×N versus N for the FPU model and theφ4model.The inset shows the same quantity for the harmonic chain and the monoatomic Toda chain.The local heatflux is defined by J i=˙x i∂Vφ4model,the head part of the profile becomes weaker and weaker.The reason is that in thefirst three cases (a-c)both the total energy and the total momentum are conserved,whereas in theφ4model the momentum con-servation breaks down due to the external potential.The inset in Fig.2(d)shows that the total momentum in theφ4model decreases at least exponentially with time. The decay of the momentum with time indicates a loss of correlation.It is thus reasonable to envisage the en-ergy transport along theφ4chain as a random walk-like scattering.The solitary waves in the FPU chain exchange energy and momentum when colliding with each other.It causes the energy loss,and the heat current decreases when the system size is increased.To show this,we start two ex-citations at the two ends of the chain with different mo-mentum,one moves to the right and another left.Let p1=6,p N=3and p i=0,i=1and N as our initial excitations.We calculate the momentums of the solitary waves(by simply summing up momentums of several lat-tices around the peaks)and investigate its change before and after the interaction.Wefind that the bigger one generally transfers part of its momentum and energy to the smaller one,as is shown in Fig.3(a).The collision takes place at t=850,where a peak is shown up.of athemo-de-tail).The horizontal line is the momentum before collision.(c)The maximal momentum gain∆p max versus initial mo-mentum p0.(d)The ratio of the heatflux of J(400)/J(800) versus the average temperature T=(T++T−)/2in the semi-logarithmic scale.The results shown here are for the FPU model.Moreover,the interaction between solitary waves is found to depend closely on a“phase”difference.Here the“phase”difference is defined as a time lag betweenthe excitations of two solitary waves.For instance,if we excite a solitary wave from the left end at time t,and another one from the right end at time t+δ,thenδis the“phase”difference.These two solitary waves,trav-eling through the chain in opposite directions,will col-lide with each other after a certain time.Although the physical meaning of the“phase”is not obvious,it is an important and good quantity to describe the interaction.We show p a L versusδfor two different kinds of collision in Fig3(b),where p a L is the momentum of the solitary wave from the left after collision.In thefirst case both left and right solitary waves have the same initial mo-mentum p L=p R=6.27,which is excited by an initial condition of p1=p N=6and p i=0for other i.In the second case,the left one has p L=6.27and the right one p R=3.28,excited by an initial condition of p1=6, p N=3and p i=0for i=1and N.Thefigure shows that P a L depends on the“phase”sinusoidally.Other interesting features of the collision of solitary waves are shown in Fig.3(c),where we plot the maxi-mum momentum gain∆p max versus the initial momen-tum p0for the FPU model.∆p max is measured by sub-tracting the initial momentum p0from the maximum p L in Fig.3b.First of all,this picture tells us that the ex-change of momentum and energy depends on the initial momentum and energy.Secondly,there exists a criti-cal momentum below which no energy exchange can take place.The critical p c0∼1.8is clearly seen in thefigure.For p0<p c0,∆p max is zero.This result is very significant, it indicates that there exists a threshold for the solitary wave interaction,below this threshold the interaction ceases,i.e.no momentum and energy exchanges between the solitary waves.A direct consequence of this fact is the existence of a threshold temperature below which the FPU chain should behave like a harmonic chain,namely, the excited waves travel freely along the chain without any energy loss,no temperature gradient can be set up, and the heat current remains a constant even though the size of the chain is changed.To testify this argument,we show the quantity J(400)/J(800)versus T=(T++T−)/2 in Fig.3(d),where J(N)is the heat currentflux for the system of size N.In the case of a size-independent J(N)one should get J(400)/J(800)=1,otherwise one would get J(400)/J(800)>1.Fig.3(d)captures this transition nicely for the FPU chain.The corresponding temperature threshold is about T c∼0.01.In the region of T∼0.001the numerical calculations do show that no temperature gradient is formed.The different scattering mechanism in the FPU chain and those chains having normal thermal conductivity leads to a different temperature dependence of ther-mal conductivityκ(T).In Fig.4we plotκ(T)for a FPU chain with an external potential of form U(x)=−γcos(2πx)for four different values ofγ=0,0.01,0.05 and0.1.The chain size isfixed at N=100.As pointed out above,in smallγregime such asγ=0and0.01,the 3energy transport is assisted by the solitary waves,the system has a largeκwhich decreases as the temperature is increased.However,in the opposite regime(γ=0.05 and0.1),the energy transport is diffusive and obeys the Forier law,κincreases with temperature,because more phonons are excited.transport process can be reached,and the heat conduc-tion obeys the Fourier law;(ii)Although the interaction of solitary waves makes it possible to set up temperature gradient in the FPU and alike nonintegrable models,the momentum conservation prohibits the diffusive transport and consequently leading to the divergent thermal con-ductivity.In addition,we have uncovered an important fact in the FPU model,namely,the existence of a thresh-old temperature,below which the FPU mode behaves like a harmonic chain.BL would like to thank G.Casati for useful discus-sions.This work was supported in part by Hong Kong Research Grant Council and the Hong Kong Baptist Uni-versity Faculty Research Grant.Note added in proof.–After submission of this pa-per we get to know the following results.Prosen and Campbell[15]proved in a more rigorous way that for a1D classical many-body lattice total momentum con-servation implies anomalous conductivity.The normal thermal conductivity in theφ4lattice has also been ob-served by Aoki and Kusnezov[16].The role of the exter-nal potential has been further studied by Tsironis et al [17].。

经济学研究范式的英文Economic research paradigms have evolved significantlyover the years, reflecting the complexity and dynamism of the field. The paradigms serve as frameworks that guideeconomists in their quest to understand and explain economic phenomena.One of the earliest and most influential paradigms is the Classical Economics paradigm, which emerged during the Industrial Revolution. It was characterized by a belief inthe self-regulating market and the 'invisible hand' that guides economic activity towards societal welfare. This paradigm emphasized the importance of laissez-faire policies and minimal government intervention.In contrast, the Keynesian Economics paradigm, which gained prominence in the mid-20th century, shifted the focus towards the role of government in managing economic cyclesand addressing unemployment. Keynes argued that aggregate demand, rather than market forces alone, determined the level of economic activity.The Neoclassical Economics paradigm, which emerged in the late 19th century, introduced the concept of marginal utility and the importance of individual choice in economic decisions. This paradigm also emphasized the role of equilibrium in markets and the efficiency of market outcomes.Behavioral Economics, a more recent paradigm, challenges the traditional assumptions of rationality in economic agents. It incorporates insights from psychology to explain anomalies in decision-making that deviate from the predictions of standard economic models.Another significant development is the Post-Keynesian Economics paradigm, which extends Keynes' insights to include issues of income distribution, financial instability, and the role of money and credit in the economy.The Institutional Economics paradigm, meanwhile, focuses on the role of social institutions and their impact on economic behavior and outcomes. It emphasizes the importanceof historical context and the evolution of economic systems.Finally, the Ecological Economics paradigm addresses the interdependence of economic systems with the environment, advocating for sustainable development and the integration of ecological concerns into economic policy.Each of these paradigms offers a unique lens throughwhich to view and interpret economic events and trends. The diversity of these paradigms reflects the multifaceted nature of economics as a discipline and the ongoing quest for a more comprehensive understanding of economic processes.。

Practical calculations usingfirst-principles QMConvergence,convergence,convergenceKeith RefsonSTFC Rutherford Appleton LaboratorySeptember18,2007Results of First-Principles Simulations (2)Synopsis (3)Convergence4 Approximations and Convergence (5)Convergence with basis set (6)Error Cancellation (7)Plane-wave cutoffconvergence/testing (8)Pseudopotentials and cutoffbehaviour (9)FFT Grid parameters (10)Force and Stress (11)Iterative Tolerances (12)K-point convergence (13)Golden rules of convergence testing (14)Structural Calculations15 Solids,molecules and surfaces (16)Convergence of Supercells for Molecules (17)Variable Volume Calculations (18)Variable Volume Calculations-II (19)Summary20 Summary (21)Results of First-Principles SimulationsFirst-principles methods may be used for subtle,elegant andaccurate computer experiments and insight into the structureand behaviour of matter.First principles results may be worthless nonsense2/21SynopsisThis aims of this lecture are1.To use the examples to demonstrate how to obtain converged results,ie correct predictions from the theory.2.How to avoid some of the common pitfalls and to avoid computing nonsense.Further reading:Designing meaningful density functional theory calculation in materials science-a primer Anne E Mattson et al.Model.Sim.Mater.Sci Eng.13R1-R31(2005).3/21Convergence4/21 Approximations and Convergences“Every ab-initio calculation is an approximate one”.s Distinguish physical approximationsx Born-Oppenheimerx Level of Theory and approximate XC functionaland convergable,numerical approximationsx basis-set size.x Integral evaluation cutoffsx numerical approximations-FFT gridx Iterative schemes:number of iterations and exit criteria(tolerances)x system sizes Scientific integrity and reproducibility:All methods should agree answer to(for example)“What is lattice constant of silicon in LDA approximation”if sufficiently well-converged.s No ab-initio calculation is ever fully-converged!5/21Convergence with basis setBasis set is fundamental approximation to shape of orbitals.How accurate need it be?s The variational principle states that E0≤<ψ|H|ψ>offprecision=COARSE/MEDIUM/FINE.s Though E is monotonic in E c it is not necessarily regular.6/21Error Cancellations Consider energetics of simple chemical reaction MgO (s)+H 2O (g)→Mg(OH)2,(s)s Reaction energy computed as ∆E =E product −P E reactants =E Mg (OH )2−(E MgO +E H 2O )sEnergy change on increasing E cut from 500→4000eVMgO -0.021eV H 2O -0.566eV Mg(OH)2-0.562eV Convergence error in ∆E -0.030eVs Errors associated with H atom convergence are similar on LHS and RHS and cancel in final result.s Energy differences converge much faster than ground-state energy.sAlways use same cutofffor all reactants and products .7/21FFT Grid parametersSome optimizations and tweaks of FFT grid dimensions...s FFT grid should be large enough to accommodate all G-vectors of density,n(r),within cutoff:G≤2G MAX.s Guaranteed to avoid”aliasing”errors in case of LDA and pseudopotentials without additional core-charge density.s In practice can often get away with1.5G MAX or1.75G MAX with little error penalty for LDA without core or augmentation charge.s GGA XC functionals give density with higher Fourier components,and need1.75G MAX-2G MAXs Finer grid may be needed to represent USP augmentation charges or core-charge densities.s CASTEP incorporates a second,finer grid for charge density to accommodate core/augmentation charges while usingG MAX for orbitals.s Set by either parameter fine scale(multiplier of G MAX)or finegmax is property of pseudopotential and transferable to other cells but fine scale is not.s(Rarely)may need to increase fine scale for DFPT phonon calculations using GGA functionals for acoustic sum rule to be accurately satisfied.10/21Iterative Tolerancess How to control the iterative solvers ?s Parameter elec tol specifies when SCF energy converged.s Optimizer also exits if max cycles reached –always check that it really did converge .sHow accurate does SCF convergence need to be?s Energetics:same accuracy of result.s Geometry/MD:much smaller energy tolerance needed to converge forces.s Cost of higher tolerance is only a few additional SCF iterations.sComing soon to a code near you –elec tol to exit SCF using force convergence criteriasInaccurate forces are common cause of geometry optimization failure.12/21Golden rules of convergence testing1.Test PW convergence on small model system(s)with same elements.2.Testing requires going to unreasonably high degree of convergence,so calculations could be expensive.Test oneparameter at a time.e knowledge of transferability of PW cutoffandfinegmax for USPs.e physical∆k spacing to scale k-point sampling from small system to large.e forces,stresses and other cheap properties as measure of convergence.7.May need small number of tests on full system andfinal property of interest(eg dielectric permittivities are verysensitive to k-point sampling).8.Write your convergence criteria in the paper,not just“250eV cutoffand10k-points in IBZ”.9.Convergence is achieved when value stops changing with parameter under test,NOT when the calculation exceedsyour computer resources and NOT when it agrees with experiment.14/21Structural Calculations15/21Solids,molecules and surfacesSometimes we need to compute a non-periodic system with a PBC code.s Surround molecule by vacuum space to model using periodic code.s Similar trick used to construct slab for surfaces.s Must include enough vacuum space that periodic images do not interact.s To model surface,slab should be thick enough that centre is“bulk-like”.s Beware of dipolar surfaces.Surface energy does not converge with slab thickness.s When calculating surface energy,try to use same cell for bulk and slab to maximise error cancellation of k-point set.s Sometimes need to compare dissimilar cells–must use absolutely converged k-point set as no error cancellation.16/21Variable Volume Calculationss Two ways to evaluate equilibrium lattice parameter-minimum of energy or zero of stress.s Stress converges less well than energy minimum with cutoff.Variable Volume Calculations-IIs Two possibilities for variable-cell MD or geometry optimization when using plane-wave basis set.s Infixed basis size calculation,plane-wave basis N PW is constant as cell changes.s Cell expansion lowers G max and K.E.of each plane wave,and therefore lowers effective E c.s Easier to implement but easy to get erroneous results.s Need very well-converged cutofffor success.sfixed cutoffcalculations reset basis for each volume,changing N PW but keeping G max and E cfixed.s This is almost always the correct method to use.18/21Summary19/21 Summarys Used with care,first principles simulations can give highly accurate predictions of materials properties.s Full plane-wave basis convergence is rarely if ever needed.Error cancellation ensure that energy differences,forces and stress converge at lower cutoff.s Convergence as a function of adjustable parameters must be understood and monitored for the property of interest to calculate accurate results.s Don’t forget to converge the statistical mechanics as well as the electronic structure!s A poorly converged calculation is of little scientific value.20/21。

FINITE DIMENSIONAL REDUCTION OF NONAUTONOMOUS DISSIPATIVESYSTEMSAlain MiranvilleUniversit´e de Poitiers Collaborators:Long time behavior of equations of the formy′=F(t,y)For autonomous systems:y′=F(y)In many situations,the evolution of the sys-tem is described by a system of ODEs:y=(y1,...,y N)∈R N,F=(F1,...,F N)Assuming that the Cauchy problemy′=F(y),y(0)=y0,is well-posed,we can define the family of solv-ing operators S(t),t≥0,acting on a subset φ⊂R N:S(t):φ→φy0→y(t)This family of operators satisfiesS(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0We say that it forms a semigroup onφQualitative study of such systems:goes back to Poincar´eMuch is known nowadays,at least in low di-mensionsEven relatively simple systems can generate very complicated chaotic behaviorsThese systems are sensitive to perturbations: trajectories with close initial data may diverge exponentially→Temporal evolution unpredictable on ti-me scales larger than some critical value→Show typical stochastic behaviorsExample:Lorenz systemx′=σ(y−x)y′=−xy+rx−yz′=xy−bzObtained by truncature of the Navier-Stokes equationsGives an approximate description of a layer of fluid heated from belowSimilar to what is observed in the atmosphereFor a sufficiently intense heating:sensitive dependence on the initial conditions,repre-sents a very irregular convection→Butterfly effectVery often,the trajectories are localized in some subset of the phase space having a very complicated geometric structure(e.g.,locally homeomorphic to the product of R m and a Cantor set)→Strange attractor(Ruelle and Takens)Main feature of a strange attractor:dimen-sionSensitivity to initial conditions:>2(dimen-sion of the phase space≥3,say,3)Contraction of volumes:its volume is equal to0→noninteger,strictly between2and3→Fractal dimensionExample:Lorenz system:dim F A=2.05...Distributed systems:systems of PDEsφis a subset of an infinite dimensional func-tion space(e.g.,L2(Ω)or L∞(Ω))Solution:y:R+→φt→y(t)x→y(t,x)If the problem is well-posed,we can define the semigroup S(t):S(t):φ→φy0→y(t)The analytic structure of a PDE is much more complicated than that of an ODE:the global well-posedness can be a very difficult problemSuch results are known for a large class of PDEs→it is natural to investigate whether the notion of a strange attractor extends to PDEsSuch chaotic behaviors can be observed in dissipative PDEsChaotic behaviors arise from the interaction of•Energy dissipation in the higher part of the Fourier spectrum•External energy income in the lower part•Energyflux from the lower to the higher modesThe trajectories are localized in a”thin”in-variant region of the phase space having a very complicated geometric structure→the global attractor1.The global attractor.S(t)semigroup acting on E:S(t):E→E,t≥0S(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0 Continuity:x→S(t)x is continuous on E,∀t≥0A set A⊂E is the global attractor for S(t)if(i)it is compact(ii)it is invariant:S(t)A=A,t≥0(iii)∀B⊂A,lim t→+∞dist(S(t)B,A)=0dist(A,B)=supa∈A infb∈Ba−b EEquivalently:∀B⊂φbounded,∀ǫ>0,∃t0= t0(B,ǫ)s.t.t≥t0implies S(t)B⊂UǫThe global attractor is uniqueIt is the smallest closed set enjoying(iii)It is the maximal bounded invariant setTheorem:(Babin-Vishik)We assume that S(t)possesses a compact attracting set K, i.e.,∀B⊂E bounded,lim t→+∞dist(S(t)B,K)=0Then S(t)possesses the global attractor A.The global attractor is oftenfinite dimen-sional:the dynamics,restricted to A isfinite dimensionalFractal dimension:Let X be a compact setdim F X=lim supǫ→0+ln Nǫ(X)ǫNǫ(X):minimum number of balls of radius ǫnecessary to cover XIf Nǫ(X)≤c(1Theorem:(H¨o lder-Ma˜n´e theorem)Let X⊂E compact satisfy dim F X=d and N>2d be an integer.Then almost every bounded linear projector P:E→R N is one-to-one on X and has a H¨o lder continuous inverse.This result is not valid for other dimensions (e.g.,the Hausdorffdimension)If A hasfinite fractal dimension,then,fixing a projector P satisfying the assumptions of the theorem,we obtain a reduced dynamical system(S),S= P(A),which isfinite dimensional(in R N)and H¨o lder continuousDrawbacks:(S)cannot be realized as a system of ODEs which is well-posedReasonable assumptions on A which would ensure that the Ma˜n´e projectors are Lipschitz are not knownComplicated geometric structure of A and AThe lower semicontinuitydist(A0,Aǫ)→0asǫ→0is more difficult to prove and may not hold It may be unobservable:∂y∂x2+y3−y=0,x∈[0,1],ν>0y(0,t)=y(1,t)=−1,t≥0A={−1}There are many metastable”almost station-ary”equilibria which live up to t⋆≡eν−12.Inertial manifolds.A Lipschitzfinite dimensional manifold M⊂E is an inertial manifold for S(t)if(i)S(t)M⊂M,∀t≥0(ii)∀u0∈E,∃v0∈M s.t.S(t)u0−S(t)v0 E≤Q( u0 E)e−αt,α>0,Q monotonicM contains A and attracts the trajectories exponentiallyConfirms in a perfect way thefinite dimen-sional reduction principle:The dynamics reduced to M can be realized as a Lipschitz system of ODEs(inertial form)Perfect equivalence between the initial sys-tem and the inertial formDrawback:all the known constructions are based on a restrictive condition,the spectral gap condition→The existence of an inertial manifold is not known for several important equations, nonexistence results for damped Sine-Gordon equations3.Exponential attractors.A compact set M⊂E is an exponential at-tractor for S(t)if(i)It hasfinite fractal dimension(ii)S(t)M⊂M,∀t≥0(iii)∀B⊂E bounded,dist(S(t)B,M)≤Q( B E)e−αt,α>0,Q monotonicM contains AIt is stillfinite dimensional and one has a uni-form exponential control on the rate of at-traction of trajectoriesIt is no longer smoothDrawback:it is not unique→One looks for a simple algorithm S→M(S)Initial construction:non-constructible and valid in Hilbert spaces onlyConstruction in Banach spaces:Efendiev, Miranville,Zelik→Exponential attractors are as general as global attractorsMain tool:Compact smoothing property on the difference of2solutionsLet S:E→E.We consider the discrete dynamical system generated by the iterations of S:S n=S◦...◦S(n times)Theorem:(Efendiev,Miranville,Zelik)We consider2Banach spaces E and E1s.t.E1⊂E is compact.We assume that•S maps theδ-neighborhood Oδ(B)of a bounded subset B of E into B•∀x1,x2∈Oδ(B),≤K x1−x2 ESx1−Sx2 E1Then the discrete dynamical system gener-ated by the iterations of S possesses an ex-ponential attractor M(S)s.t.(i)M(S)⊂B,is compact in E anddim F M(S)≤c1(ii)S M(S)⊂M(S)(iii)dist(S k B,M(S))≤c2e−c3k,k∈N,c3>0 (iv)The map S→M(S)is H¨o lder continu-ous:∀S1,S2,dist sym(M(S1),M(S2))≤c4 S1−S2 c5,c5>0, wheredist sym(A,B)=max(dist(A,B),dist(B,A))S =supSh Eh∈Oδ(B)Furthermore all the constants only depend on B,E,E1,δand K and can be computed explicitly.Remarks:1)We have a mapping S→M(S)and,due to the H¨o lder continuity,we can construct continuous families of exponential attractors2)Exponential attractors for a continuous semigroup S(t):Prove that∃t⋆>0s.t.S⋆=S(t⋆)satisfies the assumptions of the theorem→M⋆for S⋆If(x,t)→S(t)x is Lipschitz(or H¨o lder)on B×[0,t⋆],setS(t)M⋆M=∪t∈[0,t⋆]We again have a mapping S(t)→M(S)which is H¨o lder continuous3)For damped hyperbolic equations:asymp-totically smoothing property4.Finite dimensional reduction of nonau-tonomous systems.Systems of the form∂yDrawback:the uniform attractor has infinite dimension in general.Example:∂yThe family{A(t),t∈R}is a pullback attrac-tor for U(t,τ)if(i)A(t)is compact in E,∀t∈R(ii)U(t,τ)A(τ)=A(t),∀t≥τ(iii)∀B⊂E bounded,dist(U(t,t−s)B,A(t))=0lims→+∞Remarks:1)The pullback attractor is unique2)If the system is autonomous,we recover the global attractor3)In general,A(t)hasfinite fractal dimen-sion,∀t∈RDrawback:The forward convergence does not hold in generalExample:y′=f(t,y),where f(t,y)=−y if y≤0,(−1+2t)y−ty2 if t∈[0,1],and y−y2if t≥1Then A(t)={0},∀t∈R,but every trajectory starting from a neighborhood of0leaves this neighborhood never to enter it againThe forward convergence does not hold be-cause the rate of attraction is not uniform in t→This can be solved by constructing ex-ponential attractorsWe can construct a family{M(t),t∈R}, called nonautonomous exponential attractor, s.t.(i)dim F M(t)≤c1,∀t∈R,c1independent of t(ii)U(t,τ)M(τ)⊂M(t),∀t≥τ,(iii)∀B⊂E bounded,dist(U(t,τ)B,M(t+τ))≤Q( B E)e−αt,t∈R,t≥τ,α>0,Q monotonic(iii)implies the pullback attraction,but also the forward attraction→(i)and(iii)yield a satisfactoryfinite di-mensional reduction principle for nonautono-mous systemsRemarks:1)The time dependence is arbitrary2)The map U(t,τ)→{M(t),t∈R}is also H¨o lder continuous。

本期推荐本栏目责任编辑：王力基于支持向量机的弗兰克-赫兹实验曲线拟合周祉煜1,孟倩2（1.河北师范大学物理学院，河北石家庄050024；2.江苏师范大学计算机科学与技术学院，江苏徐州221116）摘要：弗兰克-赫兹实验是“近代物理实验”中的重要实验之一，数据量大且数据处理复杂。

支持向量机是一种广泛应用于函数逼近、模式识别、回归等领域的机器学习算法。

本文将支持向量机算法应用于弗兰克－赫兹实验数据的拟合，过程简单，在python 环境下验证该方法拟合精度高，效果好。

支持向量机算法还可应用于其他的物理实验曲线拟合。

关键词：支持向量机；曲线拟合；弗兰克-赫兹实验；Python 中图分类号：TP18文献标识码：A文章编号：1009-3044(2021)13-0001-02开放科学（资源服务）标识码（OSID ）：Curve Fitting of Frank Hertz Experiment Based on Support Vector Machine ZHOU Zhi-yu 1,MENG Qian 2(1.Hebei Normal University,College of physics.,Shijiazhuang 050024,China;2.School of Computer Science and technology,Jiang⁃su Normal University,Xuzhou 221116,China)Abstract:Frank-Hertz experiment is a classical experiment in modern physics experiments.It has a large amount of experimental data and a complicated data processing process.Support Vector Machine is a machine learning algorithm which widely used in function approximation,pattern recognition,regression and other fields.In this paper,support vector machine is used to do curve fitting for the experimental data of Frank-Hertz experiment.The process is simple,and the method is verified to have high curve fit⁃ting accuracy and good effect in python environment.SVM can also be applied to curve fitting in other physics experiments.Key words:support vector machine,curve fitting,Frank Hertz experiment ，python 1998年，Vapnik V N.等人[1]提出了一种新型的基于小样本和统计学习理论的机器学习方法-支持向量机(Support Vector Machine,SVM)，该方法可以从有限的训练样本出发寻找“最优函数规律”,使它能够对未知输出作尽可能准确的预测，可应用于函数逼近、模式识别、回归等领域。

INFORMATION CRITERION AND CHANGE POINTPROBLEM FOR REGULAR MODELSInformation criteria are commonly used for selecting competing statistical models. They do not favor the model which gives the best to the data and little interpretive value, but simpler models with good fit. Thus, model complexity is an important factor in information criteria for model selection. Existing results often equate the model complexity to the dimension of the parameter space. Although this notion is well founded in regular parametric models, it lacks some desirable properties when applied to irregular statistical models. We refine the notion of model complexity in the context of change point problems, and modify the existing information criteria. The modified criterion is found consistent in selecting the correct model and has simple limiting behavior. The resulting estimatorof the location of the change point achieves the best convergence rate Op(1), and its limiting distribution is obtained. Simulation results indicate that the modified criterion has better power in detecting changes compared to other methodsIntroductionOut of several competing statistical models, we do not always use the one with the best to the data. Such models may simply interpolate the data and have little interpretive value. Information criteria, such as the Akaike information criterion and the Schwarz information criterion, are designed to select models with simple structure and good interpretive value, see Akaike (1973) and Schwarz (1978). The model complexity is often measured in terms of the dimensionality of the parameter space.Consider the problem of making inference on whether a process has undergone some changes. In the context of model selection, we want to choose between a model with a single set of parameters, or a model with two sets of parameters plus the location of change. The Akaike and the Schwarz information criteria can be readily adopted to this kind of change point problems. There have been many fruitful research done in this respect such as Hirotsu, Kuriki and Hayter (1992) and Chen and Gupta (1997), to name a few.Compared to usual model selection problems, the change point problem contains a special parameter: the location of the change. When it approaches the beginning or the end of the process, one ofthe two sets of the parameter becomes completely redundant. Hence, the model is un-necessarily complex. This observation motivates the notion that the model complexity also depends on the location of the change point. Consequently, we propose to generalize the Akaike and Schwarz information criteria by making the model complexity also a function of the location of the change point. The new method is shown to have a simple limiting behavior, and favourable power properties in many situations via simulation.The change point problem has been extensively discussed in the literature in recent years.The study of the change point problem dates back to Page (1954, 1955 and 1957) which tested the existence of a change point. Parametric approaches to this problem have been studied by a number of researchers, see Chernoff and Zacks (1964), Hinkley (1970), Hinkley et.al.(1980), Siegmund (1986) and Worsley (1979, 1986). Nonparametric tests and estimations have also been proposed (Brodsky and Darkhovsky, 1993; Lombard, 1987; Gombay and Huskova, 1998). Extensive discussions on the large sample behavior of likelihood ratio test statistics can be found in Gombay and Horvath (1996) and Csorgo and Horvath (1997).The detail can be found in some survey literatures such asBhattacharya (1994), Basseville and Nikiforov (1993), Zacks (1983), and Lai (1985). The present study deviates from other studies by refining the traditional measure of the model complexity, and bydetermining the limiting distribution of the resulting test statistic under very general parametric model settings.In Section 2, we define and motivate the new informationcriterion in detail. In Section 3, we give the conditions under which the resulting test statistic has chi-square limiting distribution and the estimator τ of change point attains the best convergence rate. An application example and some simulation results are given in Section4. The new method is compared to three existing methods and found to have good finite sample properties. The proofs are given in the Appendix.Main ResultsLet X 1，X 2， ……，Xn be a sequence of independent randomvariables. It is suspected that Xi has density function 1f (,)x θ when i<k and density 2f (,)x θ for i>k . We assume that 1f (,)x θand 2f (,)x θbelong to the same parametric distribution family {f (,):}d x R θθ∈.The problem is to test whether this change has indeed occurred and if so, find the location of the change k . The null hypothesis is 0H : 12θθ= and the alternative is 1H : 12θθ≠and 1k n <<Equivalently, we are asked to choose a model from 0H or a modelfrom 1H for the data.For regular parametric (not change point) models with loglikelihood function ()n l θ, Akaike and Schwarz information criteria are defined as:2()2dim()2()2dim()log()n n AIC l SIC l n θθθθ=-+=-+where θ is the maximum point of ()n l θ. The best model according to these criteria is the one which minimizes AIC or SIC . The Schwarz information criterion is asymptotically optimal according to certain Bayes formulation.The log likelihood function for the change point problem has the form121211(,;)log (,)log (,)k nn i i i i k l k f x f x θθθθ==+=+∑∑ The Schwarz information criterion for the change point problem becomes12()2(,;)[2dim()1]log()n SIC k l k n θθθ=-++and similarly for Akaike information criterion, where 12,θθmaximize 12(,;)n l k θθfor given k . See, for example, Chen and Gupta (1997). When the model complexity is the focus, we may also write it as1212()2(,;)(,;)log()n SIC k l k complexity k n θθθθ=-+We suggest that the notion of 12(,;)complexity k θθ = 2dim()1θ+ needs re-examination in the context of change point problem. When k takes values in the middle of 1 and n , both 1θand 2θare effectiveparameters. When k is near 1 or n , either 1θor 2θbecomes redundant.Hence, k is an increasingly undesirable parameter as k getting close to 1 or n . We hence propose a modified information criterion with2122(,;)2dim()(1)an k complexity k const t nθθθ=+-+ For 1<k<n, let 21212()2(,;)[2dim()(1)]log()n k MIC k l k n n θθθ=-++- Under the null model, we define12()2(,;)dim()log()n MIC k l k n θθθ=-+If 1()min ()k nMIC n MIC k <<>, we select the model with a change point and estimate the change point by τ such that1()min ()k nMIC MIC k τ<<= Clearly, this procedure can be repeated when a second change point is suspected.The size of model complexity can be motivated as follows. If the change point is at k , the variance of 1θ would be proportional to 1k -and the variance of 2θ would be proportionalto 1()n k --. Thus, the total variance is1211111[()]42k n k n k n --+=--- The specific form in (2) reflects this important fact. Thus, if a change at an early stage is suspected, relatively stronger evidence is needed to justify the change. Hence, we should place larger penalty when k is near 1 or n . This notion is shared by many researchers.The method in Inclan and Tiao (1994) scales down the statistics heavier when the suspected change point is near 1 or n . The U-statistic method in Gombay and Horvath (1995) is scaleddown by multiplying the factor k (n-k ).To assess the error rates of the method, we can simulate the finite sample distribution, or find the asymptotic distribution of the related statistics. For Schwarz information criterion,the relatedstatistic is found to have type I extreme value distributionasymptotically (Chen and Gupta, 1997; CsÄorgÄo and Horvath 1997). We show that the MIC statistic has chi-square limiting distribution for any regular distribution family, the estimator τ achieves the best convergence rate Op (1) and has a limiting distribution expressed via a random walk.Our asymptotic results under alternative model is obtained under the assumption that the location of the change point k , Thus, {:1,2}in X i n n <<> form a triangle array. The classical results on almost sure convergence for independent and identically distributed (iid) random variables cannot be directly applied. However, the conclusions on weak convergence will not be affected as the related probability statements are not affected by how one sequence is related to the other. Precautions will be taken on this issue but details will be omitted.Let1()min ()dim()log()n k nS MIC n MIC k n θ<<=-+ where MIC (k ) and MIC (n ) are defined in (3) and (4). Note that this standardization removes the constant term dim()log()n θ in the difference of MIC (k ) and MIC (n ).常见模型信息准则和变点分析问题 信息准则通常是用来选择统计模型的优劣。

asymptotic analysis缩写Asymptotic analysis is a mathematical method used to analyze the behavior of an algorithm as the input size approaches infinity. It is widely used in computer science and engineering to compare different algorithms and design efficient algorithms. In this article, we will discuss the basics of asymptotic analysis and its important concepts.Firstly, we need to understand the importance of asymptotic analysis. In computer science and engineering, we often deal with huge datasets and complex algorithms. Therefore, it is important to know how the algorithm will behave as the input size becomes large. Asymptotic analysis helps us to estimate the computational time and space complexity of an algorithm for large inputs. This estimation can help us to choose the best algorithm for a given problem.Asymptotic analysis is based on the concept of limits. A limit is the value a function approaches as the input value approaches a certain point. We use big O, big Omega, and big Theta notations to express the growth rate of a function. These notations give us a rough idea about the behavior of the function.Big O notation: The big O notation gives the upper bound of the running time of an algorithm. We say that algorithm A has a time complexity of O(f(n)) if the running time of the algorithm does not exceed a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is less than or equal to 2n^2+3n+4, we can say that the time complexity of the algorithm A is O(n^2).Big Omega notation: The big Omega notation gives the lower bound of the running time of an algorithm. We say that algorithm A has a time complexity of Omega(f(n)) if the running time of the algorithm is not less than a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is greater than or equal to n^2/2, we can say that the time complexity of the algorithm A is Omega(n^2).Big Theta notation: The big Theta notation gives the tight bounds of the running time of an algorithm. We say that algorithm A has a time complexity of Theta(f(n)) if the running time of the algorithm is between a constant multiple of f(n) and another constant multiple of f(n) for large n. For example, if the running time of the algorithm A is between 5n^2+3n+4 and 7n^2+5n+6, we can say that the time complexity of the algorithm A is Theta(n^2).Asymptotic analysis also covers the space complexity of an algorithm. We use the same notations to express the growth rate of the space usage of an algorithm. For example, if the space used by the algorithm A is less than or equal to 3n+4, we can say that the space complexity of the algorithm A is O(n).In conclusion, asymptotic analysis is an important concept in computer science and engineering. It helps us to estimate the computational time and space complexity of an algorithm for large inputs. By using the big O, big Omega, and big Theta notations, we can compare different algorithms and choose the best algorithm for a given problem.。

Numerical Methods for Convection-Dominated Diffusion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Difference ProceduresAuthor(s): Jim Douglas, Jr. and Thomas F. RussellSource: SIAM Journal on Numerical Analysis, Vol. 19, No. 5 (Oct., 1982), pp. 871-885 Published by: Society for Industrial and Applied MathematicsStable URL: /stable/2156980Accessed: 27/04/2010 00:10Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at/action/showPublisher?publisherCode=siam.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@.Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Journal on Numerical Analysis.。

感官分析术语与分析方法有关的术语The document was prepared on January 2, 2021GB —88本标准参照采用国际标准ISO 5492/1～6感官分析──词汇.1 主题内容和适用范围本标准规定了感官分析与分析方法有关的术语.2 术语及其定义被检样品test sample被检验产品的一部分.被检部分test portion直接提交评价员检验的那部分被检样品.参照值reference point与被评价的样品对比的一个选择的值一个或几个特性值,或者某产品的值.对照样control选择用作参照值的被检样品.所有其他样品都与其作比较.参比样reference本身不是被检材料,而是用来定义一个特性或者一个给定特性的某一指定水平的物质.差别检验difference test对两种样品进行比较的检验方法.偏爱检验preference test对两种或多种样品估价更喜欢哪一种的检验方法.成对比较检验paired comparison test为了在某些规定的特性基础上进行比较,而成对地给出样品的一种差别检验方法.三点检验triangular test差别检验的一种方法.同时提供三个已编码的样品,其中有两个样品是相同的,要求评价员挑出其中单个的样品.二-三点检验duo-trio test差别检验的一种方法.首先提供对照样品,接着提供两个样品,其中之一与对照样相同,要求评价员识别.“五中取二”检验“two out of five” test差别检验的一种方法.五个已编码的样品,其中有两个是一种类型的,其余三个是另一种类型,要求评价员将这些样品按类型分成两组.“A”-“非A”检验“A”or“not A”test差别检验的一种方法.当评价员学会识别样品“A”以后,将一系列可能是“A”或“非A”的样品提供给他们,要求评价员指出每一个样品是“A”还是“非A”.排序ranking按指定指标的强度或程度排列一系列样品的分类方法.这种方法只将样品排定次序而不估计样品之间差别的大小.分类classification将样品划归到预先定义的名义类别的方法.评估rating按照类别分类的方法.每种类别按有序标度排列.这是一个排定顺序的过程.评分scoring一种使用数字标度评估的形式.在评分中所使用的数字形成等距或比率标度.分等grading由优选评价员或专家在一个或多个指标的基础上对产品按其质量分类.简单描述检验simple descriptive test对样品的各个指标定性描述的一种检验.这些指标构成了样品的整个特征.定量描述和感官剖面检验quantative descriptive and sensory profile tests用可以再现的方式评价产品感官特性的一种检验方法或理论分析的方法.这种评价是使用以前由简单描述检验确定的词汇中选择的词.稀释法dilution method以逐渐降低的浓度制备样品,然后顺序地检验.筛选screening初步的选择过程.配比matching把一对一对出现的相关的样品相等同的过程.通常用以确定对照的和未知的样品之间或两个未知的样品之间的相似程度.客观方法objective method受人为因素影响最小的方法.主观方法subjective method受人为因素影响较大的方法.量值估计magnitude estimation对指标的强度定值的过程.被定值的比率和评价员的感觉是相同的.独立评价independent assessment在没有直接比较的情况下,评价一种或多种刺激.比较评价comparative assessment对同时出现的样品的比较.质量要素quality factor被挑选用以评价某产品整体质量的因素.标度scale报告评价结果所使用的尺度.它是由顺序相连的一些值组成的系统.这些值可以是图形的,描述的或数字的形式.快感标度hedonic scale表达喜欢或厌恶程度的一种标度.单极标度unipolar scale有零端点的一种标度.例如从无味到很甜这样一种表示溶液味道的标度双极标度bipolar scale在两端点有相反刻度的一种标度.例如从硬的到柔软的这样一种质地标度顺序标度ordinal scale以预先确定的单位或以连续级数作单位的一种标度.顺序标度既无绝对零点又无相等单位,因此这种标度只能提供对象强度的顺序,而不能提供对象之间差异的大小.等距标度interval scale有相等单位但无绝对零点的标度.相等的单位是指相同的数字间隔代表了相同的感官知觉差别.等距标度可以度量对象强度之间差异的大小,但不能比较对象强度之间的比率.比率标度ratio scale既有绝对零点又有相等单位的标度.比率标度不但可以度量对象强度之间的绝对差异,又可度量对象强度之间的比率.这是一种最精确的标度.附加说明：本标准由中华人民共和国农业部提出.本标准由全国农业分析标准化技术委员会归口.本标准由中国标准化综合研究所、中国农科院分析室负责起草.本标准主要起草人毕健、陈必芳、周苏玉.附录 A汉 语 索 引参考件A“A”－“非A”检验 ……………………B被检部分 ………………………………被检样品 ……………………………………… 比较评价……………………………… 比率标度……………………………………… 标度……………………………………C参比样 ………………………………… 参照值 ………………………………………… 差别检验 ……………………………… 差别阈………………………………………… 尝味 …………………………………… 成对比较检验 ………………………………… 稠度…………………………………… 初级评价员 …………………………………… 刺激 …………………………………… 刺激阈…………………………………………D单极标度……………………………… 等距标度……………………………………… 定量描述和感官剖面检动验…………… 觉…………………………………………… 独立评价……………………………… 对比效应……………………………………… 对照样 …………………………………E二-.三点检验………………………F乏味的………………………………… 芳香…………………………………………… 分等…………………………………… 分类…………………………………………… 风味 …………………………………… 风味增强剂…………………………………… 肤觉……………………………………G感官分析 ……………………………… 感官检查 ……………………………………… 感官检验 ……………………………… 感官疲劳……………………………………… 感官评价 ……………………………… 感官适应……………………………………… 感官特性 ……………………………… 感觉 …………………………………………… 感受器 …………………………………H后味…………………………………… 厚味的…………………………………………J基本味假热效简单描述检验…………………………碱味的 ………………………………………… 拮抗效应……………………………… 结实的………………………………………… 接受…………………………………… 觉察阈…………………………………………K开胃…………………………………… 客观方法……………………………………… 可接受性……………………………… 可口性………………………………………… 口感…………………………………… 苦味的 ………………………………………… 快感标度………………………………L老的…………………………………… 量值估计………………………………………M敏感性………………………………N嫩的…………………………………P排序…………………………………… 配比…………………………………………… 偏爱检验 ……………………………… 品尝 …………………………………………… 品尝员 ………………………………… 评分…………………………………………… 评评价小评价员 ………………………………… 平味的…………………………………………Q气味…………………………………… 气味测量……………………………………… 强度…………………………………… 区别……………………………………………R柔软的…………………………………S三点检验 ……………………………… 色觉障碍……………………………………… 涩味的 ………………………………… 筛选…………………………………………… 识别阈………………………………… 视觉…………………………………………… 食欲…………………………………… 手感…………………………………………… 收敛效应………………………………双极标度……………………………………… 顺序标度……………………………… 酥的…………………………………………… 酸感 …………………………………… 酸味的 …………………………………………T特征…………………………………… 甜味的 ………………………………………… 听觉……………………………………W外观…………………………………… 味道…………………………………………… 味觉 …………………………………… 味觉缺失……………………………………… 无味的………………………………… “五中取二”检验……………………………X稀释法………………………………… 咸味的 …………………………………………协同效应……………………………… 心理物理学……………………………………嗅 ……………………………………… 嗅觉 ……………………………………………嗅觉测量……………………………… 嗅觉测量仪……………………………………嗅觉过敏……………………………… 嗅觉减退………………………………………嗅觉缺失………………………………Y掩蔽…………………………………… 厌恶……………………………………………颜色…………………………………… 硬的……………………………………………异常风味 ……………………………… 异常气味 ………………………………………异常特征……………………………… 优选评价员 ……………………………………有硬壳的……………………………… 阈上的…………………………………………余味…………………………………… 阈下的…………………………………………阈………………………………………Z最大阈………………………………… 玷染……………………………………………质地…………………………………… 滞留度…………………………………………知觉 …………………………………… 质量要素………………………………………主观方法……………………………… 专家 ……………………………………………附 录 B英文索引参考件A“A”or“not A”test ……………………………………………………………………………… acceptability………………………………………………………………………………………… acceptance …………………………………………………………………………………………… acid……………………………………………………………………………………………………… after-taste…………………………………………………………………………………………… ageusia………………………………………………………………………………………………… alkaline………………………………………………………………………………………………… anosmia………………………………………………………………………………………………… antagonism …………………………………………………………………………………………… appearance …………………………………………………………………………………………… appetising …………………………………………………………………………………………… appetite ……………………………………………………………………………………………… aroma…………………………………………………………………………………………………… assessor………………………………………………………………………………………………… astringent……………………………………………………………………………………………… auditory sensation ………………………………………………………………………………… aversion ………………………………………………………………………………………………Bbipolar scale………………………………………………………………………………………… bitter…………………………………………………………………………………………………… bland……………………………………………………………………………………………………Ccolour …………………………………………………………………………………………………comparative assessment ……………………………………………………………………………consistency……………………………………………………………………………………………contrast effect………………………………………………………………………………………control …………………………………………………………………………………………………convergence effect …………………………………………………………………………………crisp……………………………………………………………………………………………………crusty …………………………………………………………………………………………………Ddetection threshold…………………………………………………………………………………difference test ………………………………………………………………………………………difference threshold ………………………………………………………………………………dilution method………………………………………………………………………………………discrimination ………………………………………………………………………………………duo-trio test…………………………………………………………………………………………dyschromalops ia………………………………………………………………………………………Eexpert……………………………………………………………………………………………………Ffirm ……………………………………………………………………………………………………flavour …………………………………………………………………………………………………flavour enhancer ……………………………………………………………………………………flavourless……………………………………………………………………………………………Ggrading…………………………………………………………………………………………………gustation ………………………………………………………………………………………………Hhandfeel ………………………………………………………………………………………………hard ……………………………………………………………………………………………………hyperosmia ……………………………………………………………………………………………hyposmia ………………………………………………………………………………………………Iindependent assessment ……………………………………………………………………………insipid…………………………………………………………………………………………………intensity………………………………………………………………………………………………interval scale ………………………………………………………………………………………Kkinesthesis……………………………………………………………………………………………Mmagnitude estimation ………………………………………………………………………………masking…………………………………………………………………………………………………matching ………………………………………………………………………………………………mouthfeel………………………………………………………………………………………………Nnote ……………………………………………………………………………………………………Oobjective method ……………………………………………………………………………………odorimetry ……………………………………………………………………………………………odour……………………………………………………………………………………………………off-flavour ……………………………………………………………………………………………off-note ………………………………………………………………………………………………off-odour ………………………………………………………………………………………………olfactometer …………………………………………………………………………………………olfactometry …………………………………………………………………………………………olfaction ………………………………………………………………………………………………ordinal scale…………………………………………………………………………………………organoleptic attribute………………………………………………………………………………Ppaired comparison test………………………………………………………………………………palatability …………………………………………………………………………………………panel ……………………………………………………………………………………………………perception………………………………………………………………………………………………persistence……………………………………………………………………………………………preference test ………………………………………………………………………………………primary assessor………………………………………………………………………………………primary taste…………………………………………………………………………………………pseudothermal effects………………………………………………………………………………psychophysics…………………………………………………………………………………………Qquality factor ………………………………………………………………………………………quantative descriptive and sensory profile tests …………………………………………Rranking…………………………………………………………………………………………………rating …………………………………………………………………………………………………ratio scale……………………………………………………………………………………………receptor…………………………………………………………………………………………………recognition threshold………………………………………………………………………………reference ………………………………………………………………………………………………reference point ………………………………………………………………………………………residual taste ………………………………………………………………………………………Ssalty ……………………………………………………………………………………………………sapid……………………………………………………………………………………………………scale……………………………………………………………………………………………………scoring…………………………………………………………………………………………………screening………………………………………………………………………………………………selected assessor ……………………………………………………………………………………sensation ………………………………………………………………………………………………sensitivity……………………………………………………………………………………………sensory adaptation …………………………………………………………………………………sensory analysis………………………………………………………………………………………sensory evaluation……………………………………………………………………………………sensory examination …………………………………………………………………………………sensory test……………………………………………………………………………………………sensory fatigue………………………………………………………………………………………simple descriptive test……………………………………………………………………………skin sensation ………………………………………………………………………………………smell,to…………………………………………………………………………………………………soft ……………………………………………………………………………………………………sour………………………………………………………………………………………………………stimulus…………………………………………………………………………………………………stimulus threshold …………………………………………………………………………………subjective method……………………………………………………………………………………sub-threshold…………………………………………………………………………………………supra-threshold………………………………………………………………………………………sweet ……………………………………………………………………………………………………synergism………………………………………………………………………………………………Ttaint……………………………………………………………………………………………………taste ……………………………………………………………………………………………、tasteless………………………………………………………………………………………………taster………………………………………………………………………………………………………tasting ……………………………………………………………………………………………………tender ……………………………………………………………………………………………………terminal threshold ……………………………………………………………………………………test portion………………………………………………………………………………………………test sample ………………………………………………………………………………………………texture……………………………………………………………………………………………………threshold…………………………………………………………………………………………………tough………………………………………………………………………………………………………triangular test …………………………………………………………………………………………“two out of five”test………………………………………………………………………………Uunipolarscale ………………………………………………………………………………………Vvisual sensation ……………………………………………………………………………………。

解线性变分不等式的一种时滞神经网络的稳定性分析陈丰盈;高兴宝【摘要】A delayed projection neural network for the linear variational inequality is considered. Based on the theory of functional differential equations and linear matrix inequality (LMI) method, the existence and uniqueness of the solution of the model is proved and a delay- dependent criteria for globally exponential stability of this network is presented by constructing appropriate Liapunov functionals. Meanwhile, the global asymptotic stability of this network with free delay are also shown under mild conditions. Finally, the performance of the model and the obtained results are illustrated by some numerical examples.%考虑了求解线性变分不等式的一种时滞投影神经网络．利用泛函微分方程理论和线性矩阵不等式方法，通过构造恰当的Liapunov泛函，证明了该模型解的存在唯一性，并给出了确保其全局指数稳定的延时依赖准则．对任意延时，在适当条件下证明了该模型的全局渐近稳定性．用数值实例验证了模型的性能和所得结论的正确性．【期刊名称】《陕西师范大学学报（自然科学版）》【年(卷),期】2012(040)006【总页数】6页(P11-15,21)【关键词】线性变分不等式;时滞投影神经网络;全局指数稳定性【作者】陈丰盈;高兴宝【作者单位】陕西师范大学数学与信息科学学院,陕西西安710062;陕西师范大学数学与信息科学学院,陕西西安710062【正文语种】中文【中图分类】O231.2考虑如下一类线性变分不等式问题（LVI）：找向量x＊∈Ω，使其中M∈Rn×n，p∈Rn 和Ω＝｛x∈Rn｜hi≤xi≤li，i＝1，2，…，n｝（某些hi （－lj）可以是＋∞）.显然x＊∈Ω是（1）的解当且仅当它是投影方程［1］的解，其中α＞0是常数，PΩ（x）＝［PΩ（x1），PΩ（x2），…，Pn（xn）］T 且PΩ（xi）＝min｛li，max｛xi，hi｝｝（i＝1，2，…，n）.问题（1）包括线性方程组（Ω＝Rn）和线性互补问题［1］（Ω＝｛x∈Rn｜x≥0｝），并且线性和二次规划问题均可转化为它.在经济、电路和优化等科学与工程领域，问题（1）具有广泛的应用［1］.在许多实际应并在M对称半正定的条件下分析了其稳定性和收敛性.然而与文献［4-7］中的模型一样，文献［8］并未考虑时滞因素.而时滞在神经元处理过程和信号传输中不可避免，且可能导致网络的震荡或不稳定，因此在神经网络模型设计中必须考虑这一因素.近年来，许多研究者提出了求解变分不等式以及相应优化问题的时滞神经网络模型［9－12］.为求解（2），文献［9］提出了时滞投影神经网络：用中，往往要求实时求解问题（1）［1－2］.自H¨opfield神经网络成功应用于线性规划［3］后，基于电路实现的人工神经网络成为实时求解优化问题的有效途径，而且已建立了许多求解线性变分不等式的神经网络［4－10］.特别地，文献［8］提出了求解（2）的投影神经网络其中τ≥0是传输延迟，C（［－τ，0］，Rn）是从［－τ，0］到Rn的所有连续的向量值函数的集合.利用LMI（线性矩阵不等式）方法，文献［9］分别给出了确保模型（4）全局渐近稳定和指数稳定的条件.尽管该模型简单，可求解非单调问题，但其稳定性条件比较复杂，且要求I－αM（其中I为n阶单位矩阵）的非奇异性.用PΩ［x（t－τ）－αMx（t－τ）－αq］代替（4）中的x（t－τ），文献［10］提出了求解问题（1）的一种时滞投影神经网络，并在I－αM 非奇异的条件下证明了全局指数稳定性，但其结构复杂.通过引入随时间变化的时滞，文献［11］提出了求解问题（1）的一个时变时滞神经网络.由于文献［9-10］中的模型均是其特例，因此该模型结构复杂.利用矩阵分裂技术，即令M ＝A＋B（A、B∈Rn×n），文献［12］提出了求解问题（1）的时滞神经网络并通过构造合适的Liapunov泛函，给出了确保其全局指数稳定的充分条件.这些条件要求‖I－αA‖＋a‖B‖＜1（‖·‖为l1或l2范数，见文献［12］的定理3.2和3.3）.但在实际应用中，往往不存在满足该条件的M的分裂（见第2节中的注2），因而它限制了模型（5）的应用.事实上，对给定模型（5）的稳定性条件，往往难以选择满足它的M的分裂.为克服上述模型的不足，本文考虑如下时滞投影神经网络：显然该模型包含模型（3）（τ＝0），其结构比模型（4）更简单，并且避免了对M进行恰当分裂的困难.此外，尽管模型（6）可由模型（5）导出，但它并不满足模型（5）的稳定性条件［12］.事实上，在（5）中令A ＝（I＋M）／α和B ＝－I／α，即得（6）.但‖I－αA‖＋α‖B‖＝‖M‖＋1，因此模型（5）的稳定性条件［12］并不能确保模型（6）的稳定性.基于上述考虑，本文利用泛函微分方程理论和LMI方法，通过构造恰当的Liapunov泛函，证明了模型（6）解的存在唯一性，给出了确保其全局指数稳定的延时依赖准则.此外对任意延时，在适当条件下证明了模型（6）的全局渐近稳定性.由于所给稳定性条件既不需要矩阵M的半正定性，也不需要矩阵I－αM的非奇异性，所以模型（6）比模型（4）、（5）更适合于实际应用.此外，通过简化线性矩阵不等式，本文给出只需检验简单不等式的充分条件.全文假定问题（1）的解集Ω＊＝｛x∈Ω｜x＝PΩ（x－Mx－q）｝≠∅.用‖·‖表示向量或矩阵的l2 范数，D＝diag（d1，d2，…，dn）∈Rn×n（di＞0）表示正对角矩阵.对任一对称矩阵A∈Rn×n，A＞0（＜0）表示A正定（负定），λmin（A）表示A的最小特征值；称（1）的解x＊为（6）的平衡点.引理1［13］（Schur补）设Q＝QT ∈Rm×m，R＝RT ∈ Rn×n，S∈ Rm×n，则本节分析模型（6）的动态行为.首先讨论（6）的基本特性.定理1 ∀φ∈C（［－τ，0］，Rn），系统（6）在［0，＋∞）上存在唯一连续的解x（t）.证明记（6）式右端为 T［x（t）］，则由映射PΩ（·）的非扩张性［2］，易证T在C上Lipschitz连续.所以根据泛函微分方程解的存在性定理［14］，系统（6）存在满足初始条件的唯一连续的解x（t），且其最大存在区间为［0，γ）.在（10）式中令κ＝0，可得神经网络（6）全局渐近稳定的如下结果.定理3 若存在对称正定矩阵P、Q、E∈Rn×n，正对角矩阵D ∈Rn×n，使所以由条件得，当v≠0时，dV［x（t）］／dt＜0，并且dV［x（t）］／dt＝0⇔x（t）＝x＊，y（t）＝y＊和x（t－τ）＝x＊.另一方面，V［x（t）］径向无界，即当‖x（t）－x＊‖→∞时，V［x（t）］→∞.因此系统（6）的平衡点x＊全局渐近稳定.定理得证.定理2和定理3表明神经网络（6）的稳定性并不需要矩阵I－M的非奇异性和M 的半正定性.因此（6）可用来求解非单调的问题（参见第2节例1）.此外，不像（6），文献［9-10］中的模型的稳定性需要I－M的非奇异性.注1 设W1和W2定义于定理2，W3＝2DQM，则由引理1得则神经网络（6）的平衡点x＊全局渐近稳定，其中W1定义见定理2.证明为方便，使用定理2证明中的记号.类似于定理2的证明，得进一步，在注1（ⅱ）中取Q＝0，则由定理3可得神经网络（6）的如下全局渐近稳定性条件：推论1 若存在对称正定矩阵P、E∈Rn×n，正对角矩阵D ∈ Rn×n，使则神经网络（6）的平衡点x＊全局渐近稳定，其中W1定义同定理2.推论2 若存在正对角矩阵D∈Rn×n，使DM正定，则神经网络（6）的平衡点x＊全局渐近稳定.证明取E＝D，P＝MTD＋DM，则由DM的正定性和推论1立得结论.特别地，由推论2知，当M正定时，神经网络（6）的平衡点x＊全局渐近稳定.为方便检验，在注1中令P＝δI，D＝ηI，E＝βI，Q＝σI（δ、η、β、σ均为正常数），立得如下结论：推论3 （ⅰ）若存在正常数δ、η、β、κ和σ及常数τ≥0，使α＝β（2－κ）e－κτ ＞α和2ηλmin（MT＋M）＞σ‖M‖2＋4η2／a＋（1－σ／a）／［（2－κ）（β－2η）＋δ＋κ2η2／σ］，则神经网络（6）的平衡点x＊指数稳定.（ⅱ）若存在正常数δ、η、β和σ，使2β＞σ和2ηλmin（MT ＋ M）＞σ‖M‖2 ＋2η2／β ＋［1 －σ／（2β）］［2（β－2η）＋δ］，则神经网络（6）的平衡点x＊全局渐近稳定.本节用两个例子说明系统（6）的有效性及所结果的适用性.因此由定理2和注1（ⅰ）知，神经网络（6）可求解该问题，且其平衡点x＊全局指数稳定.取初始函数φ（t）＝［－sin（t），cos（t），t］T（t∈ ［－1，0］），图1显示了神经网络（6）收敛于其平衡点x＊的轨线性态.注2 例1中M的任一分裂均不满足文献［12］中给定的模型（5）的稳定性条件.事实上，∀α＞0，易证‖I－αM‖ ≥1（对l1和l∞范数，有‖I－αM‖1＝‖I－αM‖∞ ＝｜1－0.5α｜＋1.7α＞1）.所以对M的任一分裂A、B（M＝A＋B），均有‖I－αA‖＋α‖B‖≥‖I－αM‖≥1（该不等式对l1和l∞范数成立）.经计算知，定义于（12）中的矩阵~W 和＾W 均负定.因此由定理2和注1（ⅰ）知，神经网络（6）可求解该问题，且其平衡点x＊全局指数稳定.取初始函数φ（t）＝［－t2，et，cos（t）］T（t∈ ［－1，0］），图2显示了神经网络（6）收敛于其平衡点x＊（该问题的解）的轨线性态.此外，对相同的初始参数，尽管模拟显示文献［9-10］中的模型均收敛于该问题的解，然而由于I－M 奇异，所以不像神经网络（6），文献［9-10］中给定的条件并不能确保它们的稳定性.本文利用泛函微分方程理论，证明了求解线性变分不等式的一种时滞投影神经网络模型解的存在唯一性.构造了恰当的Liapunov泛函，利用LMI方法，给出了确保该时滞投影神经网络全局指数稳定的延时依赖准则.同时对任意延时，在适当条件下证明了该模型的全局渐近稳定性.由于所给稳定性条件并不需要矩阵I－M的非奇异性和M 的半正定性，因此该模型更适合于实际应用.【相关文献】［1］Harker P T，Pang J S.Finite-dimensional variationalinequality and nonlinear complementarity problems：a survey oftheory，algorithms，and applications［J］.Mathematical Programming，1990，48（1／3）：161-220.［2］Gao Xingbao.A novel neural network for solving 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network for solving linear projection equations and its analysis［J］.IEEE Transaction on Neural Networks，2005，16（4）：834-843.［10］Cheng Long，Hou Zengguang，Tan Min.A delayed projectionneural network for solving linear variational inequalities［J］.IEEE Transaction on Neural Networks，2009，20（6）：915-925.［11］Liu Zixin，Lu Shu，Zhong Shouming.A new delayed projection neural network for solving linear variational inequalities and quadratic optimization problems［C］∥2008International Symposium on Computational Intelligence and Design.L os Alamitos：IEEE Computer Society，2008，1：211-214.［12］Cheng Long，Hou Zengguang，Tan Min.Solving linear variational inequalities by projection neural network with time-varying delays［J］.Physics Letters：A，2009，373：1739-1743.［13］张化光.递归时滞神经网络的综合分析与动态特性研究［M］.北京：科学出版社，2008：17-18.［14］Kolmanovskii V，Myshkis A.Introduction to the theory and applications of functional differential equations［M］.New York：Academic，1999：94-101.。

a r X i v :c o n d -m a t /0407246v 2 [c o n d -m a t .s o f t ] 1 N o v 2004TOPICAL REVIEWNon-Equilibrium in Adsorbed Polymer LayersBen O’Shaughnessy and Dimitrios Vavylonis Department of Chemical Engineering,Columbia University,New York,NY 10027,USA E-mail:bo8@,dv35@ Abstract.High molecular weight polymer solutions have a powerful tendency to deposit adsorbed layers when exposed to even mildly attractive surfaces.The equilibrium properties of these dense interfacial layers have been extensively studied theoretically.A large body of experimental evidence,however,indicates that non-equilibrium effects are dominant whenever monomer-surface sticking energies are somewhat larger than kT ,a common case.Polymer relaxation kinetics within the layer are then severely retarded,leading to non-equilibrium layers whose structure and dynamics depend on adsorption kinetics and layer ageing.Here we review experimental and theoretical work exploring these non-equilibrium effects,with emphasis on recent developments.The discussion addresses the structure and dynamics in non-equilibrium polymer layers adsorbed from dilute polymer solutions and from polymer melts and more concentrated solutions.Two distinct classes of behaviour arise,depending on whether physisorption or chemisorption is involved.A given adsorbed chain belonging to the layer has a certain fraction of its monomers bound to the surface,f ,and the remainder belonging to loops making bulk excursions.A natural classification scheme for layers adsorbed from solution is the distribution of single chain f values,P (f ),which may hold the key to quantifying the degree of irreversibility in adsorbed polymer layers.Here we calculate P (f )for equilibrium layers;we find its form is very different to the theoretical P (f )for non-equilibrium layers which are predicted to have infinitely many statistical classes of chain.Experimental measurements of P (f )are compared to these theoretical predictions.PACS numbers:82.35.-x,68.08.-p,05.40.-a Submitted to:J.Phys.:Condens.Matter1.IntroductionHigh molecular weight polymers are extremely surface-active molecules.Even a weak interaction between a single monomer and a surface tends to be magnified into a powerful attraction or repulsion when many of these monomers are linked together to form a polymeric chain.It is a remarkable fact that surfaces contacting even extremely dilute polymer solutions can readily develop very dense polymer layers.Technologies revolving around the properties of either synthetic or biological polymer layers are many and varied,including adhesion [1,2],coating[3],colloid stabilization[4],fiber reinforced thermoplastics[5,6],flocculation processes[7],DNA microaarrays[8]and biocompatibilization[9].Motivated both by fundamental questions and by technology, understanding and predicting the structure and formation kinetics of these soft layers is a major concern of polymer science[10].A crucial aspect of experimental studies is that adsorbed polymer dynamics are typically extremely slow for long chains:an individual chain has many surface stickers and interacts with numerous other chains impeding its motion.Irreversibility and non-equilibrium effects are therefore very common.The subject of this review is experimental and theoretical work on these non-equilibrium effects, and though we consider adsorption from dilute solutions,semi-dilute solutions and melts our main emphasis is neutral homopolymer adsorption from dilute solutions.This is the simplest and most widely studied class.Polymer-surface adsorption systems are naturally classified according to the mode of adsorption. Roughly,there are two classes:chemisorption and physisorption(seefigure1).The clearest example of irreversibility arises in chemisorption(figure1(b))where the covalent polymer-surface bonds which develop are often essentially irreversible on experimental timescales.Monomer sticking free energies,ǫ,have values typical of covalent bonds which are one to two orders of magnitude greater than kT.Chemical adsorption is employed in various technologies where polymers are attached by chemical reactions to solid surfaces either from a polymer melt as in the reinforcement of polymer-polymer or polymer-solid interfaces[2,6,11,12],or from solution as in colloid stabilization by chemically grafting polymers onto particle surfaces[13–15].What is less obvious is why non-equilibrium effects are so widely observed in physisorbing systems, even for rather weak sticking energies.Available experimental evidence suggests that irreversibility effects become important as soon asǫbecomes somewhat larger than kT.For example the experiments by Schneider et al.[16,17]for polymethylmethacrylate(PMMA)adsorption onto oxidized silica via hydrogen bonding in dilute CCl4solutions(ǫ≈4kT)show essentially frozen-in adsorbed chain confirge physisorption sticking energies(ǫ>kT)originate in hydrogen bonding or other dipolar forces,dispersion forces or attractions between charged groups.Metal and silicon-based surfaces are usually oxidized and many polymer species form strong hydrogen bonds with the surface oxygen or silanol groups[18,19].Biopolymers such as proteins and DNA attach tenaciously to many surfaces due to their many charged,polar and hydrophobic groups[8,9,20].Since hydrogen bonds,for instance,typically have energies of several kT[21,22]it is apparent that strong physical bonds are very common.This suggests that whether physical or chemical bonding is involved,for long chains irreversible effects may in practice be the rule rather than the exception.Figure 1.(a)Schematic of physisorption from afluid polymer phase onto a surface.Adsorbed chainsconsist of loops,tails and sequences of bound monomers(“trains”).When non-equilibrium effects becomeimportant,layer structure depends on kinetics of adsorption.This review addresses phyisorption from dilutesolutions in sections2,3and4and physisorption from melts in section6.(b)As in(a)but for chemisorption.In this case chains carry reactive groups which can form covalent bonds(shown black)with a functionalizedsurface.Chemisorption from dilute solutions is reviewed in section5and from concentrated solutions insection6.To understand non-equilibrium layers,one must identify how they differ from equilibrium layers.The theory of fully equilibrated layers is far more advanced,at both the mean-field[23]and scaling[24–31]level of description.A main result of these theories is expressions for the decay of the monomer density profile asFigure2.The two broad classes of polymer adsorption,physisorption and chemisorption,have very differentvalues of the parameter Q,the local monomer-surface association rate.Q can be thought of as the conditionalmonomer-surface sticking probablity per unit time,given the unattached monomer contacts the surface.Though many systems are in practice mixtures of chemisorption and physisorption,a simplified view ofmonomer free energy as a function of distance between monomer and surface is shown.(a)For physisorbingpolymers,the activation barrier is very small and and monomer-surface association is very likely upon contact,i.e.Qt a is of order unity,where t a the monomer relaxation time.When the sticking energyǫexceeds a fewkT,experiment indicates that chains need large amounts of time to escape the surface,presumably due tocomplex many-chain effects.(b)Chemisorption typically involves a large activation barrier,u≫kT.Manymonomer-surface collisions are needed to traverse this barrier,Qt a≪1.The adsorbed state is also usuallystrongly favored,ǫ≫kT.a function of the distance z from the surface.For adsorption from dilute solutions for example,in the scaling picture originally developed by de Gennes[24,25],Eisenriegler et al.[26,27],and de Gennes and Pincus[28], each adsorbed chain has sequences of surface-bound monomers(trains)interspersed with portions extending away from the surface(tails and loops of size s)with distributionΩ(s)∼s−11/5[29–31]leading to a self-similar density profile c(z)∼z−4/3.Experimentally,the existence of an extended diffuse layer is well established by a large number of neutron scattering[32–37]and neutron reflectivity[38–40]studies.However a universally accepted quantitative test of the predicted density profiles has been difficult to achieve,both due to intrinsic limitations of the experimental techniques[41]and to the asymptotic nature of many of the theoretical results which are valid in the limit of very long chains.Furthermore,for experiments specifically directed at equilibrium,ensuring that equilibrium conditions are realised is difficult when the very non-equilibrium effects one wishes to avoid are poorly identified.Understanding the origin of the observed deviations from equilibrium for weakly adsorbing systems in dilute solutions is a major unresolved puzzle in polymer science.At present it is unclear how universal these non-equilibrium effects may be.Various effects have been speculated to play important roles.Kremer[42] and de Gennes[43]have suggested that if the solution temperature is below the polymer melt glass transition temperature,glassy effects may onset at the interface where polymer densities are similar to those of melts. Interactions with the surface might further enhance glassiness.Chakraborty and coworkers[44–47]suggested flattened-down chains experience strong kinetic barriers due to local steric constraints which drastically slow down dynamics.Ideas related to slow cooperative motions,mutual pinning,development of entanglements at the interface and crystalization have been proposed by Johner and Semenov[48],Sommer[49],Granick[50] and Raviv et el.[51]in a series of comments following a recent theoretical work by Raviv et al.[52]which interpreted past experiments[53,54]exhibiting non-equilibrium effects.In this review we do not attempt an exhaustive review of the vast body of past research work involving strongly physisorbing or chemisorbing polymers at interfaces.Instead,with fundamental issues in mind,our aim is to(i)assemble and classify a number of theoretical ideas and numerical simulations which articulate the community’s current level of understanding of equilibrium and non-equilibrium polymer adsorption,and (ii)summarize a number of experimental results which we believe are particularly pertinent and which still demand a theoretical explanation.The emphasis is on the simplest case:adsorption of neutralflexible polymers from dilute bulk solutions.We also review work on irreversible adsorption from semi-dilute bulk solutions and melts,motivated by ideas initiated by Guiselin[55].Polyelectrolyte solutions,polymers with complex architectures and non-flat surfaces are beyond the scope of the present review.Physisorption and chemisorption will be carefully distinguished.These are characterized by very different values of the local monomer-surface association rate,Q(seefigure2).In physisorption,monomer attachment is usually essentially diffusion-limited,Q=C/t a,where t a is monomer relaxation time and C is a system-dependent constant of order unity[56].Chemisorption is normally much slower[56–58]with Q values typically 8or more orders of magnitude smaller than those of physisorption.The origin of this difference is that chemical bond formation usually involves a large activation barrier(seefigure2).Similarly,desorption ratesafter chemisorption are usually very small and can be ignored.The effect of desorption on physisorbing systems is more subtle and is discussed in section4.The above two classes naturally lead to very different adsorption kinetics.This is analogous to bulk polymer-polymer reaction kinetics where depending on Q, polymer length N,and solvent conditions,the kinetics are described by one of a range of“diffusion-controlled”and“mean-field”kinetic regimes[58–61].Such regimes also arise for end-adsorbing polymers[56,57,62–67].In section2we briefly review the equilibrium picture for dilute solutions and in section3we discuss experimental evidence for non-equilibrium departures from this picture.Theoretical work related to physisorbing non-equilibrium layers from dilute solution is reviewed in section4.We move to chemisorption, again from dilute solution,in section5.Section6addresses irreversibility effects involving melts and semi-dilute solutions.We conclude with a brief discussion of the experimental and theoretical outlook.2.Adsorption from Dilute Solutions:The Equilibrium Picture2.1.Structure of Equilibrium LayersThis section briefly outlines some central results of equilibrium theories of adsorbed polymer layers.Much more extensive reviews can be found in refs.[41,68–71].In the scaling picture developed mainly in the 1980’s[24–28],each adsorbed chain consists of surface-bound monomers and large loops and tails generating a monomer density profile c(z)as a function of distance from the surface,z.Eisenriegler et al.[26–28]showed that very close to the surface,in the“proximal”region,the density decays as a power law,c(z)∼z−m,where the critical exponent m≈1/3represents competion between surface-sticking energy gain,chain entropy,and excluded volume interactions.The proximal region decay law crosses over to de Gennes’“self-similar grid”regime[24,25,28],c(z)∼z−4/3,beyond a certain distance h prox.For z>h prox the polymer layer can be thought of as a semi-dilute solution with continously varying local concentration c(z).In this region the correlation length measuring the range of excluded-volume interactions,ξ=a−5/4c−3/4,is proportional to the distance from the surface,z,since this is the only relevant length scale:ξ≈z.Here a is defined to be the monomer size.Expressingξin terms of c leads to†a3c(z)≈ (a/h prox)(a/z)1/3,a<z<h proxh prox=a kT/ǫ,R F=aN3/5(1)(a/z)4/3,h prox<z<R FUnless the bulk polymer concentration,c,is extremely small[72],then the equilibrium layer height is of order the Flory bulk coil radius R F as indicated in equation(1).In this same range of c the adsorption isotherm exhibits a large plateau,i.e.surface coverageΓis weakly dependent on c.Even in weakly adsorbing polymer systems,e.g.adosrption through weak van der Waals interactions,the value ofǫis usually of order kT.By studying the adsorption/desorption transition in binary solvent mictures, van der Beek et al.[18]estimated the sticking energies per monomer of many typicalflexible polymers onto silica and alumina surfaces from organic solvents to lie in the range0.5to6kT.Hence the width of the proximal region is typically of order the monomer size,h prox≈a,and a clear signature of the proximal region is hard to probe experimentally.In the following we considerǫof order kT or larger.We remark that the net monomer free energy of adsorptionǫincludes both the“stickiness”a monomer feels for the surface, but also the entropic disadvantage due to constraining local orientational degrees of freedom upon contact with the surface.Thus,crudely speaking one can say the stickiness contribution must exceed a crtical value ǫc representing the entropic disadvantage before a monomer can puter simulations showǫc is of order kT and is lattice-dependent[27].The real situation is more complex,with various contributions from electronic and structural factors such as solvent molecule entropy effects,etc[21].The density decay law of equation(1)reflects a power law distribution of loop and tail sizes.Neglecting differences between loops and tails and the details associated with the proximal region,then the loop size distribution per surface site is[29–31]Ω(s)≈a−2s−11/5.(2) Beyond this,Semenov and Joanny[31]showed that the inner region of the layer,z<z∗≡aN1/2,is dominated by loops while the outer region,z>z∗,is dominated by tails;the resulting density profile obeys a z−4/3law above and below z∗,respectively,but with different numerical prefactors.Support for the scaling conclusions of equations(1)and(2)is provided by Monte-Carlo simulations of Zajac and Chakrabarti[73], de Joannis et al.[74,75],and Cifra[76].These produce a density decay consistent with the z−4/3law for long chains.Zajac and Chakrabarti[73]additionally report agreement with equation(2).†The cross-over distance h prox and the prefactor in the proximal region density law can be determined by demanding(i)a smooth cross-over at h prox and(ii)the osmotic free energy per unit area, R F a dzkT/ξ3,balances the sticking free energy per unit area,ǫa c(a).Complementary to the scaling approach outlined above has been the numerical lattice model of Scheutjens and Fleer(SF)[77,78].This is a self-consistent meanfield theory which averages excluded volume interactions and thus treats self-avoidance in an approximate manner.This approximation however allows numerical solutions for the density profile and for the loop and tail distributions and can additionally describe chains of finite length.The meanfield description becomes more accurate for solvents near the theta temperature(a common case)where self-avoidance is a weak perturbation except for the longest loops and tails.The existence of the loop-and tail-dominated regions of the layer was in factfirst established by the SF model[78,79].The layer height in the SF theory scales as h∼N1/2[77,78]while the density profile decays as c(z)∼z−2(for sufficiently long chains),different to the z−4/3decay predicted by the scaling approach,as shown by van der Linden and Leermakers[80].Analytical meanfield solutions for the density profile in the limit of very long chains were derived within the ground-state dominance approximation[81] by Jones and Richmond[82].Going beyond the ground state dominance approximation,Semenov et al.[23] subsequently generalized this approach to account forfinite length effects to leading order and analytically describe the different contributions of loops and tails to the density profile.They found that loops dominate for z<z∗MF≡aN1/3while tails dominate for z>z∗MF,similarly to the scaling approach of Semenov and Joanny[31].These new methods have revived interest in analytical and numerical meanfield approaches to polymer adsorption[83–89].Turning now to experiment,the fundamentals of polymer adsorption at the solid/liquid and air/liquid interface have been studied in a vast number of experiments.Research prior to1993is reviewed in the book by Fleer et al.[41].Given the strong evidence for nonequilibrium effects(see below),in general one should be very cautious when comparing experimentalfindings to equilibrium expectations.Overall,experiment is consistent with the general trend predicted by equilibrium theories regarding structure of the polymer layers which were studied,although thefine differences between the meanfield and scaling pictures are hard to distinguish.Very briefly,measurements of the layer’s surface bound monomer fraction as a function of total adsorbed amount and molecular weight(MW)by techniques such as NMR[90,91],ESR[92,93],or infrared spectroscopy[94]give results fairly consistent with the predictions of the SF theory[41].The thickness, h exp,of polymer layers has been probed as a function of chain length by hydrodynamic methods[41,95–97], ellipsometry[98],and the surface force apparatus[99].Depending on the method,h exp is proportional to a certain moment of the density profile and many existent measurements are compatible with a power law,h exp∼Nα.Certain studies have favored the SF theory predictions[41,96]while others support the scaling predictions[97,99].Forflexible polymer species the total surface coverageΓas a function of bulk concentration is found to be very weakly dependent on the polymer concentration in the bulk except for very dilute solutions,in qualitative agreement with both the scaling and the SF theories[41].For a given bulk concentration,meaurements ofΓas a function of N in good solvents typically show a weak dependence on chain length for large N[41].This is consistent with the SF and scaling theories which predictΓ∼ h a dz c(z) is dominated by the lower,N-independent limit.Small angle neutron scattering(SANS)and neutron reflectivity have been used to probe the density profile.These experiments established the existence of a diffuse extended layer but at present there is no general agreement as to the exact form of the density decay.A techical difficulty intrinsic to SANS,as dicussed in ref.[41],is its limited sensitivity to the more dilute regions of the layer.Neutron reflectivity experiments are also often ambiguous,since multiple density profiles can befitted to the same data.The SANS experiments of Auvray and Cotton[35]and Hone et al.[37]are consistent with the z−4/3scaling law.However the results of Hone et al.could also be described by an exponential profile(see also[32]). SANS experiments by Cosgrove et al.[33,34]do not support the scaling predictions,but are more consistent with the SF theory.Other SANS studies by Rennie et al.[36]are inconsistent with both scaling and SF predictions,while neutron reflectivity experiments of Lee et al.[38,39]and Sun et al.[40]have generated data consistent with the scaling predictions.2.2.Single Chain Statistics and the Equilibrium Distribution of Bound FractionsSo far this section has discussed many-chain layer properties.Equally important,and characteristic of the layer-forming processes,are properties of individual surface-adsorbed chains in the layer.What is the spectrum of configurations of individual chains?According to the scaling picture,a typical chain has ND(s) loops of length s or greater,where D(s)≡ ∞s ds′Ω(s′)∼s−6/5after using equation(2).Semenov and Joanny[100]argue that because of screening effects these are essentially independent blobs and their2D spatial extent parallel to the surface is[ND(s)]1/2as3/5=aN1/2.This occurs for all scales s;in particular, a given chain has of order one loop of length N5/6,also of size aN1/2.Hence a typical chain has a lateral size of order aN1/2,the ideal result(to within logarithmic corrections[100]).Figure3.Equilibrium probabilty distribution P eq of chain bound fraction,f,in good solvents.For verylong chains the distribution is sharply peaked at a value¯f of order unity.For realistic values of N thedistribution is rather broad.A special role is played by another single chain property,directly reflecting the degree to which individual chains are bound to the surface.This is the probability distribution P eq(f)that a chain has a fraction f of its monomers touching the surface.This property plays a central role in this review,since its features closely reflect whether irreversible effects matter or not.In two independent Monte Carlo studies by Wang and Rajagopalan[101]and Zajac and Chakrabarti[73]an equilibrium distribution was found with a single peak at a value of f of order unity.To our knowledge,P eq(f)has not been calculated analytically,at least at the scaling level.In order to compare equilibrium predictions with experimental measurements of bound fractions in non-equilibrium layers in later sections,we have calculated P eq(f)by relating an adsorbed chain to a1D unidirectional walk combining an initial tail,a sequence of intermediate steps,and afinal tail.The result,which is derived in the appendix and is shown infigure3,reads:N−1/5P eq(f)≈Figure4.Schematic of new chain adsorption in an equilibrium polymer layer(shown as a self-similar grid)as described in ref.[102].(a)Entry:a bulk chain reptates into the layer and makes afirst attachment to thesurface.(b)Spreading:the incoming chain establishes an increasing number of contacts with the surface.(c)A typical adsorbed chain configuration is adopted,consisting of trains,loops and tails.A similar picturewas used in ref.[100].Chain desorption follows the same path in the reverse order.3.1.Theories of Dynamics in Equilibrium LayersCompared to static properties,much less is established theoretically about equilibrium dynamics.These have been studied for good solvents by de Gennes[43,102–105],Semenov and Joanny[100],and Baschnagel et al.[106]for bidisperse solutions.The picture emerging from these works is that the layer of bound chains has a certain characteristic equilibration timeτeq.This can be thought of as the time after which the chains following the distribution P eq(f)offigure3are completely reshuffled among themselves.The exchange of chains between the bulk and the layer was predicted to be a slower process due to the fact that incoming and outgoing chains have to pass through unfavored configurations having a small fraction of bound monomers (seefigure4).de Gennes assumed reptation dynamics(i.e.entangled layers)and found the longest relaxation time of the layer to scale as[102,104,105]τeq≈t s N3(entangled layers).(4) Here t s is the relaxation time of an adsorbed monomer which,due to wall-polymer interactions,may be much larger than the corresponding time in the bulk,t a[107].Semenov and Joanny[100]assumed unentangled layers and Rouse-Zimm dynamics and obtainedτeq≈t a N2(in their work t s≈t a was assumed).In equilibrium,layer and bulk chains desorb and adsorb at the same average rate,respectively.In de Gennes’picture bulk chains adsorb in two stages(seefigure4).During thefirst“entry”stage,the bulk chain overcomes the exclude-volume barrier presented by the layer’s loops and tails and makes itsfirst contact with the surface,i.e.f=1/N.During a second“spreading”stage the chain continues to make an increasing number of surface-contacts,f increases up to f min,and the chain becomes part of the bound layer. When entry is rate-limiting he found that the mean lifetime of an adsorbed chain before its desorption is τex≈t a N3.7/φ,whereφis the volume fraction of polymer in the bulk.Semenov and Joanny[100]described the dynamics using a similar picture,but assuming unentangled layers and Rouse-Zimm dynamics.They obtained a slighlty different chain lifetime,τex≈t a N2.42/φ(to within logarithmic corrections).Note that the exchange timescale,τex,has a weak power law dependence on N rather than exponential because the incoming/outgoing barrier is small.The scalingτex∼1/φreflects the linear dependence on concentration of the rate of chain arrival at the surface.Note also that even for the highest dilute solution concentrations,φ=φ∗,whereφ∗≡N−4/5is the chain overlap threshold concentration[81],one still has τeq≪τex.A prediction of the above works is that chain desorption into pure solvent,φ→0,is extremely slow,which is well-established experimentally[108].Now suppose one starts with a layer of labeled chains in equilibrium and replaces the bulk solution with identical but unlabeled chains of the same concentration at t=0.An important prediction of the above theories is that the decay of the surface coverage of labeled chains,Γ,is a simple exponential for all times[43,100]:Γ(t)=Γ(0)e−t/τex(5) An implicit feature of equation(5)is that there is a single observed desorption rate sinceτex≫τeq,i.e.the desporption process is slow enough to sample an average over all equilibrium chain states in the layer.Note this result in fact assumes a given desorbed labeled chain does not readsorb,i.e.memory is lost instantly. Experimentally,this necessitates a mixing process in the bulk to dilute the desorbed chains.In the absence of such mixing diffusion returns a desorbed chain to the surface repeatedly,an effect which can lead to non-exponential decay[60,109,110].The kinetics of polymer layer build up starting from empty or“starved”surfaces is more complex and has been considered in refs.[43,100,102,111].3.2.Dynamics of Adsorbed Polymer Layers:Monte Carlo SimulationsThis sub-section provides a brief review of numerical Monte Carlo simulations of dynamics in many-chain polymer layers in contact with dilute solutions(for dynamics of single chains see refs.[112–116]).The simulations reported here did not include hydrodynamic interactions.The main results of the simulations by Wang et al.[101,117]are qualitatively in agreement with the theoretical picture of the previous subsection. They found that the lateral dynamics of adsorbed chains up to N=100are consistent with Rouse dynamics. For sufficiently sticky surfaces(0<ǫ≤1.5kT withǫc=0.5kT)the value ofτex was found to be much larger than the lateral relaxation time,even though the scaling dependence on N was the same.This should be contrasted with the Semenov and Joanny prediction that the two exponents differ by a small value,0.42. Wang et al.observed non-exponential exchange kinetics arising from readsorption of desorbed chains.Lai[113,118]studied the layer dynamics as a function ofǫfor N≤80and interestingly found that for ǫ>∼1kT(withǫc≈0.9kT)the lateral chain dynamics started to slow down and to approach apparently glassy dynamics atǫ≈4kT.This result was claimed to be valid despite an omission in the implemented algorithm[113,119].This report is important since it indicates that the value ofǫis crucial in polymer adsorption.Zajac and Chakrabarti[120]studied the dynamics for N=100and N=200andǫ+ǫc=1.8kT near and aboveφ=φ∗.Their algorithm involved unphysical reptation moves in order to speed up the dynamics. In equilibrium they found a distribution of bound fractions similar to the one offigure3and observed that the internal dynamics of reshuffling of chains between different f values is complex.The timescale for the exchange of adsorbed chains by bulk chains was found to be slower than internal equilibration processes. Simple exponential exchange kinetics were observed as in equation(5).Takeuchi[121]also used the Monte Carlo method with extra reptation moves.For surfaces withǫ≈1.6kT he observed exponential exchange kinetics while forǫ≈0.9kT readsorption effects were important leading to non-exponential exchange kinetics.3.3.Experiment:Departure from Equilibrium PictureThefirst experimental studies of exchange kinetics onflexible polymers were performed by Pefferkon et al.[122–125]using radioactive labeling techniques.One study[122,124,125]involved labeled polyacrylamide (PAM)in water adsorbed through hydrogen bonding onto aluminol-grafted glass beads.The beads were exposed to a dilute solution of labeled PAM for approximately30min until time-independent coverage was achieved.The labeled solution was then replaced by a dilute unlabeled PAM solution of varying concentration c and the amount of labeled PAM which remained adsorbed,Γ,was monitored as a function of time as shown infigure5(i).An interesting result of this experiment was that the exchange rate per labeled chain,shown infigure5(ii),was time-dependent and reached a constant value after a cross-over period of≈300min which was approximately the same for every c.This asymptotic rate was found to increase linearly with c,as shown infigure5(iii).The observed spectrum of exchange times disagrees with equation(5)and this can be interpreted in many ways as follows:(i)The observed non-exponential exchange kinetics is a signature of non-equilibrium.Pefferkorn et al.[122,124]argued that the interface is populated with a spectrum of different frozen or slowly changing configurations and,consequently,different kinetic properties.(They proposed that the layer consists of aflat sublayer of tightly bound chains which exchange slowly,plus a less tightly bound population which exchange more rapidly).(ii)The layer is not in equilibrium when the exchange experiment starts but it equilibrates afterτeq≈300min which is larger than the layer’s preparationg time.The asymptotic exchange rate then becomes constant and equal to1/τex.The fact that asymptoticallyτex∼1/c as seen infigure5(iii)and the fact that τex>300min as can be seen infigure5(ii),are consistent with this interpretation and the theories reviewed in subsection3.1.Assuming reptation dynamics,equation(4),and given N≈1400,this implies a relaxation time of adsorbed monomers of order t s≈10−5s.This is much larger than monomer relaxation times in the bulk,t a≈10−10s.(iii)The layer is in fact in equilibrium but its exchange kinetics and internal equilibration processes are much more complex than assumed by existent theories,at least for this system.For example,if the equilibrium P eq(f)is very broad and chains with different f values have very different exchange times,then the intial drop inΓwill be due mainly to the most rapidly desorbing chains if their desorption times are less than τeq.Issues related to surface density of aluminol groups,polydispersity,and effect of pH(this experiment was performed at pH=4where PAM is neutral while many of the surface aluminol groups were positively。

What Determines Productivity?*Chad SyversonUniversity of Chicago Booth School of Businessand National Bureau of Economic Researchchad.syverson@April 2010AbstractEconomists have shown that large and persistent differences in productivity levels across businesses are ubiquitous. This finding has shaped research agendas in a number of fields, including (but not limited to) macroeconomics, industrial organization, labor, and trade. This paper surveys and evaluates recent empirical work addressing the question of why businesses differ in their measured productivity levels. The causes are manifold, and differ depending on the particular setting. They include elements sourced in production practices—and therefore over which producers have some direct control, at least in theory—as well as from producers’ external operating environments. After evaluating the current state of knowledge, I lay out what I see are the major questions that research in the area should address going forward.* I thank Eric Bartelsman, Nick Bloom, Roger Gordon, John Haltiwanger, Chang-Tai Hsieh, Ariel Pakes, Amil Petrin, John Van Reenen, and anonymous referees for helpful comments. This work is supported by the NSF (SES-0519062 and SES-0820307), and both the Stigler Center and the Centel Foundation/Robert P. Reuss Faculty Research Fund at the University of Chicago Booth School of Business. Contact information: University of Chicago Booth School of Business; 5807 S. Woodlawn Ave.; Chicago, IL 60637.1. IntroductionThanks to the massive infusion of detailed production activity data into economic study over the past couple of decades, researchers in many fields have learned a great deal about how firms turn inputs into outputs. Productivity, the efficiency with which this conversion occurs, has been a topic of particular interest. The particulars of these studies have varied depending on the researchers’ specific interests, but there is a common thread. They have documented, virtually without exception, enormous and persistent measured productivity differences across producers, even within narrowly defined industries.The magnitudes involved are striking. I found in Syverson (2004a) that within 4-digit SIC industries in the U.S. manufacturing sector, the average difference in logged total factor productivity (TFP) between an industry’s 90th and 10th percentile plants is 0.651. This corresponds to a TFP ratio of e0.651 = 1.92. To emphasize just what this number implies, it says that the plant at the 90th percentile of the productivity distribution makes almost twice as much output with the same measured inputs as the 10th percentile plant. Note that this is the average 90-10 range. The range’s standard deviation across 4-digit industries is 0.173, so several industries see much larger productivity differences among their producers. U.S. manufacturing is not exceptional in terms of productivity dispersion. Indeed, if anything, it is small relative to the productivity variation observed elsewhere. Chang-Tai Hsieh and Peter J. Klenow (2009), for example, find even larger productivity differences in China and India, with average 90-10 TFP ratios over 5:1.2These productivity differences across producers are not fleeting, either. Regressing a producer’s current TFP on its one-year-lagged TFP yields autoregressive coefficients on the order of 0.6 to 0.8 (see, e.g., Árpád Ábrahám and Kirk White (2006) and Foster, Haltiwanger, and Syverson (2008)). Put simply, some producers seem to have figured out their business (or at least are on their way), while others are woefully lacking. Far more than bragging rights are at2 These figures are for revenue-based productivity measures; i.e., where output is measured using plant revenues (deflated across years using industry-specific price indexes). TFP measures that use physical quantities as output measures rather than revenues actually exhibit even more variation than do revenue-based measures, as documented in Lucia Foster, John Haltiwanger, and Syverson (2008). Hsieh and Klenow (2009) also find greater productivity dispersion in their TFP measures that use quantity proxies to measure output (actual physical quantities are not available for most producers in their data). Even though it is only a component of revenue-based TFP (the other being the producer’s average price), quantity-based TFP can be more dispersed because it tends to be negatively correlated with prices, as more efficient producers sell at lower prices. Thus revenue-based productivity measures, which combine quantity-based productivity and prices, tend to understate the variation in producers’ physical efficiencies.stake here: another robust finding in the literature—virtually invariant to country, time period, or industry—is that higher productivity producers are more likely to survive than their less efficient industry competitors. Productivity is quite literally a matter of survival for businesses.1.1. How Micro-Level Productivity Variation and Persistence Has Influenced ResearchThe discovery of ubiquitous, large, and persistent productivity differences has shaped research agendas in a number of fields. Here are some examples of this influence, though by no means is it meant to be a comprehensive accounting. They speak to the breadth of the impact that answers to this paper’s title question would have.Macroeconomists are dissecting aggregate productivity growth—the source of almost all per capita income differences across countries—into various micro-components, with the intent of better understanding the sources of such growth. Foster, Halitwanger, and C.J. Krizan (2001), for example, overview the substantial role of reallocations of economic activity toward higher productivity producers (both among existing plants and through entry and exit) in explaining aggregate productivity growth. Hsieh and Klenow (2009) ask how much larger the Chinese and Indian economies would be if they achieved the same efficiency in allocating inputs across production units as does the U.S. Models of economic fluctuations driven by productivity shocks are increasingly being enriched to account for micro-level patterns, and are estimated and tested using plant- or firm-level productivity data rather than aggregates (e.g., Jeffrey R. Campbell and Jonas D. M. Fisher (2004); Eric Bartelsman, Haltiwanger, and Stefano Scarpetta (2008); Marcelo Veracierto (2008)). Micro productivity data have also been brought to bear on issues of long-run growth, income convergence, and technology spillovers. They offer a level of resolution unattainable with aggregated data.In industrial organization, research has linked productivity levels to a number of features of technology, demand, and market structure. Examples include the effect of competition (Syverson (2004b), James A. Schmitz (2005)), the size of sunk costs (Allan Collard-Wexler (2008)), and the interaction of product market rivalry and technology spillovers (Nicholas Bloom, Mark Schankerman, and John Van Reenen (2007)). Another line of study has looked at the interaction of firms’ organizational structures with productivity levels (e.g., Vojislav Maksimovic and Gordon Phillips (2002), and Antoinette Schoar (2002), and Ali Hortaçsu and Syverson (2007, 2009)).Labor economists have explored the importance of workers’ human capital in explaining productivity differences (John Abowd, Haltiwanger, Ron Jarmin, Julia Lane, Paul Lengermann, Kristin McCue, Kevin McKinney, and Kristin Sandusky (2005) and Jeremy Fox and Valerie Smeets (2009)), the productivity effects of incentive pay (Edward P. Lazear (2000)), other various human resources practices (Casey Ichniowski and Kathryn Shaw (2003)), managerial talent and practices (Bloom and Van Reenen (2007)), organizational form (Luis Garicano and Paul Heaton (2007)), and social connections among co-workers (Oriana Bandiera, Iwan Barankay, and Imran Rasul (2009)). There has also been a focus on the role of productivity-driven reallocation on labor market dynamics via job creation and destruction (Haltiwanger, Scarpetta, and Helena Schweiger (2008)).Perhaps in no other field have the productivity dispersion patterns noted above had a greater influence on the trajectory of the research agenda than in the trade literature. Theoretical frameworks using heterogeneous-productivity firms like Jonathan Eaton and Samuel Kortum (2002) and Marc J. Melitz (2003) are now the dominant conceptual lenses through which economists view trade impacts. In these models, the trade impacts vary across producers, and depend on their productivity levels in particular. Aggregate productivity gains come from improved selection and heighted competition that trade brings. A multitude of empirical studies have accompanied and been spurred by these theories (e.g., Nina Pavcnik (2002); Andrew Bernard, J. Bradford Jensen, and Peter Schott (2006); and Eric A. Verhoogen (2008)). They have confirmed many of the predicted patterns, and raised questions of their own.1.2. The Question of “Why?”Given the important role that productivity differences play in these disparate literatures, the facts above raise obvious and crucial questions. Why do firms (or factories, stores, offices, or even individual production lines, for that matter) differ so much in their abilities to convert inputs into output? Is it dumb luck, or instead something—or many things—more systematic? Can producers control the factors that influence productivity, or are they purely external products of the operating environment? What supports such large productivity differences in equilibrium?A decade ago, when Bartelsman and Mark Doms (2000) penned the first survey of the micro-data productivity literature for this journal, researchers were just beginning to ask the “Why?” question. Much of the work to that point had focused on establishing facts like thoseabove—the “What?” of productivity dispersion. Since then, the literature has focused more intensely on the reasons why productivity levels are so different across businesses. There has definitely been progress. But we’ve also learned more about what we don’t know, and this is guiding the ways in which the productivity literature will be moving. This article is meant to be a guide to and comment on this research.I begin by setting some boundaries. I have to. A comprehensive overview of micro-founded productivity research is neither possible in this format nor desirable. There are simply too many studies to allow adequate coverage of each. First, I will focus on empirical work. This is not because I view it as more important than theory. Rather, it affords a deeper coverage of this important facet of a giant literature, and it better reflects my expertise. That said, I will sketch out a simple heterogeneous-productivity industry model below to focus the discussion, and I will also occasionally bring up specific theoretical work with particularly close ties to the empirical issues discussed. Furthermore, for obvious reasons, I will focus on research that has been done since Bartelsman and Doms (2000) was written.Even within these boundaries, there are more studies than can be satisfactorily described individually. I see this article’s role as filtering the broader lessons of the literature through the lens of a subset of key studies. The papers I focus on here are not necessarily chosen because they are the first or only good work on their subject matter, but rather because they had an archetypal quality that lets me weave a narrative of the literature. I urge readers whose interests have been piqued to more intensively explore the relevant literatures. There is far more to be learned than I can convey here.A disclaimer: some of my discussion contains elements of commentary. These opinions are mine alone and may not be the consensus of researchers in the field.I organize this article as follows. The next section sketches the conceptual background: what productivity is, how it is often measured in practice, and how differences in productivity among producers of similar goods might be sustained in equilibrium. Section 3 looks at influences on productivity that operate primarily within the business. This can be at the firm level, plant level, or even on specific processes within the firm. Many of these influences may potentially be under the control of the economic actors inside the business. In other words, they can be “levers” that management or others have available to impact productivity. Section 4 focuses on the interaction of producers’ productivity levels and the markets in which theyoperate. These are elements of businesses’ external environments that can affect productivity levels. This impact might not always be direct, but they can induce producers to pull some of the levers discussed in Section 3, indirectly influencing observed productivity levels in the process. They may also be factors that affect the amount of productivity dispersion that can be sustained in equilibrium, and influence observed productivity differences through that channel. Section 5 discusses what I see as the big questions about business-level productivity patterns that still need to be answered. A short concluding section follows.2. Productivity—What It Is, How It Is Measured, and How Its Dispersion Is SustainedThis section briefly reviews what productivity is conceptually, how it is measured in practice, and how productivity differences among producers of similar goods might be supported in equilibrium. Deeper discussions on the theory of productivity indexes can be found in Douglas W. Caves, Laurits R. Christensen, and W. Erwin Diewert (1982) and the references therein. More detail on measurement issues can be found in the large literature on the subject; see, for example, G. Steven Olley and Ariel Pakes (1996); Zvi Griliches and Jacques Mairesse (1998); Richard Blundell and Stephen R. Bond (2000); James Levinsohn and Amil Petrin (2003); and Daniel Ackerberg, C. Lanier Benkard, Steven Berry, and Ariel Pakes (2007). Examples of models that derive industry equilibria with heterogeneous-productivity producers include Boyan Jovanovic (1982); Hugo A. Hopenhayn (1992); Richard Ericson and Pakes (1995); Melitz (2003); Marcus Asplund and Volker Nocke (2006); and Foster, Haltiwanger, and Syverson (2008).2.1. Productivity in ConceptSimply put, productivity is efficiency in production: how much output is obtained from a given set of inputs. As such, it is typically expressed as an output-input ratio. Single-factor productivity measures reflect units of output produced per unit of a particular input. Labor productivity is the most common measure of this type, though occasionally capital or even materials productivity measures are used. Of course, single-factor productivity levels are affected by the intensity of use of the excluded inputs. Two producers may have quite different labor productivity levels even though they have the same production technology, if one happens to use capital much more intensively, say because they face different factor prices.Because of this, researchers often use a productivity concept that is invariant to theintensity of use of observable factor inputs. This measure is called total factor productivity, or TFP (it is also sometimes called multifactor productivity, MFP). Conceptually, TFP differences reflect shifts in the isoquants of a production function: variation in output produced from a fixed set of inputs. Higher-TFP producers will produce greater amounts of output with the same set of observable inputs than lower-TFP businesses, and hence have isoquants that are shifted up and to the right. Factor price variation that drives factor intensity differences does not affect TFP, because it induces shifts along isoquants rather than shifts in isoquants.TFP is most easily seen in the often-used formulation of a production function where output is the product of a function of observable inputs and a factor-neutral (alternatively, Hicks-neutral)h s ifter:, , ,where Y t is output, F (·) is a function of observable inputs capital Y t , labor L t , and intermediate materials M t , and A t is the factor-neutral shifter. In this type of formulation, TFP is A t . It captures variations in output not explained by shifts in the observable inputs that act through F (·).3TFP is, at its heart, a residual. As with all residuals, it is in some ways a measure of our ignorance: it is the variation in output that cannot be explained based on observable inputs. So it is fair to interpret the work discussed in this survey as an attempt to “put a face on” that residual—or more accurately, “put faces on,” given the multiple sources of productivityvariation. The literature has made progress when it can explain systematic influences on output across production units that do not come from changes in observable inputs like standard labor or capital measures.2.2. Measuring Productivity While productivity is relatively straightforward in concept, a host of measurement issues arise when constructing productivity measures from actual production data. Ironically, while3I use a m lt lica ely sep b tec o to make exposition easy, but TFP can be extracted from a general ti tio K t ,L t ,M t ,). Totally differentiating this production function gives: u ip tiv ara le hn logy shift me-varying production func n Y t = G t (A t , . Without loss of generality, we can choose units to normalize ∂G /∂A = 1. Thus when observed inputs are fixed (dK t = dL t = dM t = 0), differential shifts in TFP, dA t , create changes in output dY t .research with micro production data greatly expands the set of answerable question and moves the level of analysis closer to where economic decisions are made than aggregate data does, it also raises measurement and data quality issues more frequently.The first set of issues regards the output measure. Many businesses produce more than one output. Should these be aggregated to a single output measure, and how if so? Further, even detailed producer microdata do not typically contain measures of output quantities. Revenues are typically observed instead. Given this limitation of the data, the standard approach has been to use revenues (deflated to a common year’s real values using price deflator series) to measure output. While this may be acceptable, and even desirable, if product quality differences are fully reflected in prices, it can be problematic whenever price variation instead embodies differences in market power across producers. In that case, producers’ measured productivity levels may reflect less about how efficient they are and more about the state of their local output market. Recent work has begun to dig deeper into the consequences of assuming single-product producers and using revenue to measure output. I’ll discuss this more below. In the mean time, I will go forward assuming deflated revenues accurately reflect the producer’s output.The second set of measurement issues regards inputs. For labor, there is the choice of whether to use number of employees, employee-hours, or some quality-adjusted labor measure (the wage bill is often used in this last role, based on the notion that wages capture marginal products of heterogeneous labor units). Capital is typically measured using the establishment or firm’s book value of its capital stock. This raises several questions. How good of a proxy is capital stock for the flow of capital services? Should the stock be simply the producer’s reported book value, and what are the deflators? Or should the stock be constructed using observed investments and the perpetual inventory method—and what to assume about depreciation? When measuring intermediate materials, an issue similar to the revenue-as-output matter above arises, because typically only the producer’s total expenditures on inputs are available, not input quantities. More fundamentally, how should intermediate inputs be handled? Should one use a gross output production function and include intermediate inputs directly, or should intermediates simply be subtracted from output so as to deal with a value-added production function? On top of all these considerations, one makes these input measurement choices in the context of knowing that any output driven by unmeasured input variations (due to input quality differences or intangible capital, for example) will show up as productivity.The third set of measurement concerns involves aggregating multiple inputs in a TFP measure. As described above, TFP differences reflect shifts in output while holding inputs constant. To construct the output-input ratio that measures TFP, a researcher must weight the individual inputs appropriately when constructing a single-dimensional input index. The correct weighting is easiest to see when the production function is Cobb-Douglas:.In this case, the inputs are aggregated by taking the exponent of each factor to its respective output elasticity. It turns out that this holds more generally as a first-order approximation to any production function. The input index in the TFP denominator can be constructed similarly for general production functions.4Even after determining how to construct the input index, one must measure the output elasticities αj, j∈{k,l,m}. Several approaches are common in the literature. One builds upon assumptions of cost-minimization to construct the elasticities directly from observed production data. A cost-minimizing producer will equate an input’s output elasticity with the product of that input’s cost share and the scale elasticity. If cost shares can be measured (obtaining capital costs are usually the practical sticking point here), and the scale elasticity either estimated or assumed, then the output elasticities αj can be directly constructed. If a researcher is willing to make some additional but not innocuous assumptions—namely, perfect competition and constant returns to scale—then the elasticities equal the share of revenues paid to each input. This makes constructing the αj simple. Materials’ and labor’s shares are typically straightforward to collect with the wage bill and materials expenditures data at hand. Capital’s share can be constructed as the residual, obviating the need for capital cost measures. (Though there is a conceptual problem since, as the model that follows below points out, it is unclear what makes the producer’s size finite in a perfectly competitive, constant returns world.) An important caveat is that the index approach assumes away factor adjustment costs. If they are present, the first-order conditions linking observed factor shares to output elasticities will not hold. This can be mitigated in part (but at cost) by using cost shares that have been averaged over either time or producers in order 4 While Cobb-Douglas-style approaches are probably the most common in the literature, many researchers also use the translog form (see Caves, Christensen, and Diewert 1982), which is a second-order approximation to general production functions, and as such is more flexible, though more demanding of the data. There is also an entirely nonparametric approach, data envelopment analysis (DEA), that is used in certain, somewhat distinct circles of the literature. See William W. Cooper, Lawrence M. Seiford, and Kaoru Tone (2006) for an overview of DEA methods.to smooth out idiosyncratic adjustment-cost-driven misalignments in actual and optimal input levels, but some mismeasurement could remain.A separate approach is to estimate the elasticities αj by estimating the production function. In this case, (logged) TFP is simply the estimated sum of the constant and the residual. In the Cobb-Douglas case (which again, recall, is a first-order approximation to more general technologies), the estimated equation is:ln ln ln .lnHence the TFP estimate would be , where the first term is common across production units in the sample (typically the technology is estimated at the industry level), and the second is idiosyncratic to a particular producer.This approach raises econometric issues. As first pointed out by Marschak and Andrews (1944), input choices are likely to be correlated with the producer’s productivity ωt: more efficient producers are, all else equal, likely to hire more inputs. There is also potential selection bias when a panel is used, since less efficient producers—those with low ωt—are more likely to exit from the sample. (As will be discussed below, the positive correlation between productivity and survival is one of the most robust findings in the literature.) Then there is the issue of producer-level price variation mentioned above. A substantial literature has arisen to address these issues; see Griliches and Mairesse (1998); Ackerberg, Benkard, Berry, and Pakes (2007); and Johannes Van Biesebroeck (2008) for overviews.There is debate as to which of the many available methods is best. In the end, as I see it, choosing a method is a matter of asking oneself which assumptions one is comfortable making. Certainly one cannot escape the fact that some assumptions must be made when estimating the production function.Fortunately, despite these many concerns, many of the results described in this paper are likely to be quite robust to measurement peculiarities. When studies have tested robustness directly, they typically find little sensitivity to measurement choices. The inherent variation in establishment- or firm-level microdata is typically so large as to swamp any small measurement-induced differences in productivity metrics. Simply put, high-productivity producers will tend to look efficient regardless of the specific way that their productivity is measured. I usually use cost-share-based TFP index numbers as a first pass in my own work; they are easy to construct and offer the robustness of being a nonparametric first-order approximation to a generalproduction function. That said, it is always wise to check one’s results for robustness to specifics of the measurement approach.2.3. A Model of Within-Industry Productivity DispersionGiven the large differences in productivity within an industry that I discussed above, a natural question is to ask how they could be sustained in equilibrium. The ubiquity of this dispersion suggests there must be some real economic force at work, rather than it simply being an artifact of measurement or odd chance. Here, I sketch out a simple model that shows how that is possible. The model will also prove helpful in facilitating discussion throughout this survey.d r x y i, earn profits given byIn ustry p oducers, inde ed b, , .R(⋅) is a general revenue function. A i is the producer’s productivity level, and L i is its labor input. (I assume labor is the firm’s only input for the sake of simplicity.) Productivity levels differ across producers. The specific form of R(⋅) depends on the structure of the output market and the production function. Revenues can also depend on an industry state D. This can be a vector or a scalar, and depending on the structure of output market competition, it may include industry-wide demand shocks, the number of industry producers, their productivity levels, and/or moments of the productivity distribution. Both the wage rate w and fixed cost f are common across, and taken as given by, all producers.assume R(⋅) is twice differentiable with ∂R/∂L > 0, ∂2R/∂L2 < 0, ∂R/∂A > 0, andI∂2R/∂A∂L > 0. If the industry is perfectly competitive, these conditions are satisfied given a production function that is similarly differentiable, concave in L, and where productivity and labor are complements. Further, under perfect competition, all information contained in D is reflected in the market price P that equates total demand and supply, which the producers of course take as given. In imperfectly competitive markets, the assumptions about R(⋅) place restrictions on the form of competitive interaction (be it monopolistically competitive or oligopolistic) and through this the shapes of the residual demand curves. The contents of D will also depend on the particulars of the competitive structure. For example, in a heterogeneous-cost Cournot oligopoly, D will contain the parameters of the industry demand curve and the productivity levels of the industry’s producers, as these are sufficient to determine the Nash。

a rXiv:c ond-ma t/9411013v12Nov1994Concentration and energy fluctuations in a critical polymer mixture M.M¨u ller and N.B.Wilding Institut f¨u r Physik,Johannes Gutenberg Universit¨a t,Staudinger Weg 7,D-55099Mainz,Germany Abstract A semi-grand-canonical Monte Carlo algorithm is employed in conjunction with the bond fluctuation model to investigate the critical properties of an asymmetric binary (AB)polymer mixture.By applying the equal peak-weight criterion to the concentration distribution,the coexistence curve separating the A-rich and B-rich phases is identified as a function of temperature and chemical potential.To locate the critical point of the model,the cumulant intersection method is used.The accuracy of this approach for determining the critical parameters of fluids is assessed.Attention is then focused on the joint distribution function of the critical concentration and energy,which is analysed using a mixed-field finite-size-scaling theory that takes due account of the lack of symme-try between the coexisting phases.The essential Ising character of the binary polymer critical point is confirmed by mapping the critical scaling operator distributions onto in-dependently known forms appropriate to the 3D Ising universality class.In the process,estimates are obtained for the field mixing parameters of the model which are compared both with those yielded by a previous method,and with the predictions of a mean field calculation.PACS numbers 64.70Ja,05.70.Jk1IntroductionThe critical point of binary liquid and binary polymer mixtures,has been a subject of abidinginterest to experimentalists and theorists alike for many years now.It is now well established that the critical point properties of binary liquid mixtures fall into the Ising universality class(the default for systems with short ranged interactions and a scalar order parameter)[1]. Recent experimental studies also suggest that the same is true for polymer mixtures[2]–[11].However,since Ising-like critical behaviour is only apparent when the correlation length farexceeds the polymer radius of gyrationξ≫ R g ,the Ising regime in polymer mixtures is confined(for all but the shortest chain lengths)to a very narrow temperature range nearthe critical point.Outside this range a crossover to mean-field type behaviour is seen.Theextent of the Ising region is predicted to narrow with increasing molecular weight in a manner governed by the Ginsburg criterion[12],disappearing entirely in the limit of infinite molecular weight.Although experimental studies of mixtures with differing molecular weights appear to confirm qualitatively this behaviour[5],there are severe problems in understanding the scaling of the so-called“Ginsburg number”(which marks the centre of the crossover region and is empirically extracted from the experimental data[13])with molecular weight Computer simulation potentially offers an additional source of physical insight into polymer critical behaviour,complementing that available from theory and experiment.Unfortunately, simulations of binary polymer mixtures are considerable more exacting in computational terms than those of simple liquid or magnetic systems.The difficulties stem from the problems of deal-ing with the extended physical structure of polymers.In conventional canonical simulations, this gives rise to extremely slow polymer diffusion rates,manifest in protracted correlation times[14,15].Moreover,the canonical ensemble does not easily permit a satisfactory treat-ment of concentrationfluctuations,which are an essential feature of the near-critical region in polymer mixture.In this latter regard,semi-grand-canonical ensemble(SGCE)Monte Carlo schemes are potentially more attractive than their canonical counterparts.In SGCE schemes one attempts to exchange a polymer of species A for one of species B or vice-versa,thereby permitting the concentration of the two species tofluctuate.Owing,however,to excluded volume restrictions,the acceptance rate for such exchanges is in general prohibitively small, except in the restricted case of symmetric polymer mixtures,where the molecular weights of the two coexisting species are identical(N A=N B).All previous simulation work has therefore focussed on these symmetric systems,mapping the phase diagram as a function of chain length and confirming the Ising character of the critical point[16,17,18].Tentative evidence for a crossover from Ising to meanfield behaviour away from the critical point was also obtained [19].Hitherto however,no simulation studies of asymmetric polymer mixtures(N A=N B) have been reported.Recently one of us has developed a new type of SGCE Monte Carlo method that amelio-rates somewhat the computational difficulties of dealing with asymmetric polymer mixtures [20].The method,which is described briefly in section3.1,permits the study of mixtures of polymer species of molecular weight N A and N B=kN A,with k=2,3,4···.In this paper we shall employ the new method to investigate the critical behaviour of such an asymmet-ric polymer mixture.In particular we shall focus on those aspects of the critical behaviour of asymmetric mixtures that differ from those of symmetric mixtures.These difference are rooted in the so called‘field mixing’phenomenon,which manifests the basic lack of energetic(Ising) symmetry between the coexisting phases of all realisticfluid systems.Although it is expected to have no bearing on the universal properties offluids,field mixing does engender certain non-universal effects in near-criticalfluids.The most celebrated of these is a weak energy-like critical singularity in the coexistence diameter[21,22],the existence of which constitutes afailure for the‘law of rectilinear diameter’.As we shall demonstrate however,field mixing has a far more legible signature in the interplay of the near-critical energy and concentration fluctuations,which are directly accessible to computer simulation.In computer simulation of critical phenomena,finite-size-scaling(FSS)techniques are of great utility in allowing one to extract asymptotic data from simulations offinite size[23].One particularly useful tool in this context is the order parameter distribution function[24,25,26]. Simulation studies of magnetic systems such as the Ising[25]andφ4models[27],demonstrate that the critical point form of the order parameter distribution function constitutes a useful hallmark of a university class.Recently however,FSS techniques have been extended tofluids by incorporatingfield mixing effects[28,29].The resulting mixed-field FSS theory has been successfully deployed in Monte Carlo studies of critical phenomena in the2D Lennard-Jones fluid[29]and the2D asymmetric lattice gas model[30].The present work extends this programme offield mixing studies to3D complexfluids with an investigation of an asymmetric polymer mixture.The principal features of our study are as follows.We begin by studying the order parameter(concentration)distribution as a function of temperature and chemical potential.The measured distribution is used in conjunction with the equal peak weight criterion to obtain the coexistence curve of the model.Owing to the presence offield mixing contributions to the concentration distribution,the equal weight criterion is found to break down near thefluid critical point.Its use to locate the coexistence curve and critical concentration therefore results in errors,the magnitude of which we gauge using scaling arguments.Thefield mixing component of the critical concentration distribution is then isolated and used to obtain estimates for thefield mixing parameters of the model. These estimates are compared with the results of a meanfield calculation.We then turn our attention to thefinite-size-scaling behaviour of the critical scaling op-erator distributions.This approach generalises that of previousfield mixing studies which concentrated largely on thefield mixing contribution to the order parameter distribution func-tion.We show that for certain choices of the non-universal critical parameters—the critical temperature,chemical potential and the twofield mixing parameters—these operator distri-butions can be mapped into close correspondence with independently known universal forms representative of the Ising universality class.This data collapse serves two purposes.Firstly,it acts as a powerful means for accurately determining the critical point andfield mixing param-eters of modelfluid systems.Secondly and more generally,it serves to clarify the sense of the universality linking the critical polymer mixture with the critical Ising magnet.We compare the ease and accuracy with which the critical parameters can be determined from the data collapse of the operator distributions,with that possible from studies of the order parameter distribution alone.It is argued that for criticalfluids the study of the scaling operator distri-butions represent the natural extension of the order parameter distribution analysis developed for models of the Ising symmetry.2BackgroundIn this section we review and extend the mixed-fieldfinite-size-scaling theory,placing it within the context of the present study.The system we consider comprises a mixture of two polymer species which we denote A and B,having lengths N A and N B monomers respectively.The configurational energyΦ(which we express in units of k B T)resides in the intra-and inter molecular pairwise interactions between monomers of the polymer chains:N i<j=1v(|r i−r j|)(2.1)Φ({r})=where N=n A N A+n b N B with n A and n B the number of A and B type polymers respectively. N is therefore the total number of monomers(of either species),which in the present study is maintained strictly constant.The inter-monomer potential v is assigned a square-well formv(r)=−ǫr≤r m(2.2)v(r)=0r>r mwhereǫis the well depth and r m denotes the maximum range of the potential.In accordance with previous studies of symmetric polymer mixtures[16,17],we assignǫ≡ǫAA=ǫBB=−ǫAB>0.The independent model parameters at our disposal are the chemical potential difference per monomer between the two species∆µ=µA−µB,and the well depthǫ(both in units of k B T). These quantities serve to control the observables of interest,namely the energy density u and the monomer concentrationsφA andφB.Since the overall monomer densityφN=φA+φB is fixed,however,it is sufficient to consider only one concentration variableφ,which we take as the concentration of A-type monomers:φ≡φA=L−d n A N A(2.3) The dimensionless energy density is defined as:u=L−dǫ−1Φ({r})(2.4) with d=3in the simulations to be chronicled below.The critical point of the model is located by critical values of the reduced chemical potential difference∆µc and reduced well-depthǫc.Deviations ofǫand∆µfrom their critical values control the sizes of the two relevant scalingfield that characterise the critical behaviour.In the absence of the special symmetry prevailing in the Ising model,onefinds that the relevant scalingfields comprise(asymptotically)linear combinations of the well-depth and chemical potential difference[21]:τ=ǫc−ǫ+s(∆µ−∆µc)h=∆µ−∆µc+r(ǫc−ǫ)(2.5) whereτis the thermal scalingfield and h is the ordering scalingfield.The parameters s and r are system-specific quantities controlling the degree offield mixing.In particular r is identifiable as the limiting critical gradient of the coexistence curve in the space of∆µandǫ. The role of s is somewhat less tangible;it controls the degree to which the chemical potential features in the thermal scalingfield,manifest in the widely observed critical singularity of the coexistence curve diameter offluids[1,22,31].Conjugate to the two relevant scalingfields are scaling operators E and M,which comprise linear combinations of the concentration and energy density[28,29]:M=11−sr[u−rφ](2.6) The operator M(which is conjugate to the orderingfield h)is termed the ordering operator, while E(conjugate to the thermalfield)is termed the energy-like operator.In the special caseof models of the Ising symmetry,(for which s=r=0),M is simply the magnetisation while E is the energy density.Near criticality,and in the limit of large system size L,the probability distributions p L(M) and p L(E)of the operators M and E are expected to be describable byfinite-size-scaling relations having the form[25,29]:p L(M)≃a M−1L d−λM˜p M(a M−1L d−λMδM,a M LλM h,a E LλEτ)(2.7a)p L(E)≃a E−1L d−λE˜p E(a E−1L d−λEδE,a M LλM h,a E LλEτ)(2.7b) whereδM≡M−M c andδE≡E−E c.The functions˜p M and˜p E are predicted to be universal,modulo the choice of boundary conditions and the system-specific scale-factors a M and a E of the two relevantfields,whose scaling indices areλM=d−β/νandλE=1/νrespectively.Precisely at criticality(h=τ=0)equations2.7a and 2.7b implyp L(M)≃a M−1Lβ/ν˜p⋆M(a M−1Lβ/νδM)(2.8a)p L(E)≃a E−1L(1−α)/ν˜p⋆E(a E−1L(1−α)/νδE)(2.8b) where˜p⋆M(x)≡˜p M(x,y=0,z=0)˜p⋆E(x)≡˜p E(x,y=0,z=0)(2.9) are functions describing the universal and statistically scale-invariantfluctuation spectra of the scaling operators,characteristic of the critical point.The claim that the binary polymer critical point belongs to the Ising universality class is expressed in its fullest form by the requirement that the critical distribution of thefluid scaling operators p L(M)and p L(E)match quantitatively their respective counterparts—the magneti-sation and energy distributions—in the canonical ensemble of the critical Ising magnet.As we shall demonstrate,these mappings also permit a straightforward and accurate determination of the values of thefield mixing parameters s and r of the model.An alternative route to obtaining estimates of thefield mixing parameters is via thefield mixing correction to the order parameter(i.e.concentration)distribution p L(φ).At criticality, this distribution takes the form[29,30]:p L(φ)≃a M−1Lβ/ν ˜p⋆M(x)−sa E a M−1L−(1−α−β)/ν∂∂x (˜p⋆M(x)˜ω⋆(x)),is a function characterising the mixing of the critical energy-like operator into the order parameter distribution.Thisfield mixing term is down on the first term by a factor L−(1−α−β)/νand therefore represents a correction to the large L lim-iting behaviour.Given further the symmetries of˜ω(x)and˜p⋆M(x),both of which are even (symmetric)in the scaling variable x[29],thefield mixing correction is the leading antisym-metric contribution to the concentration distribution.Accordingly,it can be isolated frommeasurements of the critical concentration distribution simply by antisymmetrising around φc= φ c.The values of s and r are then obtainable by matching the measured critical func-tion−s∂simply by cutting a B-type polymer into k equal segments.Conversely,a B-type polymer is manufactured by connecting together the ends of k A-type polymers.This latter operation is,of course,subject to condition that the connected ends satisfy the bond restrictions of the BFM.Consequently it represents the limiting factor for the efficiency of the method,since for large values of k and N A,the probability that k polymer ends simultaneously satisfy the bond restrictions becomes prohibitively small.The acceptance rate for SGCE moves is also further reduced by factors necessary to ensure that detailed balance is satisfied.In view of this we have chosen k=3,N A=10for the simulations described below,resulting in an acceptance rate for SGCE moves of approximately14%.In addition to the compositionalfluctuations associated with SGCE moves,it is also nec-essary to relax the polymer configurations at constant composition.This is facilitated by monomer moves which can be either of the local displacement form,or of the reptation(‘slith-ering snake’)variety[2].These moves were employed in conjunction with SGCE moves,in the following ratios:local displacement:reptation:semi-grandcanonical=4:12:1the choice of which was found empirically to relax the configurational and compositional modes of the system on approximately equal time scales.In the course of the simulations,a total offive system sizes were studied having linear extent L=32,40,50,64and80.An overall monomerfilling fraction of8φN=0.5was chosen, representative of a dense polymer melt[15].Here the factor of8constitutes the monomeric volume,each monomer occupying8lattice sites.The cutoffrange of the inter-monomeric√square well potential was set at r m=whereφ∗is a parameter defining the boundary between the two peaks.Well below criticality,the value of∆µcx obtained from the equal weight criterion is in-sensitive to the choice ofφ∗,provided it is taken to lie approximately midway between the peaks and well away from the tails.As criticality is approached however,the tails of the two peaks progressively overlap making it impossible to unambiguously define a peak in the manner expressed by equation3.1.For models of the Ising symmetry,for which the peaks are symmetric about the coexistence concentrationφcx,the correct value of∆µcx can nevertheless be obtained by choosingφ∗= φ in equation3.1.In near-criticalfluids,however,the imposed equal weight rule forces a shift in the chemical potential away from its coexistence value in or-der to compensate for the presence of thefield mixing component.Only in the limit as L→∞(where thefield mixing component dies away),will the critical order parameter distribution be symmetric allowing one to chooseφ∗= φ and still obtain the correct coexistence chemical potential.Thus forfinite-size systems,use of the equal weight criterion is expected to lead to errors in the determination of∆µcx near the critical point.Although this error is much smaller than the uncertainty in the location of the critical point along the coexistence curve (see below),it can lead to significant errors in estimates of the critical concentrationφc.To quantify the error inφc it is necessary to match the magnitude of thefield mixing component of the concentration distribution w(δp L),to the magnitude of the peak weight asymmetry w′(δµ)associated with small departuresδµ=∆µ−∆µcx from coexistence:w(δp L)=w′(δµ)(3.2) Now from equation2.10w(δp L)≈ φNφ∗dφδp L(φ)∼L−(1−α−β)/ν(3.3) whilew′(δµ)≈ φNφ∗dφ∂p L(φ)3 m2 2(3.7)where m2and m4are the second and fourth moments respectively of the order parameter m=φ− φ .To the extent thatfield mixing corrections can be neglected,the critical order parameter distribution function is expected to assume a universal scale invariant form. Accordingly,when plotted as a function ofǫ,the coexistence values of G L for different system sizes are expected to intersect at the critical well depthǫc[26].This method is particularly attractive for locating the critical point influid systems because the even moments of the order parameter distribution are insensitive to the antisymmetric(odd)field mixing contribution. Figure2displays the results of performing this cumulant analysis.A well-defined intersection point occurs for a value G L=0.47,in accord with previously published values for the3D Ising universality class[36].The corresponding estimates for the critical well depth and critical chemical potential areǫc=0.02756(15)∆µc=0.003603(15)It is important in this context,that a distinction be drawn between the errors on the location of the critical point,and the error with which the coexistence curve can be determined.The uncertainty in the position of the critical point along the coexistence curve,as determined from the cumulant intersection method,is in general considerably greater than the uncertainty in the location of the coexistence curve itself.This is because the order parameter distribution function is much more sensitive to small deviations offcoexistence(due tofiniteǫ−ǫcx orfinite ∆µ−∆µcx)than it is for deviations along the coexistence curve,(ǫand∆µtuned together to maintain equal weights).In the present case,wefind that the errors on∆µc andǫc are approximately10times those of the coexistence valuesǫcx and∆µcx near the critical point.The concentration distribution function at the assigned value ofǫc and the corresponding value of∆µcx,(determined according to the equal weight rule withφ∗=<φ>),is shown infigure3for the L=40and L=64system sizes.Also shown in thefigure is the critical magnetisation distribution function of the3D Ising model obtained in a separate study[37]. Clearly the L=40and L=64data differ from one another and from the limiting Ising form.These discrepancies manifest both the pure antisymmetricfield mixing component of the true(finite-size)critical concentration distribution,and small departures from coexistence associated with the inability of the equal weight rule to correctly identify the coexistence chemical potential.To extract the infinite-volume value ofφc from thefinite-size data,it is therefore necessary to extrapolate to the thermodynamic limit.To this end,and in accordance with equation3.6,we have plottedφcx(L),representing thefirst moment of the concentration distribution determined according to equal weight criterion at the assigned value ofǫc,against L(1−α)/ν.This extrapolation(figure4)yields the infinite-volume estimate:φc=0.03813(19)corresponding to a reduced A-monomer densityφc/φN=0.610(3).Thefinite-size shift in the value ofφcx(L)is of order2%.We turn next to the determination of thefield mixing parameters r and s.The value of r represents the limiting critical gradient of the coexistence curve which,to a good approxima-tion,can be simply read offfromfigure1with the result r=−0.97(3).Alternatively(and as detailed in[29])r may be obtained as the gradient of the line tangent to the measured critical energy function(equation2.11)atφ=φc.Carrying out this procedure yields r=−1.04(6).The procedure for extracting the value of thefield mixing parameter s from the concentra-tion distribution is rather more involved,and has been described in detail elsewhere[29,30]. The basic strategy is to choose s such as to satisfy∂δp L(φ)=−swhereδp L(φ)is the measured antisymmetricfield mixing component of the critical concen-tration distribution[30],obtained by antisymmetrising the concentration distribution about φc(L)and subtracting additional corrections associated with small departures from coexistence resulting from the failure of the equal weight rule.Carrying out this procedure for the L=40 and L=64critical concentration distributions yields thefield mixing components shown in figure5.The associated estimate for s is0.06(1).Also shown infigure5(solid line)is the predicted universal form of the3D order parameterfield mixing correction−∂u−rφ1−sr p L(E)=p Lsection2).This discrepancy implies that the system size is still too small to reveal the asymp-totic behaviour Nevertheless the data do afford a test of the approach to the limiting regime, via the FSS behaviour of the variance of the energy distribution.Recalling equation2.14, we anticipate that this variance exhibits the same FSS behaviour as the Ising susceptibility, namely:L d( u2 −u2c)∼Lγ/ν(3.10) By contrast,the variance of the scaling operator E is expected to display the FSS behaviour of the Ising specific heat:L d( E2 −E2c)∼Lα/ν.(3.11) Figure9shows the measured system size dependence of these two quantities at criticality.Also shown is the scaled variance of the ordering operator L d( M2 −M2c)∼Lγ/ν.Straight lines of the form Lγ/νand Lα/ν,(indicative of the FSS behaviour of the Ising susceptibility and specific heat respectively)have also been superimposed on the data.Clearly for large L,the scaling behaviour of the variance of the energy distribution does indeed appear to approach that of the ordering operator distribution.4Meanfield calculationsIn this section we derive approximate formulae for the values of thefield mixing parameters s and r on the basis of a meanfield calculation.Within the well-known Flory-Huggins theory of polymer mixtures,the mean-field equation of state takes the form:∆µ=1N Bln(1−ρ)−2zǫ(2ρ−1)+C(4.1)In this equation,z≈2.7is the effective monomer coordination number,whose value we have obtained from the measured pair correlation function.ρ=φ/φN is the density of A-type monomers and the constant C is the entropy density difference of the pure phases,which is independent of temperature and composition.In what follows we reexpressρby the concen-trationφ.The critical point is defined by the condition:∂∂φ2∆µc=0(4.2) where∆µc=∆µ(φc,ǫc).This relation can be used to determine the critical concentration and critical well-depth,for which onefindsφc1+1/√ǫc=z4N A N BN A+√where the expansion coefficients take the formr ′=−2z (2φc φN b =(1+√3√k )4φ+−φ−(4.6)where φ−and φ+denote the concentration of A monomers in the A-poor phase and A-rich phases respectively.Thus to leading order in ǫ,the phase boundary is given by :∆µcx (ǫ)=∆µc +r ′δǫc +···(4.7)Consequently we can identify the expansion coefficient r ′with the field mixing parameter r (c.f.equation 2.5)that controls the limiting critical gradient of the coexistence curve in the space of ∆µand ǫ.Substituting for ∆µc and ǫc in equation 4.7and setting k =3,we find r =−1.45,in order-of-magnitude agreement with the FSS analysis of the simulation data.In order to calculate the value of the field mixing parameter s ,it is necessary to obtain the concentration and energy densities of the coexisting phases near the critical point.The concentration of A-type monomers in each phase is given byδφ±=φ±−φc =± b −2acδǫ2−φc =−2acδǫ2 z s +z (2ρ−1)2 =−φN φN −1)2+rδφ−2z2−u (φc )=−2za5zb δǫ+···(4.11)Now since (1−rs ) M = δφ −s δu vanishes along the coexistence line,equations 4.9and4.11yield the following estimate for the field mixing parameter s :s = δφ 5zb 1+r cφN 20z √20z √[30].The sign of the product rs differs however from that found at the liquid-vapour critical point.In the present context this product is given byrs10(√NB)2N A N B(4.13)However an analogous treatment of the van der Waalsfluid predicts a positive sign rs,in agreement with that found at the liquid vapour critical point[29,30].5Concluding remarksIn summary we have employed a semi-grand-canonical Monte Carlo algorithm to explore the critical point behaviour of a binary polymer mixture.The near-critical concentration and scal-ing operator distributions have been analysed within the framework of a mixed-fieldfinite-size scaling theory.The scaling operator distributions were found to match independently known universal forms,thereby confirming the essential Ising character of the binary polymer critical point.Interestingly,this universal behaviour sets in on remarkably short length scales,being already evident in systems of linear extent L=32,containing only an average of approximately 100polymers.Regarding the specific computational issues raised by our study,wefind that the concen-tration distribution can be employed in conjunction with the cumulant intersection method and the equal weight rule to obtain a rather accurate estimate for the critical temperature and chemical potential.The accuracy of this estimate is not adversely affected by the anti-symmetric(odd)field mixing contribution to the order parameter distribution,since only even moments of the distribution feature in the cumulant ratio.Unfortunately,the method can lead to significant errors in estimates of the critical concentrationφc,which are sensitive to the magnitude of thefield mixing contribution.The infinite-volume value ofφc must therefore be estimated by extrapolating thefinite-size data to the thermodynamic limit(where thefield mixing component vanishes).Estimates of thefield mixing parameters s and r can also be extracted from thefield mixing component of the order parameter distribution,although in practice wefind that they can be determined more accurately and straightforwardly from the data collapse of the scaling operators onto their universalfixed point forms.In addition to clarifying the universal aspects of the binary polymer critical point,the results of this study also serve more generally to underline the crucial role offield mixing in the behaviour of criticalfluids.This is exhibited most strikingly in the form of the critical energy distribution,which in contrast to models of the Ising symmetry,is doubly peaked with variance controlled by the Ising susceptibility exponent.Clearly therefore close attention must be paid tofield mixing effects if one wishes to perform a comprehensive simulation study of criticalfluids.In this regard,the scaling operator distributions are likely to prove themselves of considerable utility in future simulation studies.These operator distributions represent the natural extension tofluids of the order parameter distribution analysis deployed so successfully in critical phenomena studies of(Ising)magnetic systems.Provided therefore that one works within an ensemble that affords adequate sampling of the near-criticalfluctuations,use of the operator distribution functions should also permit detailed studies offluid critical behaviour. AcknowledgementsThe authors thank K.Binder for helpful discussions.NBW acknowledges thefinancial sup-port of a Max Planck fellowship from the Max Planck Institut f¨u r Polymerforschung,Mainz.。