Joel L. Lebowitz

A new algorithm for Monte Carlo simulation of Ising spin systems

We describe a new algorithm for Monte Carlo simulation of Ising spin systems and present results of a study comparing the speed of the new technique to that of a standard technique applied to a square lattice of 6400 spins evolving via single spin flips. We find that at temperatures T < Tc, the critical temperature, the new technique is faster than the standard technique, being ten times faster at T = 0.588 Tc. We expect that the new technique will be especially valuable in Monte Carlo simulation of the time evolution of binary alloy systems. The new algorithm is essentially a reorganization of the standard algorithm. It accounts for the a priori probability of changing spins before, rather than after, choosing the spin or spins to change.

Statistical Mechanics of Rigid Spheres

An equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid sphere molecules in contact with a rigid sphere solute of arbitrary size. A number of exact relations which describe rather fully the functional form of G(r) are derived. These are based on both geometrical considerations and the virial theorem. A knowledge of G(a) where a is the diameter of a rigid sphere enables one to arrive at the equation of state. The resulting analytical expression which is exact up to the third virial coefficient gives the fourth virial coefficient within 3% and the fifth, insofar as it is known, within 5%. Furthermore over the entire range of fluid density, the equation of state derived from theory agrees with that computed using machine methods. Theory also gives an expression for the surface tension of a hard sphere fluid in contact with a perfectly repelling wall. The dependence of surface tension on curvature is also given. ...

A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics

We extend the work of Kurchan on the Gallavotti–Cohen fluctuation theorem, which yields a symmetry property of the large deviation function, to general Markov processes. These include jump processes describing the evolution of stochastic lattice gases driven in the bulk or through particle reservoirs, general diffusive processes in physical and/or velocity space, as well as Hamiltonian systems with stochastic boundary conditions. For dynamics satisfying local detailed balance we establish a link between the average of the action functional in the fluctuation theorem and the macroscopic entropy production. This gives, in the linear regime, an alternative derivation of the Green–Kubo formula and the Onsager reciprocity relations. In the nonlinear regime consequences of the new symmetry are harder to come by and the large deviation functional difficult to compute. For the asymmetric simple exclusion process the latter is determined explicitly using the Bethe ansatz in the limit of large N.

Scaled Particle Theory of Fluid Mixtures

An extension of a previous one‐component theory of hard‐sphere systems (in three, two, and one dimensions) and the surface tension of real systems is made to mixtures. The theory is based on consideration of an approximate expression for the work of adding an additional hard sphere to a mixture. Comparison between theory and molecular‐dynamics calculations of the various contributions to the virial pressure (related to contact distribution functions) of such hard‐sphere mixtures is excellent. Comparison of the theory with experimental surface tensions of mixtures of simple liquids is satisfactory.

Rigorous Treatment of the Van Der Waals‐Maxwell Theory of the Liquid‐Vapor Transition

Rigorous upper and lower bounds are obtained for the thermodynamic free‐energy density a(ρ, γ) of a classical system of particles with two‐body interaction potential q(r) + γνφ(γr) where ν is the number of space dimensions and ρ the density, in terms of the free‐energy density a0(ρ) for the corresponding system with φ(x) ≡ 0. When φ(x) belongs to a class of functions, which includes those which are nonpositive and those whose ν‐dimensional Fourier transforms are nonnegative, the upper and lower bounds coincide in the limit γ → 0 and limγ → 0 a(ρ, γ) is the maximal convex function of ρ not exceeding a0(ρ) + ½αρ2, where α ≡ ∫ φ(x) dx. The corresponding equation of state is given by Maxwells equal‐area rule applied to the function p0(ρ) + ½αρ2 where p0(ρ) is the pressure for φ(x) ≡ 0. If a0(ρ) + ½αρ2 is not convex the behavior of the limiting free energy indicates a first‐order phase transition. These results are easily generalized to lattice gases and thus apply also to Ising spin systems.The two‐body dist...

Thermodynamic Properties of Mixtures of Hard Spheres

We have investigated the thermodynamic properties of a binary mixture of hard spheres (with special reference to the existence of a phase transition) by using the recently obtained exact solution of the generalized equations of Percus and Yevick for the radial distribution functions of such a mixture. The distribution function obtained from the equations of Percus and Yevick is only an approximation and so yields two different pressures, pc and pv, when used, respectively, in the compressibility equation of Ornstein and Zernike and in the equation of state obtained from the virial theorem. Comparisons with machine calculations show that pc is slightly above and pv slightly below the true pressure but that both are close to it. Our results show that the volume change on mixing at constant pressure is negative at all densities and compositions within the fluid phase when it is calculated from pc, but that it becomes positive at high densities when calculated from pv. In neither case is there a separation in...

Aspects of the Statistical Thermodynamics of Real Fluids

By extending the ideas previously applied to the statistical mechanical theory of hard sphere fluids of Reiss, Frisch, and Lebowitz, an approximate expression has been determined for the work of creating a spherical cavity in a real fluid. In turn the knowledge of this entity permits an evaluation of properties such as the surface tension and the normal heats of vaporization of fluids and the Henrys law constants of fluid mixtures. The agreement between the calculated and experimental properties is satisfactory.

Boltzmann's Entropy and Time's Arrow

Given the success of Ludwig Boltzmanns statistical approach in explaining the observed irreversible behavior of macroscopic systems in a manner consistent with their reversible microscopic dynamics, it is quite surprising that there is still so much confusion about the problem of irreversibility. (See figure 1.) I attribute this confusion to the originality of Boltzmanns ideas: It made them difficult for some of his contemporaries to grasp. The controversies generated by the misunderstandings of Ernst Zermelo and others have been perpetuated by various authors. There is really no excuse for this, considering the clarity of Boltzmanns later writings. Since next year, 1994, is the 150th anniversary of Boltzmanns birth, this is a fitting moment to review his ideas on the arrow of time. In Erwin Schrodingers words, “Boltzmanns ideas really give an understanding” of the origin of macroscopic behavior. All claims of inconsistencies that I know of are, in my opinion, wrong; I see no need for alternate expl...

Statistical Thermodynamics of Nonuniform Fluids

We have developed a general formalism for obtaining the low‐order distribution functions nq(r1, …, rq) and the thermodynamic parameters of nonuniform equilibrium systems where the nonuniformity is caused by a potential U(r). Our method consists of transforming from an initial (uniform) density n0 to the final desired density n(r) via a functional Taylor expansion. When n0 is chosen to be the density in the neighborhood of the rs we obtain nq as an expansion in the gradients of the density. The expansion parameter is essentially the ratio of the microscopic correlation length to the scale of the inhomogeneities. Our analysis is most conveniently done in the the grand ensemble formalism where the corresponding thermodynamic potential serves as the generating functional [with U(r) as the variable] for the nq. The transition from U(r) to n(r) as the relevant variable is accomplished via the direct correlation function which enters very naturally, relating the change in U at r2 due to a change in n at r1. It ...

Properties of a Harmonic Crystal in a Stationary Nonequilibrium State

The stationary nonequilibrium Gibbsian ensemble representing a harmonic crystal in contact with several idealized heat reservoirs at different temperatures is shown to have a Gaussian r space distribution for the case where the stochastic interaction between the system and heat reservoirs may be represented by Fokker—Planck-type operators. The covariance matrix of this Gaussian is found explicitly for a linear chain with nearest-neighbor forces in contact at its ends with heat reservoirs at temperatures T 1 and T N , N being the number of oscillators. We also find explicitly the covariance matrix, but not the distribution, for the case where the interaction between the system and the reservoirs is represented by very “hard” collisions. This matrix differs from that for the previous case only by a trivial factor. The heat flux in the stationary state is found, as expected, to be proportional to the temperature difference (T 1 − T N ) rather than to the temperature gradient (T 1 − T N )/N. The kinetic temperature of the jth oscillator T(j) behaves, however, in an unexpected fashion. T(j) is essentially constant in the interior of the chain decreasing exponentially in the direction of the hotter reservoir rising only at the end oscillator in contact with that reservoir (with corresponding behavior at the other end of the chain). No explanation is offered for this paradoxical result.