The examination of the equation of state of a two-dimensional model fluid (the hard disk system) was the very first application of the importance sampling Monte Carlo method in statistical mechanics (Metropolis et al., 1953), and since then the study of both atomic and molecular fluids by Monte Carlo simulation has been a very active area of research. Remember that statistical mechanics can deal well analytically with very dilute fluids (ideal gases), and it can also deal well with crystalline solids (making use of the harmonic approximation and perfect crystal lattice periodicity and symmetry), but the treatment of strongly correlated dense fluids (and their solid counterparts, amorphous glasses) is much more difficult. Even the description of short range order in fluids in a thermodynamic state far away from any phase transition is a non-trivial matter (unlike the lattice models discussed in Chapter 5, where far away from phase transitions the molecular field approximation, or a variant thereof, is usually both good enough and easily worked out, and the real interest is generally in phase transition problems).

One longstanding limitation on the resolution of Monte Carlo simulations near phase transitions has been the need to perform many runs to precisely characterize peaks in response functions such as the specific heat. Dramatic improvements have become possible with the realization that entire distributions of properties, not just mean values, can be useful; in particular, they can be used to predict the behavior of the system at a temperature other than that at which the simulation was performed. There are several different ways in which this may be done. The reweighting may be done after a simulation is complete or it may become an integral part of the simulation process itself. The fundamental basis for this approach is the realization that the properties of the systems will be determined by a distribution function in an appropriate ensemble.

Modern Monte Carlo methods have their roots in the 1940s when Fermi, Ulam, von Neumann, Metropolis, and others began considering the use of random numbers to examine different problems in physics from a stochastic perspective (Cooper, 1989); this set of biographical articles about S. Ulam provides fascinating insight into the early development of the Monte Carlo method, even before the advent of the modern computer. Very simple Monte Carlo methods were devised to provide a means to estimate answers to analytically intractable problems. Much of this work is unpublished and a view of the origins of Monte Carlo methods can best be obtained through examination of published correspondence and historical narratives. Although many of the topics which will be covered in this book deal with more complex Monte Carlo methods which are tailored explicitly for use in statistical physics, many of the early, simple techniques retain their importance because of the dramatic increase in accessible computing power which has taken place during the last two decades. In the remainder of this chapter we shall consider the application of simple Monte Carlo methods to a broad spectrum of interesting problems.

Long time tails in Green-Kubo formulas for transport coefficients indicate that long range correlations in non-equilibrium fluids cause divergent transport coefficients in the Navier-Stokes equations for two dimensional fluids, and divergent higher order gradient corrections to these equations for three dimensional fluids. Possible resolutions of these difficulties are considered, for transport of momentum or energy in fluids maintained in non-equilibrium stationary states. The resolutions of the divergence difficulties depend on the particular flow under consideration. For stationary Couette flow in a gas, the divergences are resolved by including non-linear terms in the kinetic equations, leading to logarithmic terms in the velocity gradients for the equations of fluid flow for two dimensional gases, and fractional powers for three dimensional flows. Methods used for Couette flow do not resolve the divergence problems for stationary heat flow. Instead, the difficulties are resolved by taking the finite size of the system into account.

Kinetic theory is defined as a branch of statistical mechanics that attempts to describethe non-equilibrium properties of macroscopic systems in terms of microscopic propertiesof the constituent particles or quantum excitations. The history of kinetic theory is summarizedfrom the first understandings of the connections of temperature and pressure ofperfect gases with their average kinetic energy and with the average momentum transferto the walls by particle-wall collisions. The history continues with a discussion of the contributionsof Maxwell and Boltzmann, and the development of the Boltzmann transportequation. Modern developments include extending the Boltzmann equation to moderatelydense gases, formulation of kinetic theory for hard sphere systems, discovery of long timetail contributions to the Green-Kubo expressions for transport coefficients, applications ofkinetic theory to fluctuations in gases, to quantum gases and to granular particles. Thecontents of each chapter are then summarized.

Symmetric functions of phase variables of N particles, can be expanded, formally, in terms of a series of Ursell cluster functions. They depend successively on one, two, ..., particle variables. For equilibrium systems, cluster expansions are used to obtain virial expansions of the thermodynamic functions, The cluster expansion method can be applied to N-particle time displacement operators and to the initial distribution function for a non-equilibrium system. Assuming a factorization property of the initial distribution function, one obtains expansions of the time dependent two and higher particle distributions in terms of successively higher products of one particle functions. This expansion of the pair distribution function, together with the first hierarchy equation generalizes the Boltzmann equation to higher densities in terms of the dynamics of successively higher number of particles considered in isolation. Contributions from correlated binary collision sequences appear and for hard spheres the Enskog equation is an approximation.

The Lorentz model consists of non-interacting, point particles moving among a collection of fixed scatterers of radius a, placed at random, with or without overlapping, at density n 6 in space. This model was designed to be, and serves as, a model for the motion of electrons in solids. The kinetic equation for the moving particles must be linear, and for low scatterers density, nad >> 1, it is the Lorentz-Boltzmann equation. If external fields are absent, the Chapman-Enskog method leads to the diffusion equation. For three dimensional systems with hard sphere scatterers, the Lorentz-Boltzmann equation can be solved exactly, and the range of validity of the Chapman-Enskog solution can be examined. Electrical conduction and magneto-transport can be studied for charged, moving particles. In both cases there are unexpected results. The Lorentz model with hard sphere scatterers is a chaotic system, and one can calculate Lyapunov exponents and related dynamical quantities.

The linearized Boltzmann equation is generalized to include a fluctuating source term to account for fluctuations in the distribution function about the average behavior given by the Boltzmann equation. The result is the Boltzmann-Langevin equation with fluctuations taken to be Gaussian and correlated by space and time delta functions. Linearized Navier-Stokes equations are derived with fluctuation terms identical to those obtained by hydrodynamical arguments by Landau and Lifshitz, when applied to dilute gases. The Boltzmann-Langevin equation is used to obtain the average correlations in density fluctuations needed for the spectrum of long wavelength light scattered by a dilute gas in equilibrium. The Rayleigh-Brillouin spectrum is obtained. The method is then extended, with appropriate modifications, to obtain equations that describe the scattering of light by a fluid that is maintained in 9 a non-equilibrium stationary state, with a fixed temperature gradient. The light scattering spectrum is dramatically different from the equilibrium case.

Integrations of the Boltzmann equation lead to conservation equations for the local fluid densities. The mean free path, l; of the gas particles is assumed to be small compared to the size, L of the container. When the gas is close to local equilibrium, solutions to the Boltzmann equation are obtained by expressing conserved currents as series in powers of the gradients of the local variables starting with the local Maxwell-Boltzmann distribution. Higher terms are obtained by inserting the series in the Boltzmann equation and collecting terms of equal order. This, together with the conservation equations, leads to hydrodynamic equations. The ideal fluid equations are obtained to lowest order, and the next order is the Navier-Stokes equations, with explicit expressions for the transport coefficients, which are compared with experimental results for different model potentials. To second order the rate of entropy production agrees with predictions of non-equilibrium thermodynamics.

We consider a dilute gas of particles that collide inelastically, dissipating kinetic energy at each collision, but conserving total momentum. The collision dynamics is simplified by using a constant restitution coefficient characterizing the kinetic energy remaining after each collision. A Boltzmann equation is derived, and depends on the restitution coefficient. In isolation the gas cools, and if spatially homogeneous, it evolves to a homogeneous cooling state with a cooling rate depending on the coefficient of restitution. The distribution function then satisfies a scaling law. For many interaction potentials, this state is unstable with respect to density fluctuations. Driven granular gases are also considered, for the cases that the external forces are stochastic, or in one dimension, constant. The high energy part of the population of particles is determined for isolated and for driven gases. Rings of Saturn are discussed as an example of granular systems studied using kinetic theory.

Boltzmann’s transport equation for a dilute gas with particles interacting with central, short range forces, and with bounding walls, is derived in detail, with emphasis on the use of the Stosszahlansatz. Boltzmann’s H–theorem is presented as a microscopic derivation of the law of entropy increase in non–equilibrium processes, and the Maxwell–Boltzmann equilibrium distribution is derived. Zermelo’s and Loschmidt’s arguments that the H–theorem is incompatible with the laws of mechanics are given and discussed. The Kac ring model is presented and used as a simple way to understand the application and the limitations of the Stosszahlansatz. It is concluded that the Boltzmann equation is statistical, rather than strictly mechanical, in nature, providing a description of the most probable non–equilibrium behavior of a dilute gas.

The Enskog equation was the first extension of the Boltzmann transport equation to higher densities. It applies only to hard sphere systems and takes into account excluded volume and collisional transport effects. While useful for one component gases, it has serious shortcomings, in particular, for mixtures it leads to expressions for transport coefficients that are inconsistent with the general Onsager reciprocal relations and it has no H-theorem. The Revised Enskog equation is presented and shown to satisfy an H-theorem, and, for mixtures, to have transport coefficients that satisfy the Onsager relations. The revised equation describes spatio-temporal fluctuations in a hard sphere fluid about equilibrium. It 8 is possible to extend the Revised Enskog equation to high densities where hard sphere fluids form a crystal, and to show that this solid has transport properties appropriate for an elastic solid. Explicit expressions for the appropriate transport coefficients are given.

The Boltzmann equation for dilute gas mixtures is formulated and the solution is expanded in powers of the gradients of the local density of each species, the temperature, the local velocity, and the local temperature of the gas. Thus hydrodynamic equations for mixtures of dilute gases are obtained with explicit expressions for the transport coefficients. These expressions can be expressed in forms that satisfy the Onsager reciprocal relations, and the rate of entropy production, to second order in the gradients, agrees with the predictions of non-equilibrium thermodynamics. The expressions for the transport coefficients, assuming various model interaction potentials, are compared with experimental results for binary mixtures.

At low temperatures the De Broglie wavelength of the gas particles becomes on the order of their average separation, and the effects of their indistinguishability become important. In the absence of a phase transition in the gas, the quantum mechanical Wigner distribution function for a dilute gas of fermions or bosons satisfies the Uehling-Uhlenbeck equation. This equation satisfies an H- theorem with equilibrium solutions being ideal boson or ideal fermion distributions. Navier-Stokes equations can be derived by standard methods. A low temperature gas of weakly interacting bosons undergoes a Bose-Einstein condensation with a macroscopically occupied ground state. A different approach is required to describe the non-equilibrium processes in such a situation. A kinetic equation can be derived for the Bogoliubov excitations in the gas at very low temperatures. The associated hydrodynamic equations are the Landau-Khaltnikov, two fluid equations, and explicit expressions are obtained for the six associated transport coefficients.