Random-walk theory and correlation functions in classical statistical mechanics

As a step towards a random-process formulation for classical fluids which ivolve many-body correlations, a random-walk formulation is presented wherein, for both lattice-gas and continuum models, the Green function and weight function describing the random walk are related to the total pair-correlation function of the model and to either the direct pair-correlation function or the first element of the direct correlation matrix.     

Superstable interactions in classical statistical mechanics

We consider classical systems of particles inv dimensions. For a very large class of pair potentials (superstable lower regular potentials) it is shown that the correlation functions have bounds of the form $$\varrho (x_1 ,...,x_n ) \leqq \xi ^n$$ . Using these and further inequalities one can extend various results obtained by Dobrushin and Minlos [3] for the case of potentials which are non-integrably divergent at the origin. In particular it is shown that the pressure is a continuous function of the density. Infinite system equilibrium states are also defined and studied by analogy with the work of Dobrushin [2a] and of Lanford and Ruelle [11] for lattice gases.     

Substantiating the Gibbs method in classical statistical mechanics

An approach for the substantiation of the Gibbs method in equilibrium statistical thermodynamics is described; this approach is based not on the quasiergodicity hypothesis, but on the weaker assumption of macroscopic determinacy of thermodynamic systems. A generalized microcanonical Gibbs distribution is obtained. An electron gas in a homogeneous magnetic field is taken as an example. It is shown that the classical diamagnetism of the given system is not zero in the sense of quasimean nor of generalized Gibbs ensemble distributions. The equation of state of an electron as in a magnetic field is obtained, and hence it is shown that classical diamagnetism only vanishes if isotropy of the pressure at the vessel wall is assumed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 65–68, June, 1978.     

Spectral density method in classical statistical mechanics

The spectral density method in classical statistical mechanics is formulated and application to a linear classical spin system is made in the simplest approximation.     

The inverse problem in classical statistical mechanics

We address the problem of whether there exists an external potential corresponding to a given equilibrium single particle density of a classical system. Results are established for both the canonical and grand canonical distributions. It is shown that for essentially all systems without hard core interactions, there is a unique external potential which produces any given density. The external potential is shown to be a continuous function of the density and, in certain cases, it is shown to be differentiable. As a consequence of the differentiability of the inverse map (which is established without reference to the hard core structure in the grand canonical ensemble), we prove the existence of the Ornstein-Zernike direct correlation function. A set of necessary, but not sufficient conditions for the solution of the inverse problem in systems with hard core interactions is derived.Work partially supported by NSF grant PHY-8117463Work partially supported by NSF grant PHY-8116101 A01     

Quantum approach to classical statistical mechanics

We present a new approach to study the thermodynamic properties of d-dimensional classical systems by reducing the problem to the computation of ground state properties of a d-dimensional quantum model. This classical-to-quantum mapping allows us to extend the scope of standard optimization methods by unifying them under a general framework. The quantum annealing method is naturally extended to simulate classical systems at finite temperatures. We derive the rates to assure convergence to the optimal thermodynamic state using the adiabatic theorem of quantum mechanics. For simulated and quantum annealing, we obtain the asymptotic rates of T(t) approximately (pN)/(k(B)logt) and gamma(t) approximately (Nt)(-c/N), for the temperature and magnetic field, respectively. Other annealing strategies are also discussed.     

The initial distribution in classical statistical mechanics

This paper presents arguments proving that several kinds of experimental preparation procedures for classical systems lead in certain limits to initial distributions that are functions only of macroscopic variables.This research was supported by the U.S. Air Force Office of Scientific Research under Contract F44620-72-C-0072.     

Mean entropy of states in classical statistical mechanics

The equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated: linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with theKolmogorov-Sinai invariant of ergodic theory.     

Nonequilibrium statistical mechanics of finite classical systems-I

It is pointed out that the fine-grained probability density of statistical mechanics is of interest only through coarse-grained densities—integrals over nonzero volumes of phase space. This suggests the definition of a smoothed probability density: the unsmoothed density convoluted with a kernel having a small spread aroundzero velocity. If this kernel is of Gaussian form, the smoothed density satisfies a closed and exact equation for its evolution differing from the Liouville equation by the addition of one term. This equation is applied to the simple example of a noninteracting system. We need make no assumption about the size of the system in our discussion, though if the system is large enough, the assumption that it is infinite gives the same results. Reduced distribution functions are then discussed, and a treatment of the Landau damping of electron plasma oscillations is given that is free from the usual difficulties occasioned by the breakdown of the linearization.Formerly at Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England.     

The classical statistical mechanics of Frenkel-Kontorova models

The scaling properties of the free energy, specific heat, and mean spacing are calculated for classical Frenkel-Kontorova models at low temperature, in three regimes: near the integrable limit, the anti-integrable limit, and the sliding-pinned transition (transition by breaking of analyticity). In particular, the renormalization scheme given in previous work for ground states of Frenkel-Kontorova models is extended to nonzero-temperature Gibbs states, and the hierarchical melting phenomenon of Vallet, Schilling, and Aubry is put on a rigorous footing.     

Dobrushin uniqueness techniques and the decay of correlations in continuum statistical mechanics

Uniqueness of Gibbs states and decay properties of averaged, two point correlation functions are proved for many-body potentials in continuum statistical mechanical models via Dobrushin uniqueness techniques.     

On the relation between classical and quantum statistical mechanics

Classical and quantum statistical mechanics are compared in the high temperature limit =1/kT0. While this limit is rather trivial for spin systems, we obtain some rigorous results which suggest (and sometimes prove) different asymptotics for continuous systems, depending on the behaviour of the two-body potential for small distances: the difference between suitable classical and quantum variables vanishes as ² for smooth potentials and as for potentials with hard cores.Supported in part by FAPESP. Permanent address: Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil     

Nonequilibrium statistical mechanics of finite classical systems-II

The work of the previous paper is applied to the study of weakly interacting systems. Either by quasilinear techniques or by analyzing the perturbation series for the smoothed probability density, it is possible to derive a master equation equivalent to that of Brout and Prigogine without requiring the size of the system to become infinite. The properties of this equation are discussed. The equation is self-consistent provided the interactions are weak enough; however, examination of higher terms in the perturbation series shows that their effect might make the master equation invalid for times longer than that taken by a typical particle to cross the containing vessel. In many physical cases, the relaxation time will be shorter than this; also, further studies may show the higher terms to be less important than they seem.Formerly at Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England.     

On the analyticity in the potential in classical statistical mechanics

Some general results on strong cluster properties of connected or partially connected correlations, and their links with analyticity properties with respect to the potential or to classes of perturbations of the potential are presented.Equipe de Recherche du C.N.R.S. No. 174     

The moment problem in statistical mechanics: Correlation functions of purely dissipative systems

The question of convergence of the Mori formalism is investigated in the frame of the classical theory of moments. The autocorrelation functions (acfs) of purely dissipative systems admit of both moment and continued fraction expansions (like that obtained by Mori). New physical conditions for the convergence of the latter ones are related to the analytical behaviour of the corresponding spectral moment sequences. From discussion of the analytical properties of such continued fractions it is also proved that the knowledge of the spectral moments does not suffices for determining infinite acf eigenvalue spectra. Mori's continued fraction always converges at zero frequency, thus reproducing the value of the acf relaxation time correctly. Finally, our predictions are verified by comparison with the analytical results for the Verhulst model published recently by Jung and Risken.     

Wave function of the harmonic oscillator in classical statistical mechanics

The notion of wave function of the classical harmonic oscillator is discussed. The evolution equation for this wave function is obtained using the classical Liouville equation for the probability-distribution function of the harmonic oscillator. The tomographic-probability distribution of the classical oscillator is studied. Examples of the ground-like state and the coherent state of the classical harmonic oscillator are considered.     

On the statistical mechanics of classical Coulomb and dipole gases

A detailed, rigorous study of the statistical mechanics-screening- and critical properties, phase diagrams, etc., of classical Coulomb monopole and dipole gases in two or more dimensions is presented. The statistical mechanics of the two-dimensionalXY and Villain models is reconsidered and related to the one of two-dimensional lattice Coulomb gases. At low temperatures and moderate densities those gases behave like dipole gases. The Kosterlitz-Thouless transition is analyzed in that perspective and characterized by an order parameter. Techniques useful for a proof of existence of such a transition in a two-dimensional hard-core Coulomb gas are developed and applied to the study of dipole gases.A Sloan Fellow, and supported in part by NSF grant No. DMR 7904355.     

Euclidean invariance in statistical mechanics of classical continuous system

It is shown that equilibrium states of classical particles with short range interactions are Euclidean invariant whenever their correlation functions have a clustering which is integrable. The relation between invariance and clustering is analysed for spatial rotations and internal rotational degrees of freedom. The analysis is then extended to the case of long range interactions, including the Coulomb force and jellium systems.     

Tomographic representation of kinetic equations in classical statistical mechanics

A tomographic representation of kinetic equations is constructed using the Radon transform. Liouville’s equation is considered for one and many particles. The reduced Liouville’s equation is obtained in the tomographic representation and the Bogolyubov chain is investigated in this representation. An example of the relativistic kinetic equation in the tomographic representation is considered.