On the classical approximation in the quantum statistics of equivalent particles

It is shown here that the microcanonical ensemble for a system of noninteracting bosons and fermions contains a subensemble of state vectors for which all particles of the system are distinguishable. This IQC (inner quantum-classical) subensemble is therefore fully classical, except for a rather extreme quantization of particle momentum and position, which appears as the natural price that must be paid for distinguishability. The contribution of the IQC subensemble to the entropy is readily calculated, and the criterion for this to be a good approximation to the exact entropy is a logarithmically strengthened form of the usual criterion for the validity of classical statistics in terms of the thermal de Broglie wavelength and the average volume per particle. Thus, it becomes possible to derive the Maxwell-Boltzmann distribution directly from the ensemble in the classical limit, using fully classical reasoning about the distinguishability of particles. The entropy is additive—theN! factor of the Boltzmann count cancels out in the course of the calculation, and the N! paradox is thereby resolved. The method of correct Boltzmann counting and the lowest term of the Wigner-Kirkwood series for the partition function are seen to be partly based on the IQC subensemble, and their partly nonclassical nature is clarified. The clear separation in the full ensemble of classical and nonclassical components makes it possible to derive the classical statistics of indistinguishable particles from their quantum statistics in a controlled, explicit way. This is particularly important for nonequilibrium theory. The treatment of molecular collisions along too-literally classical lines turns out to require exorbitantly high temperatures, although there are suggestions of indirect ways in which classical nonequilibrium theory might be justified at ordinary temperatures. The applicability of exact classical ergodic and mixing theory to systems at ordinary temperatures is called into question, although the general idea of coarse-graining is confirmed. The concepts on which the IQC idea is based are shown to give rise to a series development of thermostatistical quantities, starting with the distinguishable-particle approximation.This work was supported in part by the Air Force Office of Scientific Research, through Grants No. AF-AFSOR 557-64 and 557-67.     

Brillouin statistics vindicated

Brillouin statistics have been recently derived from a general theory of indistinguishable classical particles. We derive Brillouin's formulas by a random process ruling the growth of the physical system. This growth process is concerned with distinguishable and observable events, which have nothing to do with (in)distinguishability of particles.     

Space-time ergodic properties of systems of infinitely many independent particles

We investigate the ergodic properties of the equilibrium states of systems of infinitely many particles with respect to the group generated by space translations and time evolution. The particles are assumed to move independently in a periodic external field. We show that insofar as good thermodynamic behavior is concerned these properties provide much sharper discrimination than the ergodic properties of the time evolution alone. In particular, we show that though the infinite ideal gas is mixing in the space-time framework, it has vanishing space-time entropy and fails to be a space-timeK-system. On the other hand, if the particles interact with fixed convex scatterers (the Lorentz gas) the system forms a space-timeK-system. Also, the space-time entropy of a system of the type we consider is shown to equal its time entropy per unit volume.Research supported in part by the National Science Foundation Grant No. GP-16147 A No. 1.     

Statistical mechanics of quantum mechanical particles with hard cores

The states of a quantum mechanical system of hard core particles are characterized as a convex weak *compact subset of the states over aC* algebra associated with the canonical (anti-) commutation relations. It is shown that the mean conditional entropy, i.e. entropy minus energy, can be defined as an affine upper semi-continuous function over theG-invariant hard core states whereG is an invariance group containing space translations. An abstract definition of the pressure and equilibrium states is given in terms of the maximum of the conditional entropy and it is shown that the pressureP _S obtained in this way satisfiesPP _S P whereP andP are the thermodynamic pressures obtained from the usual Gibbs formalism with elastic wall, and repulsive wall, boundary conditions respectively. A number of additional results concerning the equilibrium states are also given.     

On the indistinguishability of classical particles

If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature become small. Thus, microscopic particles like electrons must remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2)³ are reduced to points in the phase space, the partition functionTr{exp(–) for N identical bosons (fermions) approaches (2)^–3N(N!) ... d³r₁ d³p₁ ... d³r_N d³p_N exp(–H). The two factors, (2)^–3N and (N!)^–1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles.     

Entropy,information theory,and the approach to equilibrium of coupled harmonic oscillator systems

Finite segments of infinite chains of classical coupled harmonic oscillators are treated as models of thermodynamic systems in contact with a heat bath, i.e., canonical ensembles. The Liouville function for the infinite chain is reduced by integrating over the outside variables to a function _N of the variables of theN-particle segment that is the thermodynamic system. The reduced Liouville function _N which is calculated from the dynamics of the infinite chain and the statistical knowledge of the coordinates and momenta att = 0, is a time-dependent probability density in the 2N-dimensional phase space of the system. A Gibbs entropy defined in terms of _N measures the evolution of knowledge of the system (more accurately, the growth of missing pertinent information) in the sense of information theory. As ¦t ¦ , energy is equipartitioned, the entropy evolves to the value expected from equilibrium statistical mechanics, and _N evolves to an equilibrium distribution function. The simple chain exhibits diffusion in coordinate space, i.e., Brownian motion, and the diffusivity is shown to depend only on the initial distribution of momenta (not of coordinates) in the heat bath. The harmonically bound chain, in the limit of weak coupling, serves as an excellent model for the approach to equilibrium of a canonical ensemble of weakly interacting particles.     

The weak second law of thermodynamics for classical gas in rectangular box

We prove, using the methods of probability theory, that the density of particles in closed classical systems consisting of a finite numberN of non-interacting point particles constrained to move in a rectangular box of the volumeV will approach a uniform density as t , if the initial states of the systems were created by random attribution of positions and velocities to particles. The time evolution of the systems is assumed to be entirely determined by the initial state: the particle dynamics contains no element of randomness. It is shown that if the number of particlesN (V remaining constant), the systems behave thermodynamically, i.e. they do not show any fluctuations of relative density of particles. The proved behaviour serves as the first step in a new approach to mathematically rigorous derivation of the second law of thermodynamics from the classical mechanics which makes no use of thermodynamic limit.     

Temperature in Friedmann thermodynamics and its generalization to arbitrary space-times

In recent articles we have introduced Friedmann thermodynamics, where certain geometric parameters in Friedmann models were treated like their thermodynamic counterparts (temperature, entropy, Gibbs potential, etc.). This model has the advantage of allowing us to determine the geometry of the universe by thermodynamic stability arguments. In this paper, in search for evidence for the definition of gravitational temperature, we will investigate a massless conformal scalar field in an Einstein universe in detail. We will argue that the gravitational temperature of the Einstein universe is given asT _g=1/2) (c/k) (1/R ₀), where R₀ is the radius of the Einstein universe. This is in accord with our definition of gravitational temperature in Friedmann thermodynamics and determines the dimensionless constant as 1/2. We discuss the limitations of the model we are using. We also suggest a method to generalize our gravitational temperature to arbitrary space-times granted that they are sufficiently smooth.Based on three essays awarded honorable mention in the years 1987, 1988 and 1989 by the Gravity Research Foundation—Ed.     

Statistics, Symmetry, and (In)Distinguishability in Bohmian Mechanics

This paper continues an earlier work by considering in what sense and to what extent identical Bohmian-mechanical particles in many-particle systems can be considered indistinguishable. We conclude that while whether identical Bohmian-mechanical particles ace considered to be statistically (in)distinguishable is a matter of theory choice underdetermined by logic and experiment, such particles are in any case physically distinguishable.     

Time without clocks—An attempt

We try to define time intervals separating two states of systems of elementary particles and observers. The definition is founded on the notion of instant state of the system and uses no information connected with the use of a clock. Applying then the definition to a classical clock and to a sample of unstable particles, we obtain results in agreement with experiment. However, if the system contains few elementary particles, the properties of the time interval present some different features.     

Statistical mechanics of lattice systems

We study the thermodynamic limit for a classical system of particles on a lattice and prove the existence of infinite volume correlation functions for a large set of potentials and temperatures.On partial leave from the University of Aix-Marseille.     

Low-density expanison for unstable interactions and a model of crystallization

For a class of unstable pair interactions in classical continuous systems of identical particles the high-temperature thermodynamic behavior is shown to be normal by extending low-density theorems for the correlation functions. In an example we prove a transition between a translation-invariant phase at high temperatures and low densities and solid with long-range oder at low temperatures. The transition is catastropic in the sense that it is accompanied by the divergence of thermodynamic quantities. We also exhibit counterexamples of unstable interactions in any dimension which do not give rise to a low-temperature catastrophe.     

Thermodynamics and phase transitions in dissipative and active Morse chains

We study the evolution of a simple one-dimensional chain of N=4 particles with Morse interactions and periodic boundary conditions which are imbedded into a heat bath creating dissipation and noise. The investigation is concentrated on thermodynamic properties for equilibrium, near-equilibrium and far-equilibrium conditions. For the thermodynamic equilibrium, created by white noise and passive friction obeying Einsteins fluctuation dissipation relation, we find a standard phase diagram. By applying active friction forces the system is driven to stationary non-equilibrium states, creating conditions where various self-sustained oscillations are excited. Thermodynamic quantities like energy, pressure and entropy are calculated near equilibrium, around a critical distance from equilibrium and far from equilibrium. We observe maximal order (minimum entropy) in certain region of the noise temperature, a phenomenon which is reminiscent of stochastic resonance. With increasing distance from equilibrium new phases corresponding to the existence of several attractors of the dynamical stem appear.     

On the dielectric susceptibility of classical Coulomb systems. II

This paper deals with the shape dependence of the dielectric susceptibility (equivalently defined, in a canonical ensemble, by the mean square fluctuation of the electric polarization or by the second moment of the charge-charge correlation function) of classical Coulomb systems. The concept of partial second moment is introduced with the aim of analyzing the contributions to the total susceptibility of pairs of particles of increasing separation. For a diskshaped one-component plasma with coupling parameter =2 it is shown, numerically and algebraically for small and large systems, that (1) the correlation function of two particles close to the edge of the disk decays as the inverse of the square of their distance, and (2) the susceptibility is made up of a bulk contribution, which saturates rapidly toward the Stillinger-Lovett value, and of a surface contribution, which varies on the scale of the disk diameter and is described by a new law called the arc sine law. It is also shown that electrostatics and statistical mechanics with shape-dependent thermodynamic limits are consistent for the same model in a strip geometry, whereas the Stillinger-Lovett sum rule is verified for a boundary-free geometry such as the surface of a sphere. Some results of extensive computer simulations of one- and two-component plasmas in circular and elliptic geometries are shown. Anisotropy effects on the susceptibilities are clearly demonstrated and the arc sine law for a circular plasma is well confirmed.     

The van der Waals limit for classical systems. I. A variational principle

We consider the thermodynamic pressurep(, ) of a classical system of particles with the two-body interaction potentialq(r)+ ^v K(r), where is the number of space dimensions, is a positive parameter, and is the chemical potential. The temperature is not shown in the notation. We prove rigorously, for hard-core potentialsq(r) and for a very general class of functionsK(s), that the limit 0 of the pressurep(, ) exists and is given by where the limit and the supremum can be interchanged. Here is a certain class of nonnegative, Riemann integrable functions,D is a cube of volume |D|, anda ⁰() is the free energy density of a system withK=0 and density . A similar result is proved for the free energy.     

Stationary nonequilibrium states in boundary-driven Hamiltonian systems: Shear flow

We investigate stationary nonequilibrium states of systems of particles moving according to Hamiltonian dynamics with specified potentials. The systems are driven away from equilibrium by Maxwell-demon reflection rules at the walls. These deterministic rules conserve energy but not phase space volume, and the resulting global dynamics may or may not be time reversible (or even invertible). Using rules designed to simulate moving walls, we can obtain a stationary shear flow. Assuming that for macroscopic systems this flow satisfies the Navier-Stokes equations, we compare the hydrodynamic entropy production with the average rate of phase-space volume compression. We find that they are equalwhen the velocity distribution of particles incident on the walls is a local Maxwellian. An argument for a general equality of this kind, based on the assumption of local thermodynamic equilibrium, is given. Molecular dynamic simulations of hard disks in a channel produce a steady shear flow with the predicted behavior.     

Spacetime dynamical entropy of quantum systems

The definition of the dynamical entropy for single automorphisms of nuclear C ^*-algebras is extended to groups of several commuting automorphisms. This entropy of a Z ^v-action is shown to be nonzero only if all the corresponding Z -subactions (0<<v) have infinite entropy. As a simple consequence, the spacetime entropy of quantum lattice spin systems, and of one-dimensional continuous systems with physically reasonable quasifree states, vanishes.     

Induced geometry and confinement

A classical, Poincaré invariant dynamical system is developed which contains, besides the natural metric _v , an induced metricg _v that is generated by a real scalar dynamical field. It is shown that scalar fields whose dynamics are governed by the induced metric can be consistently introduced. Also, point particles which follow timelike quasi-geodesic trajectories can be introduced. The reaction forces acting ong _v due to the presence of these fields and particles are computed. A discussion of causality and geometrical confinement is given.     

Wall and boundary free energies

The asymptotic free energy of planar walls and boundaries is analyzed for scalar and vector spin systems. Under the hypothesis of correlation decay, various alternative definitions are found to be equivalent in the thermodynamic limit and independent of the associated walls. Furthermore, a torus, or box having periodic boundary conditions, is shown to have no boundary or surface free energy. For vector spin systems withn-component spins, existence of the thermodynamic limit is shown forn=2 and positive interactions.     

Lower bounds on entropy for polymer chains on a square and a cubic lattice

Rigorous lower bounds on the entropy per particle as a function of the fractiong of thegauche bonds of a system of semiflexible polymer chains is obtained in the thermodynamic limit. Only square and cubic lattices are considered. For the case of a single chain havingl monomers, the bound is obtained for all gg=2/3. For the case of p>1 chains, each havingl monomers, wherel is a multiple of 4, the bound is obtained for all gg=13/90. In both cases, it is shown that the entropy is alwaysnonzero for all 0<gm(l), whereg _m(l) =(l-2)/l. Thiscontradicts the prediction from the Flory-Huggins approximations that the entropy is zero for allgg₀, whereg ₀ is some finite nonzero number. It is also pointed out that it isnot impossible to pack a lattice with disordered configurations of rodlike chains with finite entropy, again contradicting an assertion by Flory that it is impossible to do so. Finally, it is concluded that onecannot trust the Flory-Huggins approximations at least at low temperatures. The study also casts doubts on the validity of the Gibbs-DiMarzio theory of glass transitions in polymeric systems.