Statistical Mechanics of Correlated Gases

by R. Treumann

Thermal equilibria are conventionally taken as the starting point of most physical investigations of many-particle systems. They provide the undisturbed initial state. Distorting this state leads to disturbances, the behaviour of which can be followed by adopting perturbation techniques. The most common description of such systems and their equilibrium and non-equilibrium properties is based on the particle phase-space distribution function f (q, p, t) with (q, p) the phase-space element and q, p the generalised phase-space coordinates. In the classical case this is space and momentum. The function f determines the probability of finding a group of identical particles at time t in the interval [q, p; q+Δq, p+Δp]. For indistinguishable particles it satisfies a so-called kinetic equation that determines the time variation of f under the action of the external and internal forces. Such internal forces may be gravitation and electromagnetic forces. Direct frictional interactions among the particles are collected within a collision integral on the right-hand side of this equation. All kinetic equations in the same force fields differ only in this collision integral. In thermal equilibrium the total time variation of f vanishes. This implies that the collision integral also vanishes identically. However, since the collision integral itself depends on f its vanishing defines the form of the equilibrium distribution.

The best-known kinetic equation is the Boltzmann equation. The Boltzmann-collision integral accounts for direct interparticle collisions only. Its equilibrium solution is the famous Boltzmann-Gibbs distribution function. This distribution function serves the needs of an initial distribution for all thermal equilibria and their disturbances, with the latter either relaxing towards the initial equilibrium or leading to instabilities. The latter develop (under certain favourable conditions), when free energy is available, in the medium leading to the generation of structure. The physical justification of the Boltzmann-Gibbs distribution is based on some simple physically intuitive assumptions that make use of Gibbsian rules for the definition of ensembles, physically identical mental copies of the many-particle system under consideration that differ only in the order of the particle counting. Adopting a simple definition of the probability allows finding the most probable and thus physically realistic distribution (essentially by throwing a dice). In the classical theory it turns out that this procedure in fact reproduces the Boltzmann-Gibbs distribution. The reason for this is the most probable distribution of errors discovered much earlier by Gauss under the condition that the errors are statistically independent. In a collision-free system, where the particles interact only via long-range forces, such a procedure is questionable – in particular when these forces cause correlations to exist between particles. Taking into account long-range Coulomb interactions in a perturbation approach of Boltzmann theory leads to the Landau and Lenard-Balescu collision integrals, which, however, hardly change the Boltzmann distribution. Allowing for perturbations in phase space leads to the Fokker-Planck equation. Its solutions cause deformed distributions but in many cases are unable to reproduce the observed skewed distribution functions. The reason for this inability is that the Fokker-Planck equation assumes the validity of the perturbational approach while not changing the general assumption of stochasticity on which Boltzmann-Gibbsian statistics is based.

The idea followed here adopts a different starting point. We accept the unlimited validity of the kinetic theory as the highest stage of a microscopic theory of many-particle theory in the presence of external and internal forces. We investigate the possibility of an extended version of the collision integral that may not necessarily derive from a perturbation approach. In the simplest case we can model this type of collision integral in close similarity to the Boltzmann collision term. It is, however, assumed that the particles do not have to interact directly by collisions, since for collisionless media this is unsatisfactory ab initio. Instead the interaction should proceed via long-range correlations such as those which are believed to be present in turbulence. We further assume that self-organised structures or "compounds" formed as a result of such correlations are nearly scale-invariant. It is then reasonable to assume that only compounds of similar scale interact strongly. In such a case one may replace the distribution function f in the Boltzmann collision integral with a functional g[f] of the distribution function. This functional is determined by the nature of the turbulence and correlations. However, since energy and number conservation must be imposed in every single interaction, the functionals depend in a well defined way on particle energy and chemical potential in the stationary state.

Fig. 3.12: An observational example of the κ-distribution (Christon et al., 1991) as measured in the plasma sheet in the centre of the geomagnetic tail. The figure shows the observed ion fluxes. Such fluxes and distributions are typical for ions in the stationary state. The distributions may change during disturbances both in energy, flux level and slope. At high energies the distributions have a power-law tail that decays with power κ ~ 13/2, whereas they are relatively flat at low energies. The new theory explains this behaviour.

Up to this point the theory is still arbitrary. The important assumption that relates it to the classical statistical mechanics is a condition that must be imposed in order to reproduce Boltzmann's statistics for ordinary non-correlated collisions. To this end it is expedient to introduce a control parameter κ on that the functional g[f] (and hence the distribution function as well) should depend. One then demands that e.g. for κ -> ∞ the functional g[f] = f should become the identity functional. In this limit the Boltzmann-collision integral is reproduced. This assumption allows the definition of the simplest possible general functional g[f] that can be solved for the distribution function f. The distribution function obtained in this way is a member of the family of so-called κ-distributions which are frequently observed in collisionless space plasmas. Fig. 3.12 shows an example of such a distribution function measured in the Earth's magnetospheric tail.

The new distribution has a number of interesting properties. It possesses a high-energy "tail" which to higher energies follows a power-law decay. For energies near zero the distribution is practically flat. This behaviour follows from the fact that the function f (µ, q, p, t) depends explicitly on the chemical potential µ. In Boltzmann statistics this potential appears only in the exponential and drops out by normalisation. In our correlated case the chemical potential cannot be eliminated by normalisation. This implies that adding or subtracting particles from the medium requires work that has to be done.

With the help of the new distribution function one can derive a new expression for the entropy of a correlated medium. This expression differs from the Boltzmann-Shannon entropy. Physically, this does not imply that we have found a new entropy. The new expression simply tells us that in correlated media one may have to use a different mathematical expression for calculating the irreversible part of the disorder. It is very important to note that this entropy relation satisfies an H-theorem. It thus determines the new distribution function as an actual thermodynamically stable distribution on the same level as the Boltzmann-Gibbs distribution that is valid under the conditions of correlations. A correlated system that is disturbed from equilibrium will hence return to the equilibrium that is described by the function f. Moreover the entropy is a concave function. The entropies of two systems is superadditive, i. e. the entropy of the whole system is larger than the sum of both. This is a precondition for an entropy to be a true physical entropy. The correlations guarantee a fast reaching of the equilibrium.

In the equilibrium state thermodynamics remains intact. All thermodynamic relations and functions remain valid if only the thermodynamic averages are defined, as usual, as the moments of the new distribution function. This is all very satisfactory because it is in accord with observations. There are, however, a number of distinct differences from Boltzmann's statistical mechanics. In correlated ideal gases the simple relation between temperature and mean energy ceases to hold. The correlations break this symmetry, which is encountered in Boltzmann's statistics. The temperature is a mere parameter that is defined through the entropy. The mean energy, on the contrary, is defined through the pressure-energy relation that is unbrokenly valid also here. Moreover, the equation of state of the correlated gas is not given by the simple relation PV = NkT anymore. Due to the appearance of µ, it is more complicated. On the other hand, the adiabatic relations are conserved. Finally, new expressions are obtained for the specific heats and the adiabatic constant. The latter is found to be in the interval 1 < γ < 5/3 with the right-hand value holding only for the Boltzmann transition. In ideal correlated media γ is closer to 1 in agreement with the initial assumption of multiscale turbulence. This qualifies the new statistical mechanics for the description of turbulence and chaotic systems. In chaos theory a similar entropy is used, the Rényi entropy. However, this entropy is not in accord with thermodynamics, a fact that probably makes it non-physical. This suggests that it may be replaced by the new entropy relation.

The new theory also suggests a new formulation for general statistical mechanics. The thermodynamic potential can be used to derive the grand canonical partition function. It turns out that such a generalisation can be performed successfully. One then finds new expressions for the partition functions for the correlated Bose-Einstein and Fermi-Dirac distribution functions. In the Boltzmann limit these become the ordinary Bose and Fermi distributions. The application of these distributions and the new statistical mechanics to real problems in solid-state physics or stellar interiors is still open. As an interesting preliminary result one finds that for temperatures T -> 0 both the new Fermi and Bose distributions have zero occupation numbers. This disturbing result can be easily understood when remembering that near zero temperature all states freeze, no dynamics is possible and chance reigns. Hence, the lack of dynamics inhibits the formation of correlations. The medium becomes purely stochastic. This implies that near zero the statistics must become the Boltzmann limit such that only ordinary Fermi and Bose distributions exist. This in turn implies that the control parameter κ(T) is a function of temperature. In the Fermi case it is not difficult to determine this functional dependence. It will be interesting to observe which applications will open for this new kind of statistical mechanics.