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## Statistical Mechanics of Correlated Gases
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 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
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 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 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 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 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 |

<< back Updated: 2007-10-17 Contact: Michael Kretschmer |

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