The Statistical Interpretation of Non-Equilibrium Entropy

Abstract

Boltzmann’s original scheme leading to the statistical interpretation of non-equilibrium entropy may be summarized as follows: Dynamics → Stochastic Process (kinetic equation) → Entropy. Recent computer experiments as well as spin echo experiments in dipolar coupled systems illustrate clearly the difficulties in Boltzmann’s derivation. Indeed, they display situations for which a Boltzmann type of a kinetic equation is not valid. The main purpose of this communication will be to show that we can now construct a more general microscopic model of entropy which shows the expected monotoneous approach to equilibrium even in non-Boltzmannian situations such as experiments involving “negative time evolution”.

First dynamic and thermodynamic descriptions of time evolution will be compared. The time inversion symmetry present in the dynamic equations is broken in the thermodynamic description (such as the Fourier equations). The relation between this symmetry breaking and causality will be discussed.

A brief summary of non-equilibrium statistical mechanics leading to the master equation will be given. While this master equation is rigorous it is not well suited for the discussion of the statistical interpretation of entropy mainly because of its non-local character in time. However it leads to a discussion of the dissipativity condition. Briefly this condition means, that the collision operator as defined in this theory is non-vanishing and has a part which is even in the Liouville-von Neumann operator L. As the result the time inversion symmetry of the dynamic equations is broken. Examples of simple systems for which the dissipativity condition can be rigorously verified (in an asymptotic sense when the system becomes large) will be given.

A formulation of dynamics in which the even part of L are explicitly displayed will be indicated. This formulation may be called the “causal” or “obviously causal” formulation of dynamics as causality is now incorporated into the differential equations (and not only as in the usual formulation in the integral representation of the solutions). The transformation from the initial representation to the causal representation conserves averages of all observables. It leads, therefore to equivalent (but not unitary equivalent) representations of dynamics. Examples will be given. In the causal formulation of dynamics there appears a Liapounoff function which is positive and can only decrease in time. This leads directly to a statistical model for non-equilibrium entropy. One of the important features of this new model is that it contains all non-equilibrium correlations which may be introduced through initial conditions. As an application, experiments involving “negative time evolution” will be discussed. It is shown that Loschmidt’s paradox is now solved as during each time interval in such experiments, the entropy production is now positive.

It is concluded that the second law of thermodynamics is valid for all initial value problems when formulated for mechanical systems which satisfy the dissipativity condition indicated above.