Microscopic Aspects of Entropy and the Statistical Foundations of Nonequilibrium Thermodynamics

Abstract

The aim of this paper is to show how recent work in nonequilibrium statistical mechanics leads to a mechanical interpretation of the second law of thermodynamics.

First, we recall Boltzmann’s definition of entropy and the difficulties associated with it (extension to dense systems, Loschmidt’s paradox, etc.). On a simple model (McKean—Kac’s model), we show that Boltzmann’s definition is certainly not valid for all possible evolutions but that it is, nevertheless, possible to find out a Liapounoff function which is positive and can only decrease as a result of the time evolution. This leads to a microscopic definition of entropy for this model. The entropy defined in this way reduces to Boltzmann’s entropy close to equilibrium.

We then give a short summary of the work in nonequilibrium statistical mechanics of the Brussels school and emphasise the fact that irreversibility appears as a symmetry breaking of the time reversal invariance which results from the appearance of dynamic operators which are even in the Liouville—von Neumann operator L. This symmetry breaking is a consequence of causality when applied to large systems formed by many interacting degrees of freedom. The introduction of nonunitary (called star unitary) transformation leads to a representation in which the time change of the distribution function is split into two parts, one odd and one even in the Liouville—von Neumann operators. The first corresponds to reversible changes, the second to irreversible ones. The correspondence with the second law is therefore complete. We may now introduce a Liapounoff function which leads to a general microscopic model of entropy. This definition is valid whatever the initial conditions. No additional probabilistic arguments are needed to derive the increase of entropy. On the contrary the probabilistic interpretation when valid is a consequence of dynamics. We also show that there is no Loschmidt paradox associated with our new definition of entropy, and that it reduces to Boltzmann’s definition for systems close to equilibrium.

In the concluding section we stress the fact that our approach leads to a general expression for the entropy production independently of any assumption of closeness to equilibrium. The perspectives opened by this new development which unifies dynamics and thermodynamics are briefly discussed.