Non-equilibrium statistical mechanics: partition functions and steepest entropyincrease

Abstract

On the basis of just the microscopic definition of thermodynamic entropy and the definitionof the rate of entropy increase as the sum of products of thermodynamic fluxes and theirconjugated forces, we have derived a general expression for non-equilibrium partitionfunctions, which has the same form as the partition function previously obtained by otherauthors using different assumptions. Secondly we show that Onsager’s reciprocity relationsare equivalent to the assumption of steepest entropy ascent, independently of the choice ofmetric for the space of probability distributions. Finally we show that the Fisher–Raometric for the space of probability distributions is the only one that guarantees thatdissipative systems are what we call constantly describable (describable in terms of thesame set of macroscopic observables during their entire trajectory of evolution towardsequilibrium). The Fisher–Rao metric is fundamental to Beretta’s dissipative quantummechanics; therefore our last result provides a further justification for Beretta’stheory.