Non-equilibrium statistical mechanics: partition functions and steepest entropy increase

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