Yet Another Approach to Loschmidt's Paradox

TL;DR

This paper rigorously proves that non-equilibrium states tend to be local entropy minima on average, aligning with entropy growth principles without violating microscopic time-reversal symmetry.

Contribution

It introduces a rigorous proof that non-equilibrium states are local entropy minima on average, reconciling entropy growth with microscopic reversibility.

Findings

Non-equilibrium states are local entropy minima on average.

Entropy growth does not violate microscopic time-reversal symmetry.

First-order entropy derivative is zero, second-order is non-negative.

Abstract

The works by Lev Petrovich Pitaevskii are reference points for choosing an interesting research topic. An example is the article [Phys. Usp. Vol.54, pp.625-632, 2011] https://doi.org/10.3367/UFNe.0181.201106d.0647 which promotes rigorous results in non-equilibrium statistical physics. In present paper, we rigorously prove that a non-equilibrium state, on the average, is a local entropy minimum. This statement corresponds to the "entropy growth" of statistical mechanics and does not violate time reversal symmetry of microscopic motion: the first-order time derivative of the entropy is zero $\dot{S}=0$, while the second order derivative is non-negative $\ddot{S}\ge 0$.