Dissipation and the Relaxation to Equilibrium

TL;DR

This paper derives the unique equilibrium distribution for ergodic Hamiltonian systems using the Dissipation Theorem and TFT, showing that deviations from equilibrium break time independence and that systems relax to equilibrium under general conditions.

Contribution

It provides a rigorous derivation of the equilibrium distribution and relaxation process using dissipation principles, extending ergodic theory results.

Findings

Equilibrium distribution is unique and time-independent.

Deviations from equilibrium distribution break time independence.

Systems relax to equilibrium if correlations decay and conditions are met.

Abstract

Using the recently derived Dissipation Theorem and a corollary of the Transient Fluctuation Theorem (TFT), namely the Second Law Inequality, we derive the unique time independent, equilibrium phase space distribution function for an ergodic Hamiltonian system in contact with a remote heat bath. We prove under very general conditions that any deviation from this equilibrium distribution breaks the time independence of the distribution. Provided temporal correlations decay, and the system is ergodic, we show that any nonequilibrium distribution that is an even function of the momenta, eventually relaxes (not necessarily monotonically) to the equilibrium distribution. Finally we prove that the negative logarithm of the microscopic partition function is equal to the thermodynamic Helmholtz free energy divided by the thermodynamic temperature and Boltzmann's constant. Our results complement and extend the findings of modern ergodic theory and show the importance of dissipation in the process of relaxation towards equilibrium.