Abstract
Using the recently derived dissipation theorem and a corollary of the transient fluctuationtheorem (TFT), namely the second-law inequality, we derive the unique time independent,equilibrium phase space distribution function for an ergodic Hamiltonian system incontact with a remote heat bath. We prove under very general conditions that anydeviation from this equilibrium distribution breaks the time independence of thedistribution. Provided temporal correlations decay, we show that any nonequilibriumdistribution that is an even function of the momenta eventually relaxes (not necessarilymonotonically) to the equilibrium distribution. Finally we prove that the negativelogarithm of the microscopic partition function is equal to the thermodynamicHelmholtz free energy divided by the thermodynamic temperature and Boltzmann’sconstant. Our results complement and extend the findings of modern ergodictheory and show the importance of dissipation in the process of relaxation towardsequilibrium.
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