  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.
