Following a general discussion of the approach to equilibrium of a finite system in contact with a heat bath, an illustrative calculation is presented in terms of a weakly-coupled, harmonically-bound oscillator chain. A modified Gibbs entropy is defined in terms of pN, the reduced Liouville function of the system, which is obtained from the total Liouville function of the system and heat bath by (in principle) integration over the heat-bath variables. Since the system and heat bath are mutually interacting, some structure is observable in the entropy function as the system evolves from its initial value toward equilibrium, but the entropy ultimately evolves to its correct equilibrium value, despite time-reversible dynamics, because pN spreads from an initially sharp distribution to a final one that is characteristic of the heat bath in equilibrium. The entropy function is presented as an analytically defined, conceptually accurate substitute for Boltzmann’s H.
ASJC Scopus subject areas
- Chemical Engineering(all)