Boltzmann, equilibrium, nonequilibrium statistical mechanics


A dilute gas initially in equilibrium and confined to half of an isolated box by a partition willirreversibly diffuse once the partition is removed. A new equilibrium establishes, and according tothermodynamics, entropy has increased. This paper explores Boltzmann’s work in equilibrium andnonequilibrium statistical mechanics to explain this irreversible process beginning with the underlyingtime symmetric micro-dynamics of the gas particles. Boltzmann’s ergodic hypothesis, transport equation(BE), and H-theorem are examined, as well as the reversibility and recurrence objections to the BE, andtheir resolutions through the statistical explanation of the BE. The entire pursuit is driven by thequestion: “How can one describe macroscopically irreversible phenomena from dynamics which is timereversible and recurring?”

A numerical simulation of gas in a partitioned box is developed to illustrate the discussion above.The simulation explicitly calculates the particle distribution function and Boltzmann’s H-function throughtime, and verifies the kinetic and anti-kinetic behavior as expected according to the BE and Loschmidtrespectively. We find that reversibility is unstable under perturbations, which compares with the results ofOrban and Bellemans (1967). This implies that states which are in nonequlibrium, or lead tononequlibrium, are less populated than equilibrium states, as shown by Boltzmann.

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