Approach to Thermodynamic Equilibrium

Abstract

The properties of a macroscopic classical system consisting of some 10²⁰ molecules are determined by a probability density function W of the complete Γ space of moments and coordinates of all the molecules. This probability density function is that of the ensemble representing the totality of all experimental systems prepared according to the macroscopic specifications. The entropy is always to be defined as the negative of k times the integral over the distinguishable phase space of W lnWhΓ. However, the total probability density function W, even for a thermodynamically isolated system, does not obey the Liouville equation, ∂W/∂t=LW, since small fluctuations due to its contact with the rest of the universe necessarily ``smoothes'' W, by smoothing the direct many‐body correlations in its logarithm. This smoothing is the cause of the entropy increase, and in systems near room temperature and above, in which there is heat conduction or chemical species diffusion, the smoothing keeps the true entropy numerically equal to that inferred from the local temperatures, pressures, and compositions. This, however, is by no means necessarily general. The criterion of thermodynamic isolation is not that the complete probability density function W is unaffected by the surroundings, but that reduced probability density functions w<sub>n</sub> in the Γ space of n=2,3,... molecules evolve in time as if the system were unaffected by the surroundings. This criterion is sufficient to give a mathematically definable method of ``smoothing'' the complete probability density function. The smoothing consists of replacing the direct many‐body correlations in lnW by their average n‐body values, n=2,3,..., such that the smaller reduced probability density functions w_n are unaffected.