Fluctuations and relative Boltzmann entropy

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Abstract
It is shown in this paper that the variables conjugate to extensive macroscopic variables (or their molecular expressions) fluctuate from the thermodynamic intensive variables appearing in the extended Gibbs relation for the calortropy and that the relative Boltzmann entropy introduced in an earlier work [J. Chem. Phys. 102, 7169 (1995)] is intimately related to the fluctuations of the conjugate variables mentioned. We examine the nature of the fluctuations, and their implications and meanings for the relative Boltzmann entropy in kinetic theory. Based on this analysis, we find that even if the fluctuations vanish, the time derivative of the relative Boltzmann entropy depends on the path of evolution of irreversible processes and thus makes the time derivative of the Boltzmann entropy path-dependent, that is, a non-exact differential, in the thermodynamic space. The evolution equations for fluctuations are also presented which provide the solution of the Boltzmann equation beyond the level of description by the nonequilibrium canonical form for the distribution function. It is shown that, when the fluctuations are treated statistically, there follow thermodynamic uncertainty (thermodynamic complementarity) relations between thermodynamic conjugate variables in the case of nonequilibrium.