Boltzmann entropy, relative entropy, and related quantities in thermodynamic space

Abstract

The conventional solution methods for the Boltzmann kinetic equation such as the Chapman–Enskog method or the moment method provide a thermodynamic branch of the distribution function evolving through macroscopic variables under the functional hypothesis. Such a distribution function is different in nature from the phase-space distribution function obtained by directly solving the Boltzmann kinetic equation subject to initial and boundary conditions in the phase space without the functional hypothesis. The Boltzmann entropy is an information entropy enumerated with the phase-space distribution function in the phase space, whereas the compensation function introduced in the previous work is a representation of the former in thermodynamic space. The two quantities are not generally the same. By using the concept of relative entropy which is the difference between the two quantities, we examine their relations and significance for the mathematical structure of thermodynamics of irreversible processes. By using the balance equations for the compensation function and the relative entropy, we investigate the limiting behavior of the rate of relative entropy as the thermodynamic branch of the distribution function becomes convergent in the sense of means (i.e., weakly converges) to the phase-space distribution function. The time derivative of the relative entropy does not vanish in the limit, but tends to a limit associated with energy dissipation. Such a limit represents a contraction of information as the description of irreversible processes is made in the thermodynamic space contracted from the phase space of 1023 particles. On the basis of such results, it is indicated that, as was found in earlier work, a thermodynamic theory of irreversible processes can be erected on the compensation function since it is an integral of a one-form in the thermodynamic space whereas the Boltzmann entropy is not.