Simple molecular mechanisms of heat transfer: Debye relaxation versus power-law

We study a simple molecular model (at a coarse-grain level) as a basis of irreversible heat transfer through a diathermic partition. The partition separates into two adjacent parts a box containing ideal point particles that communicate only through this partition. We provide the basic mechanism of energy transfer between the left- and right-hand side gas samples by assuming equipartition of kinetic energy of all outgoing particles colliding with the partition at a given time. We analyse and compare two essentially different cases: (A) the reference one, where we assume that the border walls of the box and the diathermic partition can randomize the direction of motion of rebounding particles, and (B) the case where we assume the mirror collisions of particles with the border walls and the partition. In both cases the rebounding of the particles from border walls is elastic. The above introduced assumptions allow us to numerically simulate and analytically consider, for example, the relaxation of temperatures of both gas samples and the entropy of the system. However, in both cases the long-time relaxation is essentially different since in case (A) it is an exponential one, while in case (B) it seems to be a power-law relaxation. The obtained results well agree in case (A) with the predictions of the phenomenological, linear theory of irreversible thermodynamic processes. However, to describe case (B) a version of this theory had to be developed which assumes time-dependence of heat conductivity; it describes the relaxation of the system far from equilibrium. The explanation of the results obtained in this case is, nevertheless, an intriguing problem.