A Definition of the H-Functional for Heat Conduction

In this Note a mathematical formulation of a monotonically decreasing functional with respect to time (the Hfunctional) is derived for physical systems described by the diffusion equation. This returns a condition which is equivalent to the second law of thermodynamics. The H-theorem is certainly the most debated (and probably the most important) part of Boltzmann’s scientific heritage [1,2]. As it is well known, this theorem states that, given a function f(t,x,v) which is a solution of Boltzmann’s equation, one can define a functional which is monotonically decreasing in time (the H-functional): = n v n x f f f H log ) ( (1)