Remarks on entropy and equilibrium states Original Research Article

We show here that a large class of density functions 0 ≤ ƒ∞ (v) = ƒ∞ (|ψ|2(v)), v ∈ Ω ⊆View the MathML sourceN induces a unique “natural” entropy. As a consequence, given the equilibrium distribution View the MathML source of a kinetic equation which conserves the mass, one obtains at once the unique entropy functional monotone nonincreasing with time along the solution, independently of the dynamics. Application of this result to classical theory of rarefied gases permits to improve a theorem of McKean in [1].