Cercignani's conjecture is almost true for Smoluchowski equation. (February 2019) - Xie, Mingliang; Liu, Hong
Abstract: The asymptotic approach to thermodynamic equilibrium for the Smoluchowski equation describing Brownian coagulation, is examined using the Taylor-series expansion method of moments and the principle of entropy production. The equality of the relationship between the deviation of the entropy from its equilibrium value and the entropy production rate, similar to that in Cercignani's conjecture, allows one to infer that Cercignani's conjecture is almost true for the Smoluchowski equation with soft sphere collision. In this paper, we provide an explicit expression for the evolution of growth rate of the entropy to equilibrium for the spatially homogeneous Smoluchowski equation of Brownian coagulation. Highlights: Cercignani's conjecture is introduced into the Smoluchowski equation. The rate of entropy production for coagulation has similar structure as the conjecture for Boltzmann equation. The explicit rate of entropy production to equilibrium for coagulation is proposed. Cercignani's conjecture is almost true for Smoluchowski equation.