Vol. 10, No. 7, 2017

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Trend to equilibrium for the Becker–Döring equations: an analogue of Cercignani's conjecture

José A. Cañizo, Amit Einav and Bertrand Lods

Vol. 10 (2017), No. 7, 1663–1708
Abstract

We investigate the rate of convergence to equilibrium for subcritical solutions to the Becker–Döring equations with physically relevant coagulation and fragmentation coefficients and mild assumptions on the given initial data. Using a discrete version of the log-Sobolev inequality with weights, we show that in the case where the coagulation coefficient grows linearly and the detailed balance coefficients are of typical form, one can obtain a linear functional inequality for the dissipation of the relative free energy. This results in showing Cercignani’s conjecture for the Becker–Döring equations and consequently in an exponential rate of convergence to equilibrium. We also show that for all other typical cases, one can obtain an “almost” Cercignani’s conjecture, which results in an algebraic rate of convergence to equilibrium.

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Keywords
Becker–Döring, nucleation, exponential convergence, entropy method
Mathematical Subject Classification 2010
Primary: 34D05, 35Q82, 82C05, 82C40
Milestones
Received: 24 December 2016
Revised: 28 March 2017
Accepted: 29 May 2017
Published: 1 August 2017
Authors
José A. Cañizo
Departamento de Matemática Aplicada
Universidad de Granada
Av. Fuentenueva S/N
18071 Granada
Spain
Amit Einav
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge
CB3 0WB
United Kingdom
Bertrand Lods
Department of Economics and Statistics & Collegio Carlo Alberto
Università degli Studi di Torino
Corso Unione Sovietica, 218/bis
10134 Torino
Italy