Trend to Equilibrium for the Becker-D\"oring Equations: An Analogue of Cercignani's Conjecture

TL;DR

This paper proves exponential convergence to equilibrium for the Becker-Döring equations under certain conditions, confirming Cercignani's conjecture in this context and providing algebraic decay rates in other cases.

Contribution

It establishes the validity of Cercignani's conjecture for the Becker-Döring equations with specific coefficients and introduces an 'almost' version for other typical cases, also linking decay rates to moment conditions.

Findings

Proves exponential convergence to equilibrium under linear coagulation coefficients.

Establishes an 'almost' Cercignani's conjecture with algebraic decay rates.

Shows exponential decay rate recovery under exponential moment assumptions.

Abstract

In this work we investigate the rate of convergence to equilibrium for subcritical solutions to the Becker-D\"oring 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-D\"oring 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 that results in an algebraic rate of convergence to equilibrium. Additionally, we show that if one assumes an exponential moment condition one can recover Jabin and Niethammer's rate of decay to equilibrium, i.e. an exponential to some fractional power of $t$.