Trend to Equilibrium for the Becker-Döring Equations: An Analogue of Cercignani's Conjecture

Abstract

In this work 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 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 $t$.

References in corpus (2)

• Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance
• Exhaustion of Nucleation in a Closed System

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• About the use of entropy production for the Landau-Fermi-Dirac equation
• Uniform moment propagation for the Becker-Döring equation
• The Becker-Döring process: pathwise convergence and phase transition phenomena