## Trend to equilibrium for the Becker-Döring equations : an analogue of Cercignani’s conjecture.(English)Zbl 1373.34021

Summary: 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.

### MSC:

 34A33 Ordinary lattice differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 82C05 Classical dynamic and nonequilibrium statistical mechanics (general) 82C40 Kinetic theory of gases in time-dependent statistical mechanics
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