Polynomial convergence to equilibrium for a system of interacting particles

We consider a stochastic particle system in which a finite number of particles interact with one another via a common energy tank. Interaction rate for each particle is proportional to the square root of its kinetic energy, as is consistent with analogous mechanical models. Our main result is that the rate of convergence to equilibrium for such a system is $\sim t^{-2}$, more precisely it is faster than a constant times $t^{-2+\varepsilon}$ for any $\varepsilon>0$. A discussion of exponential vs polynomial convergence for similar particle systems is included.

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