Unusual equilibration of a particle in a potential with a thermal wall

Abstract

We consider a particle in a one-dimensional box of length $L$ with a Maxwell bath at one end and a reflecting wall at the other end. Using a renewal approach, as well as directly solving the master equation, we show that the system exhibits a slow power law relaxation with a logarithmic correction towards the final equilibrium state. We extend the renewal approach to a class of confining potentials of the form $U(x) \propto x^α$, $x>0$, where we find that the relaxation is $\sim t^{-(α+2)/(α-2)}$ for $α>2$, with a logarithmic correction when $(α+2)/(α-2)$ is an integer. For $α<2$ the relaxation is exponential. Interestingly for $α=2$ (harmonic potential) the localised bath can not equilibrate the particle.