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

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 $U(x)\propto x^{\alpha }$, $x>0$, where we find that the relaxation is $\sim t^{-(\alpha +2)\mathrm{/}(\alpha -2)}$ for $\alpha >2$, with a logarithmic correction when $(\alpha +2)\mathrm{/}(\alpha -2)$ is an integer. For $\alpha <2$ the relaxation is exponential. Interestingly for $\alpha =2$ (harmonic potential) the localised bath can not equilibrate the particle.