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)∝xα, x>0, where we find that the relaxation is ∼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.