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

Abstract

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) ∝ x α , 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. Unusual equilibration of a particle in a potential with a thermal wall 2 Figure 1. A point particle moves between a thermal reservoir kept at x = 0 and a reflecting wall at x = L. Whenever particle approaches the reservoir it renews its velocity and comes with velocity chosen from Rayleigh distribution while it gets reflected once it approaches the wall. Arrows with solid and dashed lines respectively indicate the velocity of the particle before and after collision, either with the reservoir on the left end or with the reflecting wall on right end of the cylinder.

Work, entropy production, and thermodynamics of information under protocol constraints

“…This model can be justified by imagining a box that contains a large colloidal particle, as well as a medium of small solvent particles to which the vertical partition is permeable. Note that this model differs from Szilard's original proposal [71], in which the box contains a single particle in a vacuum, which has been analyzed in [72][73][74] 6) and ( 14), apply to nonadiabatic EP, rather than overall EP.…”

Non-equilibrium dynamics of the piston in the Szilard engine

“…The possible reasons for this breakdown are the following: i) the perturbation theory here implicitly assumes a separation of time scales. As pointed out in a recent paper, for the case of a fixed piston the small particle shows a slow power-law relaxation to equilibrium [14]. This suggests that there may be no time scale separation between the particle and the piston; ii) secondly, we have not taken the multiple collisions into account.…”

“…This dynamics takes the system to the Gibb's equilibrium state P eq = Z −1 exp [−β(mv 2 /2 + M V 2 /2 + U(X))]θ(x)θ(X − x), where Z is the partition function. However the dynamics of the relaxation process is non-trivial, as can be seen even when the piston is held fixed [13].…”

“…, where Z is the partition function. However the dynamics of the relaxation process is nontrivial, as can be seen even when the piston is held fixed [14].…”