The Mpemba effect likes to hit a wall

Abstract

The historical Mpemba effect involves a first-order phase transition. This has prompted the experimental realization of microscopic proxies in the form of a colloidal particle trapped in an asymmetric double well, for which the Mpemba effect has indeed been observed. We establish that the existence of the one-dimensional Mpemba effect for a polynomial potential is driven solely by the presence of a hard enough boundary, irrespective of the potential's double-well shape. We then show that the physics of the underlying Mpemba effect is governed not only by single-well physics but also by the high-temperature initial regime.

Figures (6)

• Figure 1: The presence ([regular]) or absence ([regular]) of the Mpemba effect is shown for an asymmetric potential with a right wall far away (a,b,c) and close enough (d). If the left wall (absent in (a)) is located far away, in units of the distance to the wall, the inner structure of the potential is shrunk, and the effective (b) potential looks like (c). The inner structure of the potential plays no role in the Mpemba effect itself.
• Figure 2: The inverse temperature $\beta_\mathcal{M}$ is shown as a function of $L_-$ for various bath temperatures in a quartic (left) and a sextic (right) potential. The analytical result obtained for large $\beta$ and $L_-$ is shown as a dashed line for guidance. No notable deviations are observed, even very far from the regime in which the analytical expression was derived. We have used $L_+=2L_-$.
• Figure 3: The temperature $T_{\mathcal{M}}$ is shown as a function of $L_+-L_-$ for $L_-=20$ (black), $L_-=50$ (red) and $L_-=100$ (blue) when the bath is at $\beta=10$. The transition temperature $T_{\mathcal{M}}$ diverges as the right wall gets close enough to the left wall, signaling the disappearance of the Mpemba effect. Our prediction Eq. \ref{['eq:transition2walls']} is consistent with the behavior away from $L_+\simeq L_-$.
• Figure 4: The function $\partial_x\ell_2$ exhibits a sharp peak at low bath temperature that is robust in the high temperature regime. Regardless of the temperature, it is flatly structureless far from the barrier.
• Figure 5: For a piecewise-quadratic potential, the transition temperature increases quadratically with distance from the wall.