Mpemba effect in a two-dimensional bistable potential

Abstract

We present an exactly solvable model of the Mpemba effect in an overdamped Langevin system confined in a two-dimensional radially symmetric bistable potential. The potential is constructed as a piecewise quadratic-logarithmic function that is continuous and differentiable at the matching radii, enabling an exact mapping of the corresponding Fokker-Planck operator to a Schroedinger-type eigenvalue problem. The relaxation spectrum and eigenmodes are obtained analytically in each region in terms of confluent hypergeometric functions, with eigenvalues determined from matching conditions. Focusing on isotropic equilibrium initial states at inverse temperature $β_{\rm ini}$ quenched to a bath at inverse temperature $β$, we derive explicit expressions for the mode amplitudes governing long-time relaxation. We demonstrate that the coefficient of the slowest mode exhibits non-monotonic dependence on $β_{\rm ini}$ and identify a sufficient crossing condition for the Kullback-Leibler divergence in terms of the two slowest modes, if the global minimum of the potential is located far away from the origin and the second minimum exists near the origin. For corresponding parameters, we demonstrate that the Mpemba effect can be realized. Our results provide a rare example of an analytically tractable two-dimensional model exhibiting anomalous relaxation without any confining walls, extending previous one-dimensional constructions with a hard wall and clarifying the role of radial geometry in nonequilibrium relaxation phenomena.

Figures (13)

• Figure 1: A schematic of a bistable radially symmetric potential.
• Figure 2: Phase diagram of the sign of $V(\alpha)$ in the parameter space $(-k_\mathrm{mid},\alpha)$ for $k_\mathrm{in}=k_\mathrm{out}=\xi=1$, where the solid line represents the boundary defined by $V(\alpha)=0$.
• Figure 3: Plots of $V(r)$ and $V_S(r)$ as functions of $r$ for the set of parameters in Eq. \ref{['eq:parameters']}.
• Figure 4: Plot of $\det \mathcal{M}(\lambda)$ against $\lambda$ using Eq. \ref{['eq:parameters']} for $T=1$, $k_\mathrm{in}=1.0$, $k_\mathrm{mid}=-0.5$, $k_\mathrm{out}=0.8$, $\xi=1.0$, and $\alpha=3.0$.
• Figure 5: Plots of (a) $a_{2}(\beta_\mathrm{ini}, \beta)$ and (b) $a_{m}(\beta_\mathrm{ini}, \beta)$ ($m=3, 4,5$) against the inverse temperature $\beta_\mathrm{ini}$ for $T=1$, $k_\mathrm{in}=1.0$, $k_\mathrm{mid}=-0.5$, $k_\mathrm{out}=0.8$, $\xi=1.0$, and $\alpha=3.0$.