Mpemba as an Emergent Effect of System Relaxation

Abstract

The Mpemba effect (MpE), where a far-from-equilibrium state of a system relaxes faster compared to a state closer to it, is a well-known counterintuitive phenomenon in classical and quantum systems. Various system-specific theories have been proposed to explain this anomalous behavior in driven systems, though the fundamental mechanism of MpE in undriven systems, where MpE was first observed, remains unresolved. This paper provides a generic model of MpE for a quantum system following Markovian relaxation dynamics, regardless of system structure or environment. The key lies in the overlap of initial states with the fast relaxation mode; here, the constituents create a fast decay mode via interaction through the shared environment to show MpE, indicating MpE happens due to the collective behavior of the system. I also show that a system with anisotropic relaxation naturally exhibits MpE, even without a shared environment among the particles.

Figures (4)

• Figure 1: (color online) A schematic of MpE, where a far-from-equilibrium state ($\rho_h$) relaxes faster than a state closer ($\rho_c$) to equilibrium ($\rho_{ss}$). Here, $D$ represents a metric to measure distance; crossing of the trajectories indicates MpE in action.
• Figure 2: (color online) MpE in action. The evolution of metrics (a) $D$ and (b) $S(\rho (t) || \rho_{ss})$, with respect to equilibrium, for different initial states; these show crossover trajectories, i.e., MpE. The colors represent the initial state's distance (or relative entropy) from the equilibrium state. The insets show the evolution of the same for a system without a shared environment; in this case, we do not see MpE. The parameters used here are $T_2 = T_1$, $n = 10^3$, and $M_\circ = 0.5$; while $r$ and $\theta$ are chosen from the line $-\frac{5}{6} r + \frac{\theta}{\pi} = \frac{1}{2}$.
• Figure 3: (color online) (a) and (b) show thermalization time, $\tau_{TH}$, for different initial states for systems with and without a shared environment, respectively (in polar coordinates). $\tau_{TH}$ is numerically calculated as when $S(\rho(t) || \rho_{ss})$ becomes less than a cutoff ($10^{-4}$ here). Equivalently, one can use $D$ to find $\tau_{TH}$ instead of relative entropy. The green dot represents the equilibrium state, $\rho_{ss}$. The states closer to $\rho_{ss}$ thermalize earlier in (b), i.e., no MpE; while states with completely different initial states thermalize in (a), thus showing MpE. (c) and (d) represent the trajectories of states to $\rho_{ss}$ for systems with and without a shared environment, respectively. $\rho_{ss}$ is the stable fixed point for the system, represented here by the green dot. The color of the arrows denotes the speed of relaxation, $v$, following the trajectories. The speed increases with the distance from $\rho_{ss}$ and the trajectories are radial towards $\rho_{ss}$ in (d); whilst in (c), the trajectories are curved and speed increases with the distance from $\theta = 0$ or $\theta = \pi$ lines. Thus, (c) and (d) explain the behaviors seen in (a) and (b), respectively. Here, the used parameters are $T_2 = T_1$, $n = 10^2$, and $M_\circ = 0.5$.
• Figure 4: (color online) MpE seen using both the metrics (a) $D$ and (b) $S(\rho(t) || \rho_{ss})$. Though $D_z$ or $D_\perp$ do not show crossover trajectories. The colors represent the initial state’s distance (or relative entropy) from $\rho_{ss}$; $r = 0.75$ and $\theta$ is varied between $0$ and $\pi/3$ to generate (a) and (b). (c) shows that $\tau_{TH}$ does not depend on the initial state's distance from $\rho_{ss}$ (green dot), rather on the distance from the $r \cos \theta = M_\circ$ line. The trajectories of states, with colors representing the speed following them, are shown in (d), which explains the behavior of (c). The used parameters are $T_2 = 0.01 T_1$ and $M_\circ = 0.5$.