Non-Markovian Mpemba effect in mean-field systems

Under certain conditions, a counterintuitive behavior—an initially hotter sample freezes faster when quenched to a cold bath than an identical system initialled at a lower temperature—is known as the Mpemba effect (ME). Here we identify the existence of the ME in mean-field systems (MFS). Specifically, the thermal contact between MFS and a large thermal reservoir is built up using the microcanonical Monte Carlo algorithm. The simulation results unambiguously demonstrate that an initial hotter system undergoes the paramagnetic-ferromagnetic phase transition faster than the initial cooler one. The ME here originates from the back-reaction of the MFS system on the reservoir, which is thus an embodiment of non-Markovianness in relaxation. In addition, we confirm that the ME survives in the thermodynamic limit. And the significance of reservoir size is also explored: A smaller heat reservoir facilitates the overall relaxation process. In general, this work establishes a theoretical non-Markovian ME framework, which may shed light on widening the understanding of the mechanism behind the ME in other substances, including water.

Figure 1

Thermodynamic properties of our mean-field system. (a) The entropy diagram of our mean-field model in the $m-\varepsilon$ plane, where the blank regions are inaccessible. The equilibrium states and metastable states are highlighted by the solid and dotted lines, respectively. The black and gray of those lines indicate the nonmagnetized state and the ferromagnetic state, respectively. The dashed line is the dividing of spontaneously evolving region and nonspontaneously evolving region. (b) The corresponding final entropy $s$ as a function of energy $\varepsilon$. (c) The corresponding caloric curve.

Figure 2

The microcanonical Monte Carlo simulation of the heat exchange between the thermal reservoir and two MFSs with different initial temperatures. (a) The schematic diagram of the simulation. (b) The simulation result of (absolute value of) magnetization evolution, where the initial hotter and cooler system are represented by the black and gray lines, respectively. (c) The simulation result of energy evolution. (c) Inset: The energy changing near the phase transition point. (d) The evolution paths for two MFSs in the $m-\varepsilon$ plane. (d) Inset: The illustration of energy barriers $\mathrm{\Delta }{\varepsilon }_1$ and $\mathrm{\Delta }{\varepsilon }_2$ for HTMFS and LTMFS, respectively. And white star and square markers represent the corresponding metastable states.

Figure 3

The average lifetime of the metastable state, shown in log scale, as a function of the full system size. The black and gray circles represent the transition time of the MFSs initialled at {$T=22.56$, $\varepsilon =-0.4845$} and {$T=7.51$, $\varepsilon =-0.5035$}, respectively. Two solid lines are the guides to the eye. The ratio $\alpha =N/\left(N+N_{\mathrm{bath}}\right)=1/21$.

Figure 4

The relationships between metastable state lifetime $\tau$ and energy barrier $\mathrm{\Delta }\varepsilon$ versus the initial temperature $T$ when $N_{\mathrm{bath}}/N_{}=15$ and $=20$. $\tau$ is measured by 150 independent simulations and marked as discrete yellow circle markers (mean values) with vertical error bars. $\mathrm{\Delta }\varepsilon$ from the theoretical calculation is shown as continuous solid black line. The left ordinate represents the MC step and is given on a logarithmic scale. While the right ordinate represents energy and is exhibited as a linear scale. According to Eq. (5), $\tau$ and $\mathrm{\Delta }\varepsilon$ should be overlapped.