Theoretical model for the Mpemba effect through the canonical first-order phase transition

Sheng Zhang and Ji-Xuan Hou

Phys. Rev. E 106, 034131 – Published 23 September 2022

Abstract

The Mpemba effect is the phenomenon in which the system with high initial temperature cools faster than the system with low initial temperature when all other conditions are the same. A theoretical model of the Mpemba effect through the canonical first-order phase transition is proposed in this paper, which shows that in the cooling processes, the path of the first-order phase transition of the system with the high initial temperature does not pass through any metastable state, while the path of the first-order phase transition of the system with the low initial temperature passes through a metastable state, which leads to the occurrence of the Mpemba effect. Then an example of the theoretical model is given in the Blume-Emery-Griffiths model. The Monte Carlo algorithm is adopted to calculate the estimated times for both systems with different initial temperature to cool down and undergo a first-order phase transition. The simulation results demonstrate a Mpemba effect in the system. Moreover, the evolution paths of the first-order phase transitions of the systems with high and low initial temperatures are given, respectively. The theoretical model presented here may help explain the Mpemba effect in water.

Received 15 May 2022
Accepted 9 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.034131

Physics Subject Headings (PhySH)

First order phase transitions Nonequilibrium & irreversible thermodynamics Nonequilibrium statistical mechanics Phase transitions

Statistical Physics & Thermodynamics

Authors & Affiliations

Sheng Zhang and Ji-Xuan Hou ^*

School of Physics, Southeast University, Nanjing 211189, China

^*jxhou@seu.edu.cn

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Issue

Vol. 106, Iss. 3 — September 2022

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Images

Figure 1
Schematic diagram of the theoretical model. (a) The surface of free energy at the temperature $T_{f}$ . (b) The equilibrium line of $(x_{1}, x_{2})$ obtained by Eq. (1) of different temperature. (c) The free-energy surface at the temperature $T_{h}$ . The red dots correspond to the parameters $\vec{x} (T_{h})$ of the equilibrium state of the system with high temperature $T_{h}$ , and the blue dots correspond to the parameters $\vec{x} (T_{c})$ of the equilibrium state of the system with low temperature $T_{c}$ . The green dot in (a) and brown dots in (a) and (b) represent the metastable state and the equilibrium state of the system with temperature $T_{f}$ , respectively. The purple lines and white lines with an arrow in (a) represent the nonequilibrium cooling path from the temperature $T_{h}$ to $T_{f}$ and $T_{c}$ to $T_{f}$ , respectively. The direction of the arrows in (b) is the direction of temperature drop. $T_{p}$ in (b) is the temperature of the canonical first-order phase transition.
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Figure 2
The free energy per spin $\tilde{f} (β, m, q)$ vs the magnetization per spin $m$ and the quadrupole moment per spin $q$ for four different temperatures $T$ : (a)–(d) the equilibrium energy per spin $ɛ$ vs the temperature of the system $T$ , (e) the equilibrium quadrupole moment per spin $q$ , and (f) the magnitude of the equilibrium magnetization per spin $| m |$ vs the temperature of the system $T$ . (a) $T = T_{h} = 20$ , corresponding to the system with the high initial temperature. (b) $T = T_{c} = 0.623$ , corresponding to the system with the low initial temperature. (c) $T = 0.621$ , which is the temperature of the canonical first-order phase-transition point. (d) $T = T_{f} = 0.05$ , corresponding to the final temperature that the system will reach after cooling down, i.e., the temperature of the environment. The black mark ( $★$ ) in (a)–(d) represents the global minimum point of $\tilde{f} (β, m, q)$ . The black mark ( $▪$ ) in (b)–(d) represents the local minimum point of $\tilde{f} (β, m, q)$ . The red mark ( $▪$ ) in (e) and (f) represents the system with the high initial temperature. The orange mark ( $★$ ) in (e) and (f) represents the system with the low initial temperature. The blue mark ( $•$ ) represents the final equilibrium state that the system reaches after cooling down.
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Figure 3
The magnetization per spin $m$ vs the cooling time of the system $t$ during the cooling process of two systems that have different initial temperature obtained by a Monte Carlo simulation by setting $N = 20$ (a). The canonical first-order phase transition estimated time $τ$ vs the total number of the spin $N$ for two systems which have different initial temperature, (b),(c). The orange line in (a) represents the system with the high initial temperature, while the green line represents the system with low initial temperature. The red dots in (b) and (c) represent the average first-order phase transition estimated time from multiple Monte Carlo simulations, and they are all approximately in a straight blue line, respectively.
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Figure 4
The average evolution paths taken by the systems of two different initial temperatures during the cooling process, characterized by the energy per spin $ɛ$ , the magnetization per spin $m$ , and the quadrupole moment per spin $q$ of the system, from two perspectives, (a) and (b). The cooling paths of the systems with two different initial temperatures obtained by a single Monte Carlo simulation, characterized by the energy per spin $ɛ$ , the magnetization per spin $m$ , and the quadrupole moment per spin $q$ of the system, (c). The dark blue line in (a) and (b) represents the equilibrium energy of the system at different temperatures. The orange line with an arrow in (a) and (b) represents the average cooling path of the system with the high initial temperature $T_{i} = T_{h} = 20$ . The green line with an arrow in (a) and (b) represents the average cooling path of the system with the low initial temperature $T_{i} = T_{c} = 0.623$ . The brown line with an arrow and the light blue line with an arrow in (c) represent the cooling paths of the system with the high initial temperature $T_{i} = T_{h} = 20$ and the low initial temperature $T_{i} = T_{c} = 0.623$ , respectively, which are both obtained by a single Monte Carlo simulation.
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