Mpemba meets Newton: Exploring the Mpemba and Kovacs effects in the time-delayed cooling law

Despite extensive research, the fundamental physical mechanisms underlying the Mpemba effect, a phenomenon where a substance cools faster after initially being heated, remain elusive. Although historically linked with water, the Mpemba effect manifests across diverse systems, sparking heightened interest in Mpemba-like phenomena. Concurrently, the Kovacs effect, a memory phenomenon observed in materials such as polymers, involves rapid quenching and subsequent temperature changes, resulting in nonmonotonic relaxation behavior. This paper probes the intricacies of the Mpemba and Kovacs effects within the framework of the time-delayed Newton's law of cooling, recognized as a simplistic yet effective phenomenological model accommodating memory phenomena. This law allows for a nuanced comprehension of temperature variations, introducing a delay time ($\tau$) and incorporating specific protocols for the thermal bath temperature, contingent on a defined waiting time ($t_{\text{w}}$). Remarkably, the relevant parameter space is two-dimensional ($\tau$ and $t_{\text{w}}$), with bath temperatures exerting no influence on the presence or absence of the Mpemba effect or on the relative strength of the Kovacs effect. The findings enhance our understanding of these memory phenomena, providing valuable insights applicable to researchers across diverse fields, ranging from physics to materials science.

Figure 1

Number of publications and citations in the period 2006–2023 obtained from the Web of Science [98] with the search query “topic=(Mpemba effect).” Note the vertical logarithmic scale.

Figure 2

Sketch of supposed cooling curves for two similar systems in the same environment, adapted from Ref. [8].

Figure 3

Plot of the $\tau$-exp function $\mathcal{E}\left(t\right)$ with, from top to bottom, $\tau =0$, 0.1, 0.2, 0.3, and $e^{-1}\simeq 0.368$. Inset: The graph in semilogarithmic scale.

Figure 4

Plot of the damping coefficient $\kappa$ and the amplitude $A_{\mathcal{E}}$ as functions of $\tau$. The circle represents the damping coefficient $\kappa =e\simeq 2.718$ at the maximum delay time, ${\tau }_{max}=e^{-1}\simeq 0.368$.

Figure 5

Schematic representation of the decomposition of the (a) double-quench protocol into the sum of the (b),(c) single-quench protocols.

Figure 6

Schematic representation of the protocols A and B for (a) the direct Mpemba effect and (b) the inverse Mpemba effect.

Figure 7

Plot of $T_{\text{A}}\left(t\right)$ and $T_{\text{B}}\left(t\right)$, including their history for $t<0$. Here, $T_{\text{b}}^{\text{h}}=2, T_{\text{b}}^{\text{c}}=1$, and $\tau =0.36$, with (a) $t_{\text{w}}=0.2$, (b) $t_{\text{w}}=0.4$, and (c) $t_{\text{w}}=0.6$. In each panel, the dashed line depicts $T_{\text{B}}\left(t\right)$ if the sample were not quenched to $T_{\text{b}}^{\text{c}}$ at $t=0$. Note that a Mpemba effect exists in the case of (b) ($t_{\text{w}}=0.4$).

Figure 8

Plot of the difference function $\mathrm{\Delta }\left(t\right)$ for $\tau =0.36$ and $t_{\text{w}}=0.2$ (solid curve), 0.4 (dash-dotted curve), and 0.6 (dashed curve). The circle in the curve corresponding to $t_{\text{w}}=0.4$ denotes the corresponding crossover time $t_{\times }$.

Figure 9

Plot of the amplitude $A_{\mathrm{\Delta }}$ as a function of $t_{\text{w}}/t_{\text{w}}^{max}$ for $\tau =0$ (solid curve), 0.30 (dash-dotted curve), and 0.36 (dashed curve). The circles in the curves corresponding to $\tau =0.30$ and 0.36 denote the respective values of $t_{\text{w}}^{min}/t_{\text{w}}^{max}$.

Figure 10

Plot of the crossover time $t_{\times }$ as a function of $t_{\text{w}}/t_{\text{w}}^{min}$ for $\tau =0.25$ (solid curve), 0.30 (dash-dotted curve), and 0.36 (dashed curve). The circles denote the respective values of $t_{\text{w}}^{max}/t_{\text{w}}^{min}$. Inset: $t_{\times }$ vs ${log}_{10}(t_{\text{w}}/t_{\text{w}}^{min}-1)$.

Figure 11

Phase space for the Mpemba effect. The lower and upper curves represent $t_{\text{w}}^{min}$ and $t_{\text{w}}^{max}$, respectively, as functions of the delay time $\tau$. If $t_{\text{w}}<t_{\text{w}}^{min}$, then $\mathrm{\Delta }\left(t\right)>0$ for $t\ge 0$. If, on the other hand, $t_{\text{w}}>t_{\text{w}}^{max}$, then $\mathrm{\Delta }\left(t\right)<0$ for $t\ge 0$. Therefore, the Mpemba effect occurs if and only if $t_{\text{w}}^{min}<t_{\text{w}}<t_{\text{w}}^{max}$ (shaded region). The dashed line represents the locus defined by the condition $\mathrm{\Delta }\left(0\right)=|\mathrm{\Delta }\left(t_{\text{M}}\right)|$.

Figure 12

Plot of $T_{\text{A}}\left(t\right)$ and $T_{\text{B}}\left(t\right)$ for $t>0$. (a),(d) $(\tau ,t_{\text{w}})=(0.25,0.52)$, (b),(e) $(\tau ,t_{\text{w}})=(0.30,0.49)$, and (c),(f) $(\tau ,t_{\text{w}})=(0.36,0.47)$. (a)–(c) The direct Mpemba effect (with $T_{\text{b}}^{\text{c}}=1$ and $T_{\text{b}}^{\text{h}}=2$); (d)–(f) the inverse Mpemba effect (with $T_{\text{b}}^{\text{c}}=0.5$ and $T_{\text{b}}^{\text{h}}=1$).

Figure 13

Schematic representation of the protocols for (a) the direct Kovacs effect and (b) the inverse Kovacs effect.

Figure 14

Kovacs effect for $\tau =0.36$ and $t_{\text{w}}=0.1$, 0.2, ..., 1. (a) The direct effect (with $T_{\text{b}}^{\text{h}}=2$ and $T_{\text{b}}^{\text{c}}=1$); (b) the inverse effect (with $T_{\text{b}}^{\text{c}}=0.5$ and $T_{\text{b}}^{\text{h}}=1$). In each panel, the dashed line represents $T\left(t\right)$ if the sample were not quenched to $T\left(t_{\text{w}}\right)$ at $t=t_{\text{w}}$.

Figure 15

(a) Plot of the Kovacs hump function $K\left(t\right)$ for $\tau =0.36$ and $t_{\text{w}}=0.1$, 0.2, ..., 1. (b) Plot of the maximum value $K_{max}$ of $K\left(t\right)$ as a function of $t_{\text{w}}$ for, from bottom to top, $\tau =0.25$, 0.3, and 0.36. The dashed lines represent the asymptotic values, $1-\tau -{\kappa }^{-1}$.