Necessary conditions for the Markovian Mpemba effect

Abstract

The Mpemba effect is a thermodynamic anomaly in which a system farther away in temperature from equilibrium thermalizes before one that is initially closer. The effect has been experimentally observed across a wide range of systems, including water, colloids, and trapped ions. It has recently been the focus of numerous studies aimed at understanding its mechanisms and developing multiple applications. Despite extensive work in the field, clearly determining which types of systems exhibit the Mpemba effect remains an open question. To address this, we derive simple necessary conditions on the transition rates for the Mpemba effect in a Markovian 3-level system and show that they can be applied to study the Mpemba effect in an N-level system. Multiple time scales govern thermalization in these systems. This allows the evolution to occur more quickly across larger temperature differences, explaining the Mpemba effect. We apply our protocol to evaluate which types of systems exhibit the Mpemba effect and, in doing so, explain why the Mpemba effect in Markovian systems remains a thermodynamic anomaly. In particular, due to the maximum entropy principle, our conditions allow us to discard the sub-Ohmic and Ohmic spectra. The latter describes a wide range of physical and chemical phenomena, which will not exhibit the Mpemba effect. Moreover, our results provide a clear path to determine the minimal physical requirements for the Mpemba effect, and we apply them to understand its underlying mechanisms better. Finally, our protocol could help identify relevant parameters for experiments, numerical simulations and diverse applications.

Figures (4)

• Figure 1: Testing the Mpemba effect for a 3LS formed from rotational energy-levels (left, $r>1$) vs a 3LS formed from hydrogen atom energy-levels (right, $r<1$). A system exhibits Mpemba if the fast eigenvector (red arrow) is parallel to a tangent of the quasistatic locus (orange/green line). As an illustration, we assume that the fast vector has the same angle in both cases. Here, the rotational case exhibits the Mpemba effect, and the hydrogen atom example does not. This is a consequence of the different energy-level spacing, quantified by $r$, which determines the shape of the quasistatic locus.
• Figure 2: Probability for an $N-$level system to exhibit the Mpemba effect for random values of transition rates, energies and temperatures. The x-axis represents the percentage of triplets keeping Eq. \ref{['eq:3lsc']}. Convergence of the values was assessed by doubling the number of tested cases and requiring a change of less than $0.5\%$. For the transition rates, we considered values from 0.05 to 1 and for $\beta_b E_i$ from $0$ to $1$.
• Figure 3: Testing the Mpemba necessary conditions for a 3LS as function of the energy level spacing $\Delta_{jk}=E_j-E_k$. The blue region indicates compliance with conditions (\ref{['eq:3lsc']}). Therefore, $N-$level systems with triplets exclusively in the white regions will not exhibit the Mpemba effect. Above (below) the diagonal dashed line, the energy levels correspond to increasing (decreasing) energy-level spacing. The plot shows an asymmetry favoring systems with decreasing energy-level spacing ($r<1$, e.g., hydrogen energy levels) for the conditions (\ref{['eq:3lsc']}). Plot parameters: super-Ohmic bath spectra with $s=3$, $\beta_b \Delta_c=1$.
• Figure S1: Maximum angle between the tangent vector and the base of the triangle as a function of the energy gap ratio Eq. \ref{['e relation']}.