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arXiv:2410.06686 · Physical Review Letters

Thermomajorization Mpemba Effect

Tan Van Vu, Hisao Hayakawa

Published Mar 10, 2025• 38 citations
arXivHTMLPDF

Abstract

The Mpemba effect is a counterintuitive physical phenomenon where a hot system cools faster than a warm one. In recent years, theoretical analyses of the Mpemba effect have been developed for microscopic systems and experimentally verified. However, the conventional theory relies on a specific choice of distance measure to quantify relaxation speed, leading to several theoretical ambiguities. In this Letter, we derive a rigorous quantification of the Mpemba effect based on thermomajorization theory, referred to as the thermomajorization Mpemba effect. This approach resolves all existing ambiguities and provides a unification of the conventional Mpemba effect across all monotone measures. Furthermore, we demonstrate the generality of the thermomajorization Mpemba effect for Markovian dynamics, rigorously proving that it can occur in any temperature regime with fixed energy levels.

DOI
10.1103/physrevlett.134.107101

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The Mpemba effect is a counterintuitive physical phenomenon where a hot system cools faster than a warm one. In recent years, theoretical analyses of the Mpemba effect have been developed for microscopic systems and experimentally verified. However, the conventional theory relies on a specific choice of distance measure to quantify relaxation speed, leading to several theoretical ambiguities. In this Letter, we derive a rigorous quantification of the Mpemba effect based on thermomajorization theory, referred to as the thermomajorization Mpemba effect. This approach resolves all existing ambiguities and provides a unification of the conventional Mpemba effect across all monotone measures. Furthermore, we demonstrate the generality of the thermomajorization Mpemba effect for Markovian dynamics, rigorously proving that it can occur in any temperature regime with fixed energy levels.
The Mpemba effect refers to the counterintuitive phenomenon that an initially hot system can freeze faster than an initially warm one. Recent years have brought major advances in both classical and quantum realizations, with the asymmetric bistable potential emerging as the paradigmatic classical benchmark because its mechanism is especially transparent and controllable. Yet for precisely this benchmark problem, the impact of quantization remains unexplored. Here we show that quantization shifts Mpemba behavior to qualitatively new regimes, moving it to ultra-cold temperatures that are orders of magnitude lower than those relevant for classical thermal barrier crossing. In addition, quantization produces inverse and double inverse Mpemba effects that are absent in the corresponding classical dynamics. Our results establish quantization as a robust route to quantum-enabled Mpemba effects inaccessible in classical regimes.
The Mpemba effect is a counterintuitive relaxation phenomenon that a system which begins hot can cool down faster than the identical system which is initially cold. Here, we investigate heat and work in a quantum open system with being alternatively driven by hot-cold heat reservoirs, which operates as an Otto cycle in finite time, either with or without the Mpemba effect. Based on the quantum master equation, the time duration of the cooling relaxation process is determined by exploring the distance function which can be used to observe the appearance of the Mpemba effect. We then numerically evaluate and compare the average and variance of the power of two scenarios with and without the Mpemba effect. We demonstrate that, given the efficiency, the Markovian Mpemba effect leads to improved performance by significantly increasing the power without sacrificing the stability.
In recent years, the quantum Mpemba effect (QME) which occurs when an out-of-equilibrium system reaches equilibrium faster than another that is closer to equilibrium has attracted significant attention from the scientific community as an intriguing and counterintuitive phenomenon. It generalizes its classical counterpart, extending the concept beyond thermal equilibration. This paper approaches the QME for Markovian open quantum systems from different perspectives. Firstly, we propose a physical mechanism based on decoherence-free subspaces. Secondly, we show that a macroscopic enhancement of the decay rate that scales linearly with system size can be obtained, leading to an extreme version of the phenomenon on Markovian open quantum systems. Thirdly, we study the Mpemba effect through the unravelings of Davies maps, revealing subtleties in the figures of merit which are chosen for obtaining the QME. Lastly, we propose a microscopic model for a better understanding of thermal bath dynamics in this context.
The Mpemba effect originally referred to the observation that, under certain thermalizing dynamics, initially hotter samples can cool faster than colder ones. This effect has since been generalized to other anomalous relaxation behaviors even beyond classical domains, such as symmetry restoration in quantum systems. This work demonstrates that resource theories, widely employed in information theory, provide a unified organizing principle to frame Mpemba physics. We show how the conventional thermal Mpemba effect arises naturally from the resource theory of athermality, while its symmetry-restoring counterpart is fully captured by the resource theories of asymmetry. Leveraging the framework of modes of asymmetry, we demonstrate that the Mpemba effect due to symmetry restoration is governed by the initial overlap with the slowest symmetry-restoring mode, mirroring the role of the slowest Liouvillian eigenmode in thermal Mpemba dynamics. Through this resource-theoretical formalism, we uncover the connection between these seemingly disparate effects and show that the dynamics of thermalization naturally splits into a symmetry-respecting and a symmetry-breaking term.
The counterintuitive Mpemba effect, wherein a hotter system cools faster, critically lacks a general macroscopic theory. Here, starting from linear irreversible thermodynamics, we formulate a generalized Newton's cooling law, dT/dt=−[γ0+MQ(t)](T−Tr)\mathrm{d}T/\mathrm{d}t = -[γ_0 + \mathcal{M}Q(t)](T-T_r)dT/dt=−[γ0​+MQ(t)](T−Tr​), for a system at temperature TTT relaxing in a thermal reservoir at TrT_rTr​, where the bare relaxation rate γ0γ_0γ0​ is modified by an initial-state memory term, Q(t)∝T(0)−TrQ(t) \propto T(0)-T_rQ(t)∝T(0)−Tr​. Arising from the interplay between heat flux and structural evolution, the coefficient M\mathcal{M}M governs anomalous relaxation behaviors, where M>0\mathcal{M} > 0M>0 (M<0\mathcal{M} < 0M<0) induces the (inverse) Mpemba effect. This universal thermodynamic framework maps out phase diagram to provide general criteria for the Mpemba effect in complex systems, offering a macroscopic picture that bridges disparate microscopic approaches.
The Mpemba effect refers to faster relaxation of states that are initially farther from equilibrium, yet its characterization is often tied to a chosen distance or resource measure. We introduce resource-Markovianity, an extended concept of quantum Markovianity to quantum resource theories, and formulate the resource Mpemba effect operationally as the breaking of resource-Markovianity by a relaxation operation. This yields a measure-independent operational characterization of resource Mpemba effects in general resource theories, together with quantitative characterizations based on resource-non-Markovianity measures. We illustrate the framework with the Mpemba effect for distinguishability of states, due to its relation to quantum Markovianity, and with the thermomajorization Mpemba effect from an operational perspective. These results reveal a deep interplay between quantum resources, non-Markovianity, and the Mpemba effect.