Thermomajorization theory provides new framework for quantifying mysterious Mpemba effect

The Mpemba effect is an intriguing physical phenomenon that causes some systems to cool faster when they are hot than when they are warm or colder. This effect was observed in various systems, including water, which sometimes freezes faster when it is hot than when it is cold.

While it is now well-documented, the Mpemba effect remains poorly understood from a theoretical standpoint. This is partly due to the methods that conventional physics theories offer for quantifying the speed with which a system relaxes (i.e., its relaxation speed).

Researchers at Kyoto University recently devised a new rigorous and unified approach for quantifying the Mpemba effect. The criterion they devised, presented in Physical Review Letters, is rooted in thermomajorization theory, a mathematical framework for comparing the disorder (i.e., entropy) between different thermodynamic states.

“The Mpemba effect—the counterintuitive phenomenon where a hotter system cools faster than a colder one—has fascinated scientists for decades,” Tan Van Vu, co-author of the paper, told Phys.org.

“Despite recent extensive theoretical and experimental studies, a fundamental challenge remained: the detection of this effect depends on the choice of a specific distance measure used to evaluate relaxation speed. This led to inconsistencies, as the effect observed with one measure might not appear within finite time when assessed with another.”

Vu and his colleague Hisao Hayakawa devised a unified approach for quantifying the Mpemba effect without relying on any specific distance measure, thus preventing the ambiguities reported in previous literature. Ultimately, they introduced the thermomajorization Mpemba effect, a rigorous criterion to determine the occurrence of the Mpemba effect within finite time, irrespective of what classical distances are used to measure the relaxation speed.

“The thermomajorization Mpemba effect provides a rigorous and unified way to quantify the Mpemba effect by simultaneously considering all possible monotone distance measures—those that do not increase over time—to evaluate relaxation speed,” explained Vu. “This approach eliminates ambiguities in conventional methods, which rely on choosing a single measure.”

The researchers showed that the thermomajorization Mpemba effect is equivalent to the occurrence of the Mpemba effect within finite time for any monotone measures. This means that it provides a way to consistently and generally characterize the Mpemba effect, at least for all classical Markovian stochastic processes.

“To quantify this effect, we employ thermomajorization theory, a mathematical framework widely used in statistical physics and quantum thermodynamics,” said Vu. “This framework allows us to determine whether a hotter system relaxes to equilibrium faster than a colder one in a way that is independent of any particular distance measure.”

The new criterion devised by this research team could soon pave the way for further explorations of the intriguing Mpemba effect, which is still not fully understood. Notably, using their criterion, Vu and Hayakawa also showed that the Mpemba effect is not limited to specific temperatures but can emerge at any temperature.

“Since thermal relaxation plays a fundamental role in many physical systems and technological applications, we anticipate that our findings will have broader implications beyond theoretical physics,” said Vu. “For example, our results could help the optimization of thermal processes such as acceleration of relaxation dynamics in heat engines, refrigeration technologies, and even quantum computing, where rapid initialization of quantum states is crucial.”

So far, Vu and Hayakawa have focused on attaining probability distribution-based quantifications of the Mpemba effect in classical stochastic processes. However, the approach employed in their recent study could eventually be extended to other systems, such as open quantum systems described by quantum master equations (e.g., the so-called Lindblad equation).

“In our future studies, we may also explore generalizations to non-Markovian systems,” added Vu. “Moving forward, an exciting direction is to investigate the relationship between our approach and energy-based quantifications, which could provide deeper insights into the fundamental physical mechanisms underlying the effect.

“Another intriguing open question is: What is the minimum timescale at which the thermomajorization Mpemba effect can occur? Investigating this question through the perspective of speed limits could help establish fundamental constraints on relaxation dynamics.”

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