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The Mpemba effect is the observation that hot liquids or colloids (such as ice cream) can freeze more quickly than colder ones, for similar volumes and surrounding conditions. Physicists remain divided on the effect's reproducibility, precise definition, and underlying mechanisms.[1] It is named after Erasto Mpemba, a Tanzanian teenager who studied it scientifically in the 1960s for the first time, along with Denis Osborne.[2]

The Mpemba effect was initially observed in ice cream and water, and later in other colloids.[3][4][5] It has been studied extensively in water, with mixed results, and some experiments finding no reproducible effect.[6] It has also been studied in magnetic alloys, nanomechanical systems,[7] and quantum systems.[8][9][10]

Definition

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The definition of the Mpemba effect used in theoretical studies varies,[11] making it difficult to compare experiments.[12] Monwhea Jeng proposed this definition for the effect in water: "There exists a set of initial parameters, and a pair of temperatures, such that given two bodies of water identical in these parameters, and differing only in initial uniform temperatures, the hot one will freeze sooner."[13] Even this definition does not specify whether "freezing" refers to the point at which a visible surface layer of ice has formed, or the point at which the liquid is completely frozen.[12]

A generalized definition is "When a hotter system equilibrates faster than a colder one when both are quenched to the same low temperature."[14] In water, this would be when the liquid has completely frozen.

Mpemba's observation

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The effect is named after Tanzanian student Erasto Mpemba, who described it in 1963 in Form 3 of Magamba Secondary School, Tanganyika; when freezing a hot ice cream mixture in a cookery class, he noticed that it froze before a cold mixture. He later became a student at Mkwawa Secondary (formerly High) School in Iringa. The headmaster invited Dr. Denis Osborne from the University College in Dar es Salaam to give a lecture on physics. After the lecture, Mpemba asked him, "If you take two similar containers with equal volumes of water, one at 35 °C (95 °F) and the other at 100 °C (212 °F), and put them into a freezer, the one that started at 100 °C (212 °F) freezes first. Why?" When Osborne experimented on the issue back at his workplace, he confirmed Mpemba's finding. They later published the results together in 1969, while Mpemba was studying at the College of African Wildlife Management.[15]

Mpemba and Osborne described placing 70 ml (2.5 imp fl oz; 2.4 US fl oz) samples of water in 100 ml (3.5 imp fl oz; 3.4 US fl oz) beakers in the icebox of a domestic refrigerator on a sheet of polystyrene foam. They showed the time for freezing to start was longest with an initial temperature of 25 °C (77 °F) and that it was much less at around 90 °C (194 °F). They ruled out loss of liquid volume by evaporation and the effect of dissolved air as significant factors. In their setup, most heat loss was found to be from the liquid surface.[15]

History

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Various effects of heat on the freezing of water were described by ancient scientists, including Aristotle: "The fact that the water has previously been warmed contributes to its freezing quickly: for so it cools sooner. Hence many people, when they want to cool water quickly, begin by putting it in the sun."[16] Aristotle's explanation involved antiperistasis: "...the supposed increase in the intensity of a quality as a result of being surrounded by its contrary quality."[citation needed]

Francis Bacon noted that "slightly tepid water freezes more easily than that which is utterly cold."[17] René Descartes wrote in his Discourse on the Method, relating the phenomenon to his vortex theory: "One can see by experience that water that has been kept on a fire for a long time freezes faster than other, the reason being that those of its particles that are least able to stop bending evaporate while the water is being heated."[18]

Scottish scientist Joseph Black in 1775 investigated a special case of the phenomenon by comparing previously boiled with unboiled water.[19] He found that the previously boiled water froze more quickly, even when evaporation was controlled for. He discussed the influence of stirring on the results of the experiment, noting that stirring the unboiled water led to it freezing at the same time as the previously boiled water, and also noted that stirring the very-cold unboiled water led to immediate freezing. Joseph Black then discussed Daniel Gabriel Fahrenheit's description of supercooling of water, arguing that the previously boiled water could not be as readily supercooled.[citation needed]

Modern experimental work

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Studies of the effect in water

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Modern studies using freezers with well-understood properties have observed the Mpemba effect where water supercools before freezing. Water that starts out cooler tends to reach a lower supercooled temperature before freezing. Some studies measure the time it takes for a sample to begin to freeze (the start of recalescence, the moment where the heat of freezing first starts to be released) the time it takes to completely freeze, or the difference: the time from the onset of recalescence to the completion of freezing. Some also measure the time it takes for a sample to reach the freezing point of the fluid, before any freezing has begun.

In 1995, David Auerbach studied glass beakers placed into a liquid cooling bath, where the water supercooled to −6 to −18 °C (21 to 0 °F; 267 to 255 K) before freezing. In some cases water which started off hotter began freezing first.[20] Considerable random variation was observed in the time required for spontaneous freezing to start, and Auerbach observed the Mpemba effect more frequently when the ambient temperature was between –6 and −12 °C (10 °F). James Brownridge later studied a variety of initial conditions and containers, measuring the time to the onset of recalescence, and found that while hot samples sometimes froze first this was also affected by properties of the container holding the liquid.[21]

Writing for New Scientist, Mick O'Hare recommended starting the experiment with containers at 35 and 5 °C (95 and 41 °F; 308 and 278 K), respectively, to maximize the effect.[22]

In 2021, John Bechhoefer described a way to reliably reproduce the effect.[23] In 2024, Argelia Ortega, et al. studied the freezing of small (1-20mL) drops in a Peltier cell with a thermographic camera, and found that hot drops consistently froze faster than cold ones, with a more pronounced difference for larger drops. In particular, hot drops finished freezing sooner after the onset of recalescence, and experienced less of a temperature spike during the freezing process.[24]

Criticisms of experiments with water

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Some researchers have criticized studies of the Mpemba effect for not accounting for dissolved solids and gasses, and other confounding factors.[11] Even among experiments that agree on a definition and observe the Mpemba effect for some experimental setups, they often do not observe it for all setups and starting conditions.[12]

In 2006, Philip Ball, a reviewer for Physics World wrote: "Even if the Mpemba effect is real — if hot water can sometimes freeze more quickly than cold — it is not clear whether the explanation would be trivial or illuminating."[12] Ball wrote that investigations of the phenomenon need to control a large number of initial parameters (including type and initial temperature of the water, dissolved gas and other impurities, and size, shape and material of the container, the method of cooling, and the temperature of the refrigerator) and the need to settle on a particular method of establishing the time of freezing. Ball described simple ways in which the effect might be observed, such as if a warmer temperature melts the frost on a cooling surface, thereby increasing thermal conductivity between the cooling surface and the water container.[12]

In 2016, Burridge and Linden studied a slightly different measure, the time it took water samples to reach 0 °C but not freeze. They carried out their own experiments, and reviewed previous work by others. Their review noted that the large effects observed in early experiments had not been replicated in other studies of cooling to the freezing point, and that studies showing small effects could be influenced by variations in the positioning of thermometers: "We conclude, somewhat sadly, that there is no evidence to support meaningful observations of the Mpemba effect."[25]

Studies of the effect in colloids and other systems

[edit]

Classical systems

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The original classroom observations of the Mpemba effect were of fresh ice cream, a colloid, freezing in a freezer.

A generalized version of the Mpemba effect is "when a hotter system equilibrates faster than a colder one when both are quenched to the same low temperature." This has been modeled theoretically for simple systems such as single particles under Brownian motion.[14]

The possibility of a "strong Mpemba effect" where exponentially faster cooling can occur in a system at particular initial temperatures was predicted in 2019 by Klich, Raz, Hirschberg and Vucelja. If they existed, the effect in such systems would be easy to observe experimentally.[26] In 2020 the strong Mpemba effect was demonstrated experimentally by Avinash Kumar and John Boechhoefer in a single-particle colloidal system.[3] In 2022, that group also demonstrated an "inverse Mpemba effect" in a single-particle colloid where a cold system heats up much faster than a warmer one, under the right conditions.[27]

Quantum systems

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Since 2020, quantum researchers have studied Mpemba effects in quantum systems as an example of how initial conditions of a system affects its thermal evolution. In 2024, a team in John Goold's lab at Trinity College described their quantum-mechanical analysis of an abstract problem wherein "an initially hot system is quenched into a cold bath and reaches equilibrium faster than an initially cooler system."[9] They included computational studies of spin systems which exhibit the effect,[9] concluding that certain initial conditions of a quantum system can lead to a simultaneous increase in the thermalization rate and the free energy.[10]

In 2025, experimental observations by Zhang, et al. found a quantum strong Mpemba effect for a single trapped ion,[28] and Chatterjee, et al. found the Mpemba effect occurs naturally during the cooling of nuclear spin states.[29]

Theoretical explanations

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While the definition of the Mpemba effect used in theoretical studies varies,[11] several explanations have been offered for its occurrence.

In 2017, two research groups independently and simultaneously found a theoretical Mpemba effect and also predicted a new "inverse" Mpemba effect in which heating a cooled, far-from-equilibrium system takes less time than another system that is initially closer to equilibrium. Zhiyue Lu and Oren Raz yielded a general criterion based on Markovian statistical mechanics, predicting the appearance of the inverse Mpemba effect in the Ising model and diffusion dynamics.[30] Antonio Lasanta and co-authors also predicted the direct and inverse Mpemba effects for a granular gas in a far-from-equilibrium initial state.[31] Lasanta's paper also suggested that a very generic mechanism leading to both Mpemba effects is due to a particle velocity distribution function that significantly deviates from the Maxwell–Boltzmann distribution.[31]

In 2024, building on Kumar's work, Isha Malhotra simulated single colloids placed in a different, double-well potential, and predicted conditions under which two specific ranges of hotter starting temperatures lead to faster freezing, but not temperatures in between those ranges.[4]

Supercooling is a component of many theoretical explanations, particularly the tendency of hot liquids that are cooled rapidly to start freezing at a higher supercool temperature.[32][21] Several molecular dynamics simulations have also supported that changes in hydrogen bonding during supercooling play a role in the process.[33][34] In 2017, Yunwen Tao and co-authors suggested that the vast diversity and peculiar occurrence of different hydrogen bonds could contribute to the effect. They argued that the number of strong hydrogen bonds increases as temperature is elevated, and that the existence of the small strongly bonded clusters facilitates in turn the nucleation of hexagonal ice when warm water is rapidly cooled down. The authors used vibrational spectroscopy and modelling with density functional theory-optimized water clusters.[6]

The following additional explanations have been proposed:

  • Solutes: Calcium carbonate, magnesium carbonate, and other mineral salts dissolved in water can precipitate out when water is boiled, leading to an increase in the freezing point compared to non-boiled water that contains all the dissolved minerals.[35]
  • Dissolved gases: Cold water can contain more dissolved gases than hot water, which may somehow change the properties of the water with respect to convection currents.[13]
  • Microbubble-induced heat transfer: Boiling may induce microbubbles in water that remain stably suspended as the water cools, then act by convection to transfer heat more quickly as the water cools.[36][37]
  • Convection, accelerating heat transfers: Reduction of water density below 4 °C (39 °F) tends to suppress the convection currents that cool the lower part of the liquid mass; the lower density of hot water would reduce this effect, perhaps sustaining the more rapid initial cooling. Higher convection in the warmer water may also spread ice crystals around faster.[38] Colder temperature may freeze more readily from the top, reducing further heat loss by radiation and air convection; while warmer water freezes from the bottom and sides because of water convection. Some experiments account for this factor.[13]
  • Frost: Frost has insulating effects. A container of hotter liquid may melt through a layer of frost that is acting as an insulator around a cooler container, allowing the container to work more effectively within the same refrigeration setup.[21]
  • Crystallization: A relatively higher population of water hexamer states in warm water might be responsible for the faster crystallization.[33]
  • Distribution function: Strong deviations from the Maxwell–Boltzmann distribution can result in a Mpemba effect in gases and granular fluids.[31]

Andrei A. Klimov and Alexei V. Finkelstein state that there is a great variability in freezing speed, explaining the Mpemba effect as of stochastic origin.[39]

See also

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References

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Bibliography

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Mpemba effect

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The Mpemba effect is the observation that, under certain conditions, hot water can freeze faster than cold water when both are subjected to the same subfreezing environment, despite the initially hotter sample having more to lose. This counterintuitive phenomenon was first systematically documented in a 1969 paper by Tanzanian student and physicist Denis G. Osborne, stemming from Mpemba's high school experiment in 1963 where he noticed that a hot mixture froze more rapidly than a cooler one in a domestic freezer. Earlier anecdotal reports date back to ancient times, with philosophers like and later figures such as and René Descartes noting similar anomalies in water freezing. The effect has been replicated in various laboratory settings, but its occurrence depends on specific factors including initial temperatures (typically hot around 70–90°C and cold around 0–25°C), water purity, container materials, and ambient conditions. Proposed mechanisms include enhanced evaporation from the hotter sample reducing its volume and mass, differences in convection currents promoting faster heat loss, expulsion of dissolved gases during heating that alters freezing nucleation, and reduced supercooling in hot water due to impurities or surface effects. For instance, introducing nucleation sites like roughened container walls can induce the effect even when the hot sample's enthalpy exceeds the cold one's by over 50%. Despite these explanations, the Mpemba effect remains controversial, with some rigorous experiments failing to observe it under strictly controlled conditions and attributing positive reports to measurement biases or non-ideal setups. Ongoing research continues to explore its validity, particularly in pure systems and broader thermodynamic contexts, highlighting its value in scientific education and outreach.

Definition and Phenomenon

Core Observation

The Mpemba effect is the observation that, under identical freezing conditions, a sample of initially at a higher can freeze more quickly than an identical sample initially at a lower . This has been noted in controlled settings where equal volumes of are subjected to the same sub-zero environment. In basic experimental setups demonstrating this effect, two equal volumes of pure water—one heated to approximately 70–100°C and the other at 20–35°C—are placed in identical containers within the same freezer or cooling apparatus, ensuring no additives or external disturbances alter the process. The containers are typically shallow to promote uniform cooling, and measurements track the time until the onset of freezing or complete formation, with care taken to minimize by gentle agitation if needed. This result is counterintuitive, as it challenges the expectation from classical that cooler water, being closer to the freezing point, should solidify faster. Specifically, it contrasts with , which posits that the rate of temperature change for an object is directly proportional to the difference between its temperature and that of its surroundings, implying that hotter water should take longer to cool initially. Representative examples include household tests where hot (around 80°C) in trays forms solid sooner than room-temperature (about 25°C) in an ordinary freezer, or laboratory demonstrations using at 90°C versus 25°C in a controlled -20°C chamber.

Influencing Factors

The Mpemba effect, where hot freezes faster than cold under certain conditions, is modulated by the initial temperature difference between the samples. The phenomenon is most reliably observed when hot water starts at approximately 80°C and cold water at around 10–20°C, as these differentials maximize the relative contributions of various loss mechanisms. Container material plays a significant role due to variations in thermal conductivity and insulation properties. The volume of water also affects the outcome, as larger volumes can enhance the effect by allowing more pronounced differences in cooling dynamics, while very small volumes may make it harder to detect. Proposed factors include differences in currents, which may be stronger in hot water; variations in dissolved gases, with hot water containing fewer gases that could lead to bubble formation in cold water; higher rates in hot water reducing volume; and greater tendency for in cold water delaying . Environmental variables further modulate the effect, including , which impacts —lower accelerates water loss from hot samples and amplifies the phenomenon.

Historical Background

Early Reports

The earliest documented observation of the phenomenon where hot water appears to freeze faster than cold water dates to the BCE, when described it in his Meteorologica. He wrote that "the fact that has previously been warmed contributes to its freezing quickly; for so it cools sooner," suggesting this occurs because previously heated water possesses a different quality of heat, or , that facilitates solidification more readily than unheated water. During the , similar accounts emerged in scientific discourse. In his 1620 Novum Organum, noted that slightly warm water freezes more quickly than cold water, which he cited as evidence for the importance of empirical testing over hasty conclusions. Around the same period in the 1640s, René Descartes attributed the effect to the greater motion of particles in hot water, arguing that this agitation allows the water to congeal into ice more efficiently than stagnant cold water. In the 18th and 19th centuries, reports continued but often yielded inconsistent results under rudimentary conditions. Joseph Black's late-1700s investigations into provided indirect context for heat dynamics in freezing but did not explicitly address the hot-versus-cold disparity. These accounts were typically anecdotal, drawn from natural environments like ponds or baths, lacking standardized setups or precise measurements.

Mpemba's Observation

In 1963, Erasto Bartholomeo Mpemba, a 13-year-old student at Magamba Secondary School in , observed an unusual phenomenon while preparing to sell for extra income. He noticed that a hot mixture of boiled milk and sugar froze more quickly in the school kitchen freezers than an identical cold mixture, despite the expectation that cooler liquids would solidify first. Mpemba raised the issue with his physics teacher, who dismissed it as impossible and ridiculed him for the observation. Undeterred, Mpemba later questioned , a of physics at University College , during one of Osborne's lectures at the school. Osborne initially expressed skepticism but agreed to investigate the claim privately. Osborne later conducted tests using pure samples in a domestic freezer, confirming that hot could indeed begin freezing before cold under certain conditions. This collaboration led to a joint publication in 1969 titled "Cool?" in the journal , which detailed their experiments and brought the effect to broader scientific notice. The study involved placing equal volumes of water at various initial temperatures into domestic freezers and timing the process until complete ice formation. They found the effect most pronounced when comparing water starting at approximately 90°C to that at 25°C, with the hotter sample achieving full freezing in less time.

Experimental Investigations

Classical Experiments on Water

The seminal laboratory investigation of the Mpemba effect was conducted by and Denis G. Osborne in 1969, who systematically tested the phenomenon using identical samples of water placed in polished cans immersed in a bath maintained at -20°C. They observed that water initially at 70–90°C froze in approximately 25–30 minutes, compared to 40–50 minutes for water starting at 25°C, under controlled conditions that minimized external variables such as container differences and ambient disturbances. In 1996, researcher C. A. performed experiments to isolate potential contributing factors, such as , by using lidded containers to cover the samples during freezing. While the Mpemba effect was diminished in these setups due to reduced evaporative cooling from the hot , Knight still detected differences in freezing times for the hotter samples, attributing this primarily to variations in . A comprehensive review by Monwhea Jeng in 2006 analyzed over 20 classical studies on the Mpemba effect in , revealing inconsistent results across experiments but confirming positive observations in some cases, particularly when mechanical agitation was applied to promote mixing and prevent stratification. These studies often employed temperature-time measurements using thermistors or similar sensors to track cooling profiles, highlighting how the effect's manifestation depended on initial conditions like purity and starting temperatures. To address variability in freezing initiation, many classical setups incorporated nucleating agents, such as deliberate scratches on the inner surfaces of containers, to inhibit and ensure consistent formation onset across hot and cold samples. Such variations underscored the importance of standardized protocols in demonstrating the effect's reliability under controlled laboratory conditions focused solely on .

Modern Replications and Variations

In the 2010s, simulations provided deeper insights into the Mpemba effect in by examining microscopic structural changes. A 2015 study using revealed that the effect arises from differences in networks, where hotter initial states lead to a more favorable restructuring for formation during cooling, reducing the time to freeze compared to colder starts. Complementary theoretical work by Lu and Raz in 2017 generalized the phenomenon through , predicting the Markovian Mpemba effect in systems like , where hot systems relax faster due to initial state dependencies, with experimental implications for controlled cooling setups. Experimental replications advanced with precise imaging techniques. A 2024 thermographic study employed to monitor temperature profiles in freezing drops, confirming the Mpemba effect under controlled conditions by visualizing non-uniform cooling and faster solidification from hot samples, attributing it to enhanced dynamics. Recent simulations from 2025 further validated the effect using advanced models. In a Communications Physics study, with the TIP4P/Ice model demonstrated that hot configurations relax to solid states faster than cold ones due to metastable structures favoring rapid phase transitions. Variations of the effect have been explored in supercooled regimes and with impurities. Studies on supercooled show the Mpemba effect can persist, where hot samples exhibit differences in supercooling behavior. Impurities like salts can influence the effect by affecting and the freezing point.

Reproducibility Challenges

The reproducibility of the Mpemba effect in has been a significant point of contention, with numerous studies failing to consistently observe the under controlled conditions. A key investigation by Burridge and Linden in 2016 conducted experiments designed to eliminate potential artifacts, such as and , by using insulated containers and precise monitoring in a freezer at -18°C and on a cooling plate at 0.3°C; they found no that hotter water cools or freezes faster than colder water when these variables are strictly controlled. This work surveyed prior literature and reanalyzed classic datasets, concluding that apparent effects in earlier reports were likely due to uncontrolled factors rather than an intrinsic property of . Sources of variability in experimental outcomes include several uncontrolled environmental and methodological factors that can mimic or obscure . Container geometry influences convection currents, potentially accelerating cooling in hotter samples through enhanced mixing, while ambient affects rates, which remove mass more rapidly from hot and thus shorten apparent freezing times. Measurement errors in defining and timing the "freezing point"—such as the onset of versus full solidification—further contribute to inconsistencies across setups. These variables highlight why is highly sensitive to experimental design, often leading to irreproducible results without rigorous . Criticisms of the Mpemba effect's validity center on its poor replication in systematic studies, with Jeng's review emphasizing that while historical anecdotes abound, modern experiments rarely confirm it without methodological flaws, attributing many positive reports to observational biases or errors. Proponents have countered such critiques by arguing that the effect is condition-specific and not a claim, as in a reply to Burridge and Linden, which clarified that the phenomenon involves freezing dynamics rather than simple cooling rates and can occur when phase changes are properly accounted for. Statistical challenges exacerbate reproducibility issues, particularly in early investigations that relied on small sample sizes, limiting the ability to distinguish true effects from random fluctuations in freezing times. For instance, reanalyses of foundational studies like Mpemba and Osborne's 1969 work reveal that their observed differences could plausibly arise from statistical noise given the limited trials conducted. The lack of standardized protocols, akin to ISO guidelines for thermal testing, has hindered comparability, underscoring the need for larger, replicated datasets with predefined controls to resolve ongoing debates.

Theoretical Explanations

Mechanisms in Classical Systems

One proposed mechanism for the Mpemba effect in classical systems involves and reduced frost formation. Hot evaporates more rapidly than cold due to higher , leading to a loss of mass and thus reducing the volume to be cooled. Additionally, less frost accumulates on the surface of hotter containers because removes potential frost precursors, thereby minimizing insulation and enhancing conductive to the surroundings. Convection currents provide another explanation, as the density differences in hot water create stronger thermal gradients that drive more vigorous mixing and circulation compared to cold water. This enhanced convection expels heat from the interior more efficiently, accelerating overall cooling rates, particularly in open containers where buoyancy effects are prominent. Studies have shown that suppressing convection, such as by using narrow tubes, diminishes the observed effect, underscoring its role in classical setups. The presence of dissolved gases and solutes also contributes, with hot water expelling dissolved air more readily through , which reduces the formation of insulating gas bubbles that could impede in cold water. Furthermore, the lower of hot water facilitates better flow and mixing, aiding in the removal of heat without the drag from gas pockets or concentrated solutes that might occur in initially colder samples. Supercooling and nucleation dynamics offer a key classical pathway, as cold water is more prone to —cooling below 0°C without freezing—due to fewer nucleation sites for ice crystals. In contrast, prior heating of water disrupts weak molecular clusters, priming the system for more rapid upon subsequent cooling, which shortens the time to solidification. Experimental evidence indicates that introducing nucleation promoters, like rough container surfaces, reduces in hot water more effectively, supporting faster freezing. Changes in bonding structures represent a molecular-level mechanism, where heating breaks down transient -bond networks, increasing and creating a more disordered state that forms stable precursors more quickly during recooling. simulations confirm that this reconfiguration leads to faster relaxation toward the frozen state, with hot exhibiting altered bond distributions that favor over the clustered forms in . This effect persists in pure systems, highlighting its intrinsic role independent of external factors.

Mathematical Models

One approach to modeling the Mpemba effect involves extending to account for nonlinear effects such as , which can lead to a shorter effective cooling for hotter initial states. The standard Newton's law is given by dTdt=−k(T−Tenv)\frac{dT}{dt} = -k (T - T_{\text{env}}), where TT is the , TenvT_{\text{env}} is the environmental , and kk is a cooling constant; however, incorporating introduces a mass-loss term, modifying the energy balance to ddt(mcT)=−hA(T−Ta)−Lvdmdt\frac{d}{dt} (m c T) = -h A (T - T_a) - L_v \frac{dm}{dt}, where mm is , cc is specific heat, hh is the , AA is surface area, TaT_a is air , and LvL_v is the of vaporization. This results in a nonlinear whose solution approximates an T(t)≈Tenv+(T0−Tenv)e−t/τT(t) \approx T_{\text{env}} + (T_0 - T_{\text{env}}) e^{-t/\tau}, with the τ\tau satisfying τhot<τcold\tau_{\text{hot}} < \tau_{\text{cold}} under conditions of significant from the hotter sample, as the reduced accelerates subsequent cooling. Stochastic models provide a probabilistic framework for understanding the Mpemba effect through the evolution of probability distributions in nonequilibrium systems. In the work of Lu and Raz, the Fokker-Planck equation describes the time evolution of the probability density p(x,t)p(x,t) for a system's state xx: ∂p∂t=−∂∂x[μ(x)p]+12∂2∂x2[σ2(x)p],\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x} [\mu(x) p] + \frac{1}{2} \frac{\partial^2}{\partial x^2} [\sigma^2(x) p], where μ(x)\mu(x) is the drift term and σ(x)\sigma(x) is the diffusion coefficient. This equation reveals that, for certain nonlinear drift and diffusion functions, initial hot states project more favorably onto the equilibrium distribution, leading to faster relaxation times compared to colder initial states, as quantified by the non-monotonicity of the projection coefficient onto the slowest-relaxing mode. Energy balance considerations highlight how phase changes and mass loss influence the total heat removal required for freezing. The heat QQ to cool and freeze a sample is Q=mcΔT+mLfQ = m c \Delta T + m L_f, where LfL_f is the latent heat of fusion, which dominates the process near the freezing point; however, for hot water, enhanced evaporation reduces mm more than for cold water, effectively lowering the latent heat term mLfm L_f and allowing the system to bypass extended sensible heat removal phases in the liquid state. Simulation-based models using molecular dynamics further quantify structural relaxation differences. Employing the TIP4P water potential, simulations show that quenching from higher initial temperatures results in shorter bond autocorrelation times, indicating faster reconfiguration of hydrogen bonds toward the ice-like structure, with the autocorrelation function ⟨r(0)⋅r(t)⟩/⟨r2(0)⟩\langle \mathbf{r}(0) \cdot \mathbf{r}(t) \rangle / \langle \mathbf{r}^2(0) \rangle decaying more rapidly for hot starts due to a density of states closer to that of ice. Recent theoretical frameworks have advanced the mathematical understanding of the classical Mpemba effect. For instance, thermomajorization theory provides a quantification by analyzing how nonequilibrium states relax via majorization relations in thermodynamic state spaces, demonstrating conditions under which hotter initial states approach equilibrium faster than cooler ones. Additionally, resource theories of athermality unify the classical thermal Mpemba effect with broader nonequilibrium phenomena, showing it arises naturally from resource constraints in thermal operations as of 2025.

Extensions to Other Systems

Non-Water Fluids and Colloids

The Mpemba effect was first observed in a colloidal mixture used for making ice cream, consisting of milk, sugar, and flour. Erasto Mpemba noted that the hot mixture, heated to near boiling, froze faster than an identical cold mixture when both were placed in a freezer at the same temperature. This observation, conducted under identical volumes and conditions, highlighted the potential for the effect in complex fluids beyond pure water. Similar behavior has been extended to other colloids, where phase transitions or non-monotonic relaxation dynamics lead to accelerated cooling from higher initial temperatures. The phenomenon has also been documented in non-water liquids, particularly those capable of forming hydrogen bonds. Experimental investigations tested glycerol and ethanoic acid alongside water, finding that hot samples of these liquids froze faster than cold ones under comparable conditions, with the strength of the effect linked to initial temperature differences. Glycerol, a polyol, exhibited a pronounced Mpemba effect, attributed to its hydrogen-bonding network influencing supercooling and thermal conductivity. Ethanoic acid showed similar results, where density variations and hydrogen bonding contributed to faster freezing from elevated starting temperatures. These findings indicate that the effect is not unique to water but occurs in molecular liquids with comparable intermolecular interactions. In fluids exhibiting temperature-dependent physical properties, such as viscosity or solubility, the Mpemba effect can manifest through enhanced convective heat transfer or altered nucleation rates. For instance, higher initial temperatures reduce viscosity more significantly in some liquids, promoting faster cooling before solidification. Despite these observations, the Mpemba effect is not universal across all non-water fluids and colloids. It is absent in systems lacking suitable phase transitions or hydrogen bonding, such as pure metals or gases, where cooling follows standard thermodynamic expectations without anomalous acceleration. Overall, the effect in these systems emphasizes the role of initial thermal history in influencing relaxation pathways.

Microscopic and Complex Systems

The Mpemba effect has been observed in microscopic systems involving Brownian particles, where non-equilibrium dynamics lead to faster relaxation from hotter initial states. In a 2023 study, simulations of a Brownian particle trapped in a single well potential demonstrated that particles starting at higher temperatures equilibrate faster to a lower bath temperature than those starting cooler, attributed to the interplay between inertial and frictional forces that enhances the projection onto the equilibrium distribution. This behavior arises from the potential well's confinement, which amplifies thermal memory effects during quenching, allowing hotter states to bypass slower relaxation pathways. In biological systems, analogous effects appear in processes like protein folding and enzyme reactions, where initial hot or denatured states can refold or react more rapidly due to persistent memory in non-equilibrium configurations. For instance, spin glass models, often used to simulate protein folding landscapes, exhibit a Mpemba-like persistent memory effect, where hotter glasses relax faster to equilibrium upon cooling, reflecting disordered energy minima that favor quicker reconfiguration from high-energy starts. Similarly, in enzymatic reactions, increasing enzyme concentration can paradoxically slow biochemical rates in a manner akin to an inverse Mpemba effect, as hotter initial conditions accelerate product formation through optimized transition states in non-equilibrium thermodynamics. These examples highlight how biological complexity introduces relaxation "shortcuts" via structural memory. The effect extends to complex mixtures and heterogeneous media, such as granular materials and slurries, where particle interactions under cooling amplify anomalous thermalization. In granular fluids, both driven and freely cooling systems show hotter assemblies cooling faster than cooler ones, due to inelastic collisions that preserve velocity correlations as a form of thermal memory, enabling collective modes to decay more efficiently. Slurries and dense suspensions, including biological fluids like blood plasma, exhibit similar behaviors through viscosity gradients and particle clustering, where hot states induce faster sedimentation or phase separation during quenching. This framework underscores non-equilibrium thermodynamics as the core mechanism, where initial conditions imprint a "memory" that shortcuts relaxation in complex environments, with implications for nanoscale heat management and biophysical modeling.

Quantum and Advanced Effects

Quantum Mpemba Effect

The quantum Mpemba effect describes a counterintuitive phenomenon in quantum systems where an initial state farther from equilibrium—such as one with higher energy or effective temperature—relaxes to the steady or ground state faster than an initial state closer to equilibrium. This defies classical intuition, as one might expect systems nearer equilibrium to approach it more rapidly. In open quantum systems, the effect manifests during thermalization processes governed by dissipative dynamics, where "hotter" preparations exhibit accelerated decay rates toward the target state. The concept was theoretically proposed in 2019 by Klich, Raz, Hirschberg, and Vucelja, who analyzed it within the framework of Markovian open quantum dynamics described by the Lindblad master equation. Their work introduced the "Mpemba index" to quantify the strength of the effect, revealing that it stems from the non-normal structure of the Liouvillian superoperator, which governs the time evolution of the density matrix. This non-normality leads to transient amplifications in relaxation rates for specific non-thermal initial states, termed Mpemba states, enabling faster equilibration. The proposal highlighted how such anomalous relaxation can occur systematically in quantum settings without requiring fine-tuned conditions. Examples of the quantum Mpemba effect include simple models like driven harmonic oscillators and spin systems coupled to thermal baths. In a quantum harmonic oscillator under Lindblad dynamics, an initial coherent state with higher excitation energy can return to the ground state more quickly than a lower-energy thermal state, due to optimal alignment with the dissipation channels. Similarly, in spin-1/2 systems or small spin chains, "hot" initial configurations—such as inverted populations—demonstrate faster relaxation rates compared to cooler ones, as simulated in Markovian master equations. These cases illustrate the effect's generality across low-dimensional quantum systems. In distinction from the classical Mpemba effect, which often involves macroscopic mechanisms like evaporation or convection in fluids, the quantum analog arises fundamentally from quantum-specific features such as coherences and entanglement in the initial state. These quantum correlations allow the system to exploit transient behaviors in the evolution operator, bypassing slower diffusive paths and achieving speedup through interference-like effects in the density matrix dynamics, rather than purely thermal gradients.

Strong Mpemba Effect

The strong Mpemba effect represents an extreme variant of anomalous relaxation in quantum systems, characterized by an exponential acceleration in the cooling or equilibration process. Specifically, the relaxation time τ\tau scales as τ∝exp⁡(−αEinitial)\tau \propto \exp(-\alpha E_\text{initial}), where EinitialE_\text{initial} denotes the initial energy (or "hotness") of the system and α>0\alpha > 0 ensures that higher initial energies lead to proportionally faster relaxation toward equilibrium. This contrasts with typical , as the effective rate constant increases dramatically with initial excitation, enabling super-accelerated thermalization under certain conditions. The theoretical foundation of this effect draws from the concept of thermomajorization, a framework for comparing the thermal disorder of quantum states relative to a reference . In this approach, hotter initial states possess a higher thermomajorization order, meaning they are "further" from the equilibrium Gibbs state in a thermodynamically meaningful sense, which paradoxically shortens the path to equilibration by suppressing slower relaxation modes. Seminal work in the has unified this under the thermomajorization Mpemba effect, demonstrating its universality across monotone distance measures and providing bounds on crossover times for equilibration. These insights stem from analyzing the spectrum of the system's Liouvillian operator, where initial states aligned with faster-decaying eigenmodes dominate the dynamics. This phenomenon applies primarily to open quantum systems weakly coupled to a thermal bath, where Markovian dynamics approximate the evolution without strong memory effects. Theoretical models, such as those involving single or few qubits interacting via dissipative channels, illustrate the effect through exact diagonalization of the , showing how weak coupling preserves the exponential speedup while avoiding non-perturbative regimes. In these setups, the strong Mpemba effect emerges when the initial has vanishing projections onto the slowest-relaxing eigenspace of the Liouvillian. Unlike the general quantum Mpemba effect, which may involve polynomial or milder speedups through mechanisms like symmetry restoration, the strong variant demands precise initial state preparation—often pure or highly excited states orthogonal to the equilibrium manifold—to realize the full exponential dependence on initial energy. This specificity highlights its role as a tailored optimization in quantum thermalization, distinct from broader anomalous behaviors.

Recent Developments

In early 2025, researchers demonstrated the quantum strong Mpemba effect using a single trapped system, observing exponentially accelerated relaxation dynamics where an initially hotter state reached equilibrium faster than a cooler one under identical conditions. This experiment, conducted with precise control over the ion's vibrational modes, highlighted the role of non-normal dynamics in enabling such paradoxical thermalization. Building on this, a team from reported in October 2024 the observation of the quantum Mpemba effect in programmable quantum simulators, where quantum transformations paradoxically sped up relaxation in far-from-equilibrium states, extending the phenomenon beyond classical water systems to controlled quantum . These findings resolved some reproducibility challenges by emphasizing condition-specific initial states and environmental couplings. Further advancing experimental insights, a June 2025 study revealed a quantum Mpemba effect arising solely from initial entanglement between the system and its reservoir, where entangled configurations led to faster cooling in hotter initial states compared to separable ones, even without strong non-normality. This work underscored the influence of quantum correlations in open systems, with extensions to entangled baths showing enhanced effect magnitudes under coherent interactions. Theoretically, a January 2025 framework based on thermomajorization unified classical and quantum Mpemba effects by quantifying non-equilibrium states via orders, predicting the phenomenon's occurrence for any monotone distance measure within bounded times and providing a resource-theoretic perspective on restorations. These insights suggest practical implications for , where exploiting the effect could enable faster state preparation protocols by initializing systems in "hotter" non-equilibrium configurations to minimize relaxation times. In September 2025, researchers reported the quantum Mpemba effect in systems lacking global symmetries, demonstrating its occurrence through local dissipation mechanisms in spin lattice models. Additionally, late August 2025 work explored the effect in parity-time symmetric systems coupled to bosonic baths, showing enhanced relaxation speeds near exceptional points.

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