Mpemba Effect in Nonequilibrium Systems

• Mpemba Effect is an anomalous cooling phenomenon where a state initially farther from equilibrium reaches equilibrium faster than one closer due to non-monotonic overlaps with slow relaxation modes.
• Spectral analyses reveal that suppression or complete absence of the slowest mode—measured by the coefficient a₂—drives faster cooling, highlighting the role of system asymmetries and boundaries.
• Observations across overdamped Langevin systems, granular gases, and quantum as well as photonic lattices underscore its broad relevance and potential for engineered relaxation processes.

Searching arXiv for recent and foundational Mpemba-effect papers to ground the article. The Mpemba effect is the anomalous relaxation phenomenon in which a state initially farther from equilibrium can approach the final stationary or equilibrium state faster than a state initially closer to it. In its thermal form, this is the statement that a hotter preparation may cool faster than a warmer one when both are quenched to the same bath; in modern nonequilibrium statistical physics, the effect is formulated more generally through crossings of relaxation trajectories, suppression of the slowest relaxation mode, or measure-independent orderings such as thermomajorization (Klich et al., 2017, Vu et al., 2024, Ares et al., 12 Feb 2025). The effect has been analyzed in Markovian master equations, overdamped Langevin systems, granular gases, spin glasses, inertial suspensions, photonic lattices, and both open and isolated quantum systems, and the literature now distinguishes several inequivalent but related notions: weak, strong, inverse, metastable, and operational versions of Mpemba behavior (Walker et al., 2022, Chétrite et al., 2021, Jin et al., 8 Jun 2026).

1. Definitions and classifications

A standard Markovian formulation expands relaxation around the bath equilibrium in eigenmodes of the generator. For thermal initial conditions, the relevant long-time coefficient is the overlap with the slowest nontrivial mode, commonly denoted $a_2(T)$. The ordinary long-time Mpemba effect occurs when this overlap is non-monotonic in the initial temperature, so that for some $T_h>T_c>T_b$,

$|a_2(T_h)|<|a_2(T_c)|.$

A stronger version occurs when the slowest mode is completely absent,

$a_2(T_M)=0,$

which yields exponentially faster relaxation governed by the next mode rather than the slowest one (Klich et al., 2017, Walker et al., 2022).

The literature uses several related criteria.

Formulation	Criterion	Interpretation
Weak Mpemba effect	$|a_2(T_h)|<|a_2(T_c)|$	Hotter state overtakes colder one at long times
Strong Mpemba effect	$a_2(T_M)=0$	Exponentially faster equilibration
Inverse Mpemba effect	Heating analogue	Colder state heats faster than a warmer one
Metastable Mpemba effect	Non-monotonic $W_{\rm ex}(\beta,\beta_b)$	Faster stage-2 relaxation after local equilibration
Thermomajorization Mpemba effect	$p_t^h \prec_\pi p_t^w$	Hot trajectory is closer for all monotone measures

The strong effect motivated the introduction of the Mpemba index, defined as the number of special initial temperatures for which the slowest mode coefficient vanishes. Its parity is a topological property of the system, and the same work tied the strong effect to thermal overshoot and showed that it can survive the thermodynamic limit (Klich et al., 2017). Later work generalized the notion further: Tan Van Vu and Hisao Hayakawa defined a thermomajorization Mpemba effect that is independent of an arbitrary distance choice and equivalent to crossover for all monotone measures within finite time (Vu et al., 2024), while the operational resource-theoretic formulation identifies a resource Mpemba effect with the breakdown of resource-Markovianity under relaxation (Jin et al., 8 Jun 2026).

2. Spectral mechanisms and mode selection

Much of the modern theory reduces the effect to spectral structure. In overdamped Langevin dynamics or finite-state Markov chains, one writes

$$p(x,t)=\sum_{n\ge 1} e^{-\lambda_n t} a_n\, r_n(x),$$

with $\lambda_1=0$ and the first nonzero mode controlling the asymptotic approach. At long times,

$T_h>T_c>T_b$0

so the question becomes whether the initial preparation suppresses the slow mode more effectively when it starts farther from equilibrium (Liu et al., 2 Jun 2026).

This spectral picture underlies several distinct mechanisms. In the 2025 proposal “Mpemba as an Emergent Effect of System Relaxation,” the central object is not only the slow mode but also the fast one: a far-from-equilibrium state can relax faster if it overlaps strongly with a fast relaxation direction, while a nearer state projects more heavily onto slower directions. In the shared-environment spin model of that work, collective coupling produces an enhanced decay channel with Lyapunov exponent

$T_h>T_c>T_b$1

so the effect strengthens with system size $T_h>T_c>T_b$2. The same paper also shows that anisotropy alone, $T_h>T_c>T_b$3, can generate Mpemba behavior even without a shared bath (Das, 10 Dec 2025). This suggests that the effect is not tied to a specific microscopic substrate but to directional structure in the relaxation generator.

The 2026 analysis of Markovian $T_h>T_c>T_b$4-level systems sharpens that observation. Multiple time scales are necessary but not sufficient: the transition network must be sufficiently asymmetric that hotter and colder initial Gibbs states project differently onto slow and fast modes. That work derives explicit necessary conditions on the rates, extends the logic to $T_h>T_c>T_b$5-level systems via triplet analysis, and argues that sub-Ohmic and Ohmic spectra can be discarded because the requisite asymmetry in rates is absent there (Avitan et al., 4 Mar 2026). A plausible implication is that “Mpemba-capable” Markovian systems occupy a structurally restricted part of rate-matrix space rather than forming a generic class.

The same spectral language reappears in optics, granular models, and quantum systems. In each case the phenomenon is traced to the geometry of initial conditions relative to nontrivial eigendirections of the relaxation operator rather than to temperature alone (Longhi, 2024, Biswas et al., 2020, Ares et al., 12 Feb 2025).

3. Energy landscapes, metastability, and the role of boundaries

Early intuition often linked the Mpemba effect to rugged landscapes and metastable trapping. The current literature qualifies that intuition substantially. In double-well systems, relaxation commonly separates into two stages: fast intra-well equilibration to a local-equilibrium manifold, followed by slow inter-well population exchange. The metastable Mpemba effect in that setting corresponds to a non-monotonic dependence of the extractable work

$T_h>T_c>T_b$6

on the initial temperature. The initially hotter system can enter stage 2 closer to global equilibrium than an initially cooler one, even though it started farther away at $T_h>T_c>T_b$7 (Chétrite et al., 2021). This interpretation was developed explicitly for the double-well setting connected to the colloidal experiment of Kumar and Bechhoefer referenced in that work.

A complementary small-diffusion analysis rewrites the effect in mean-first-passage-time terms. In a high-barrier double well, the strong Mpemba effect occurs at leading order when the initial probability of occupying a well matches the bath-equilibrium probability of that well. The slow inter-well mode is then absent, so the remaining relaxation is exponentially faster (Walker et al., 2022). This formulation makes the role of barrier-crossing kinetics explicit without abandoning the eigenmode criterion.

At the same time, several exact results show that metastability is neither necessary nor sufficient. In the exact Langevin analysis of a piecewise-linear double well, neither metastability nor asymmetry in the potential is a necessary or sufficient condition for observing the effect (Biswas et al., 2023). More strongly, a Brownian particle in a piecewise-linear single-well potential devoid of any metastable minima still shows direct, inverse, and strong Mpemba effects over a wide parameter range (Biswas et al., 2023). These papers shift the emphasis from “barrier trapping” to the temperature dependence of spectral overlaps.

The 2026 classification of one-dimensional overdamped Langevin systems goes further and identifies boundaries—hard or soft—as the primary structural ingredient. In that analysis, the derivative of the first nontrivial left eigenmode behaves as a Dirac delta peak at low bath temperature,

$T_h>T_c>T_b$8

so the overlap coefficient $T_h>T_c>T_b$9 depends mainly on how cumulative equilibrium population shifts across $|a_2(T_h)|<|a_2(T_c)|.$0. The presence and placement of a wall can force that population shift to become non-monotonic with initial temperature, whereas without boundaries the overlap is monotonic and the effect disappears. The same framework unifies single- and double-well cases and permits engineering of multistage Mpemba transitions by suitable landscape design (Liu et al., 2 Jun 2026). This substantially revises the older claim that metastability or multiple minima are essential.

4. Classical many-body realizations and phase-transition settings

Spin glasses provide a distinct mechanism centered on persistent memory. Using the Janus II supercomputer, the three-dimensional Edwards-Anderson Ising spin glass was studied over eleven orders of magnitude in time. There the key nonequilibrium variable is the coherence length $|a_2(T_h)|<|a_2(T_c)|.$1, whose growth below the glass transition is extremely slow and obeys

$|a_2(T_h)|<|a_2(T_c)|.$2

The effect occurs when the bath temperature lies in the glassy phase, and the decisive relation is

$|a_2(T_h)|<|a_2(T_c)|.$3

rather than initial energy alone. Larger initial $|a_2(T_h)|<|a_2(T_c)|.$4 implies faster subsequent relaxation, so the Mpemba effect is interpreted as a persistent memory effect governed by hidden structural aging (Collaboration et al., 2018).

Granular systems exhibit several further variants. In the exactly solvable driven granular Maxwell gas, a mono-dispersed system shows the effect only for non-stationary initial conditions, whereas a bi-dispersed gas can display it even for steady-state initial conditions. The bi-dispersed case also supports a strong Mpemba effect, in which the slow mode coefficient vanishes and equilibration proceeds at the faster exponential rate (Biswas et al., 2020). In a related study of driven granular gases, however, the very presence of the effect was shown to depend on the chosen distance measure: total kinetic energy, Euclidean distance in energy space, Manhattan distance, and KL divergence lead to different phase diagrams and even different assignments of which state is “hotter” or “closer” (Biswas et al., 2023).

In inertial suspensions under shear, the effect is organized by the interplay of viscous heating and thermal relaxation. Two classes were identified: a generic normal Mpemba effect, where the initially hotter quasi-equilibrium state cools faster because it lacks initial viscous heating, and an anomalous one, where a nonequilibrium sheared state can eventually overtake a colder quasi-equilibrium state despite initially slower cooling. The same framework also yields inverse and mixed heating/cooling processes (Takada et al., 2020).

Phase-transition-based formulations are more delicate because the relevant observable can be the transition time rather than a distance to equilibrium. In Landau theory for second-order transitions, the transition time is defined dynamically by the loss of stability of the order parameter,

$|a_2(T_h)|<|a_2(T_c)|.$5

A one-dimensional order-parameter theory does not show a Mpemba effect in this sense, because the post-quench instability occurs at $|a_2(T_h)|<|a_2(T_c)|.$6 for all initial temperatures above $|a_2(T_h)|<|a_2(T_c)|.$7. A two-dimensional Landau theory with a hidden variable $|a_2(T_h)|<|a_2(T_c)|.$8 can show it, because the hotter initial state may start closer in $|a_2(T_h)|<|a_2(T_c)|.$9 to the instability manifold and therefore reach the phase transition sooner (Holtzman et al., 2022).

The water-freezing problem occupies a separate category. In carefully controlled experiments on pure water, the effect was observed only when the freezer temperature was very close to the ice nucleation temperature. In that regime the spread in freezing times was so broad that it exceeded the deterministic cooling delay of the hotter sample, so hot water sometimes froze first. The proposed resolution is explicitly stochastic: the paradox is rooted in the random timing of ice nucleation, a first-order phase-transition event, rather than in a reversal of ordinary cooling laws (Klimov et al., 7 Aug 2025).

5. Quantum, photonic, and optical forms

Quantum work distinguishes open-system and isolated-system versions of the effect. The review “The quantum Mpemba effects” organizes open quantum examples in terms of distances between density matrices—trace distance, Frobenius distance, and quantum relative entropy—and isolated-system examples in terms of symmetry restoration, often measured through entanglement asymmetry,

$a_2(T_M)=0,$0

The review emphasizes the roles of quantum fluctuations, integrability, and symmetry, and treats open and isolated dynamics as distinct manifestations rather than a single mechanism (Ares et al., 12 Feb 2025).

In isolated chaotic dynamics, charge-preserving random circuits provide a particularly clean realization. For tilted ferromagnetic initial states, the larger the initial asymmetry, the faster the system restores symmetry and approaches the grand-canonical ensemble; tilted antiferromagnetic states do not show the effect. The proposed mechanism is operator spreading: nonconserved operators emit conserved densities, and the corresponding asymmetry decays at a rate controlled by diffusive spreading. The Mpemba time scales as

$a_2(T_M)=0,$1

with $a_2(T_M)=0,$2 the subsystem size and $a_2(T_M)=0,$3 the diffusion constant (Turkeshi et al., 2024).

Open-system optics now provides an explicitly photonic realization. Stefano Longhi showed that in finite photonic lattices under incoherent dephasing dynamics, highly localized initial light distributions can diffuse faster than broader initial distributions. The proposed reason is again spectral: a localized state can have no overlap with the slowest decaying Liouvillian mode and thus bypass the relaxation bottleneck. In the strong-dephasing regime, the dynamics reduce to a classical random walk with hopping rate

$a_2(T_M)=0,$4

Fiber-based temporal mesh lattices, with random phase modulation and time-resolved intensity readout, were proposed as an experimentally accessible platform (Longhi, 2024).

The 2025 shared-environment model extends this logic to open quantum many-body systems. There the effect emerges from collective relaxation through a common environment, which produces a fast mode unavailable for independently relaxing particles. The same study also shows that anisotropic relaxation, $a_2(T_M)=0,$5, suffices even without collective coupling, so the relevant structure is again the existence of privileged fast directions in state space (Das, 10 Dec 2025).

6. Ambiguities, misconceptions, and unifying perspectives

Several common claims about the Mpemba effect now require qualification. First, the effect is not reducible to rugged landscapes or metastable traps. Exact single-well and double-well analyses, together with the 2026 boundary classification, show that metastability is neither necessary nor sufficient, and that boundary-induced population shifts can be more fundamental than the number of minima (Biswas et al., 2023, Biswas et al., 2023, Liu et al., 2 Jun 2026).

Second, the effect is not always independent of the notion of “distance from equilibrium.” In driven granular gases, different distance measures lead to different crossover regions and, in some cases, different judgments about which state is farther from the steady state (Biswas et al., 2023). This ambiguity motivates thermomajorization-based and resource-theoretic formulations. The thermomajorization Mpemba effect demands that the hot trajectory thermomajorize the warm one, $a_2(T_M)=0,$6, which is equivalent to saying that after some finite time the hot trajectory is closer to equilibrium for all monotone measures, not just for a chosen metric (Vu et al., 2024). The operational resource formulation generalizes this further: the resource Mpemba effect is the breakdown of resource-Markovianity under a relaxation operation, and its magnitude can be quantified through resource backflow measures (Jin et al., 8 Jun 2026).

Third, the effect is not generic in simple Markovian thermodynamics. The explicit necessary conditions derived for three-level systems show that multiple time scales alone do not suffice; rather, particular asymmetries in the transition rates are required. The same analysis uses bath spectra to explain why the effect remains anomalous: sub-Ohmic and Ohmic spectra can be discarded, while only sufficiently super-Ohmic structures can satisfy the inequalities (Avitan et al., 4 Mar 2026).

At the same time, the effect is broader than thermal physics narrowly construed. The mechanical analogue proposed in 2025 gives a purely Newtonian example: a particle descending a hill with a steep upper segment and a gentle lower segment can reach the bottom sooner when released from a greater height, because inertia acquired on the steep section shortens the total descent time on the lower section (Finkelstein, 28 Mar 2025). This does not reproduce thermodynamic relaxation, but it does illustrate the generalized statement that a state initially farther from equilibrium may arrive sooner.

Across these disparate settings, a stable picture has emerged. The Mpemba effect is best understood as anomalous ordering of relaxation trajectories generated by nontrivial mode structure, hidden variables, phase-transition timing, or operational orderings. What differs from platform to platform is not the existence of such order reversals, but the precise object whose relaxation is being compared: occupation probabilities, extractable work, coherence length, kinetic temperature, entanglement asymmetry, thermomajorization order, or a resource monotone.