Inverse Mpemba Effect: Anomalous Heating

• The inverse Mpemba effect is an anomalous heating phenomenon where a colder system reaches equilibrium faster than a warmer one under identical heat protocols.
• It relies on a nonmonotonic dependence of relaxation modes, as revealed by spectral decomposition techniques and the analysis of slow-mode amplitudes.
• The effect spans classical, quantum, and nonequilibrium systems, offering practical insights for designing engineered thermal protocols.

The inverse Mpemba effect is an anomalous heating phenomenon in which, under a common hot environment or target steady state, an initially colder preparation approaches the final state faster than an initially warmer one, despite starting farther from it. In modern treatments, the effect is defined through a monotone distance to the final equilibrium or nonequilibrium steady state, or through an equivalent relaxation-time criterion, and it is understood as a property of nonequilibrium relaxation spectra rather than a violation of thermodynamic monotonicity (Lu et al., 2016, Teza et al., 3 Feb 2025).

1. Definition and operational criteria

In the equilibrium-heating setting introduced by Lu and Raz, two identical copies of a system are prepared at temperatures $T_h$ and $T_c$ with $T_b > T_h > T_c$, then both are coupled to the same hot bath at temperature $T_b$. The inverse Mpemba effect occurs if there exists a time $t_m$ such that the initially colder system becomes closer to the bath equilibrium than the initially warmer one and remains so thereafter. Using a distance-to-equilibrium functional $D[p(t);T_b]$, the operational criterion is

$$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$

while initially

$D[p^c(0);T_b] > D[p^h(0);T_b].$

An equivalent relaxation-time definition uses

$$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$

with inverse Mpemba behavior when $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$ over an appropriate $T_c$0 range (Lu et al., 2016).

The same logic extends beyond equilibrium baths. The 2025 review formulates inverse Mpemba, also called anti-Mpemba, as a final-relaxation speedup toward either equilibrium or a NESS: for $T_c$1, the colder initial condition $T_c$2 exhibits inverse Mpemba behavior relative to $T_c$3 if there exists $T_c$4 such that for all later times

$T_c$5

Within this taxonomy, a weak inverse Mpemba effect means that the colder preparation has a smaller but nonzero overlap with the slowest decaying mode, whereas a strong inverse Mpemba effect means that the slowest mode is completely suppressed for one initial condition (Teza et al., 3 Feb 2025).

Different communities implement this definition with different observables. In finite-state Markov models and many classical stochastic settings, the preferred objects are the entropic distance, the Kullback–Leibler divergence, or the $T_c$6 distance (Lu et al., 2016). In the trapped-ion qubit experiment, the relevant quantity is the Euclidean distance in Bloch space to the final NESS, $T_c$7, which for qubits equals twice the trace distance (Shapira et al., 2024). In the prethermal Floquet setting, the defining observable is the energy-based relaxation time toward a prethermal plateau, supplemented by crossings in the normalized inverse participation ratio and entanglement entropy (Sugimoto et al., 7 Jul 2025). In driven granular gases, inertial suspensions, and related kinetic theories, the criterion is often a finite crossing time of total-energy trajectories under a common heating protocol (Biswas et al., 2021, Takada et al., 2020).

2. Markovian spectral framework

The foundational generic mechanism was formulated for finite-state, ergodic, continuous-time Markov processes satisfying detailed balance with respect to the bath temperature $T_c$8. The probability vector obeys

$T_c$9

with transition rates

$T_b > T_h > T_c$0

and stationary Boltzmann distribution

$T_b > T_h > T_c$1

Detailed balance guarantees real eigenvalues and diagonalizability (Lu et al., 2016).

A central requirement is the choice of a valid distance functional $T_b > T_h > T_c$2. The paper demands three properties: monotonic non-increase along relaxation, monotonic ordering along the quasi-static locus $T_b > T_h > T_c$3, and continuity plus convexity in $T_b > T_h > T_c$4. The entropic distance used explicitly is

$T_b > T_h > T_c$5

with $T_b > T_h > T_c$6 and $T_b > T_h > T_c$7. The Kullback–Leibler divergence and the $T_b > T_h > T_c$8 distance also satisfy the same criteria in this framework (Lu et al., 2016).

The spectral decomposition is the key to the effect. Writing the eigenvalues as

$T_b > T_h > T_c$9

with $T_b$0, one expands

$T_b$1

If $T_b$2, the long-time behavior is controlled by the projection onto the slowest nonzero mode, $T_b$3. For heating, a sufficient condition for inverse Mpemba is

$T_b$4

Then the initially warmer state lags along the slow direction, and the colder state eventually becomes and remains closer to equilibrium (Lu et al., 2016).

The same structure appears in the broader review literature. For a Markov generator $T_b$5 with spectral decomposition

$T_b$6

asymptotic relaxation is governed by the slowest amplitude $T_b$7. Hence inverse Mpemba is equivalent, at long times, to a nonmonotonic dependence of $T_b$8 on the initial temperature below the bath temperature, and the strong inverse effect corresponds to a zero crossing $T_b$9 (Teza et al., 3 Feb 2025).

3. Mechanisms in energy landscapes and exactly solvable models

The physical interpretation of the spectral criterion is formulated in terms of rugged energy landscapes with metastable basins. In heating to a high-temperature bath, fast intra-basin equilibration can coexist with slow inter-basin transport over barriers. The slow eigenvector $t_m$0 encodes this bottleneck. If the colder initial Boltzmann state $t_m$1 is better aligned with the bath’s coarse-grained equilibrium profile than $t_m$2, then it has a smaller projection onto the slow pathway and heats faster even though it starts farther from the final equilibrium (Lu et al., 2016).

The minimal three-state example in the original theory makes this mechanism explicit. With energies

$t_m$3

barriers

$t_m$4

and temperatures

$t_m$5

the colder initial state has a very small amplitude along the slow mode while the warmer one has a substantially larger one. The entropic distances cross, and the colder system’s $t_m$6 remains lower thereafter, constituting an inverse Mpemba effect (Lu et al., 2016).

Continuous-state analogues obey the same logic but can be more delicate. In the exactly solvable two-dimensional radially symmetric bistable Langevin model, relaxation is analyzed via a Schrödinger-type mapping of the Fokker–Planck operator and a two-mode expansion of the Kullback–Leibler divergence. For heating, the necessary condition is that the slowest-mode amplitude $t_m$7 have an extremum at some $t_m$8, where $t_m$9 is the final inverse bath temperature. A sufficient crossing condition in the two-mode approximation is

$D[p(t);T_b]$0

The paper shows that nonmonotonicity of $D[p(t);T_b]$1 is necessary but not sufficient: in an equal-depth case with $D[p(t);T_b]$2, $D[p(t);T_b]$3 peaks at $D[p(t);T_b]$4, yet no inverse crossing occurs because the second-mode inequality is not satisfied (Hayakawa et al., 25 Mar 2026).

This restriction clarifies a recurrent misconception. A nonmonotonic dependence of the slowest-mode projection on initial temperature is the enabling ingredient, but not every extremum produces a measurable crossing. The hierarchy of subleading modes and spectral gaps can suppress the overtaking window even when the slow projection alone appears favorable (Hayakawa et al., 25 Mar 2026).

A complementary negative result appears in inverse-engineering studies of simple relaxation models. In the baseline Newtonian law with constant $D[p(t);T_b]$5, in a discrete two-level system with monotone bath-controlled rates, and in an overdamped Brownian harmonic oscillator with a single exponential mode, the ordering remains normal; anomalous heating requires either initial-state-dependent effective rates, multimode relaxation, nonlinearity, delay, or memory (Löwen, 13 Apr 2026).

4. Classical nonequilibrium realizations

Inverse Mpemba behavior has been established in several nonequilibrium classical systems whose target state is a NESS rather than a Gibbs state.

System	Operational target	Enabling mechanism
Anisotropically driven inelastic Maxwell gas (Biswas et al., 2021)	Higher-energy NESS	Coupling of total energy $D[p(t);T_b]$6 to anisotropy mode $D[p(t);T_b]$7 through $D[p(t);T_b]$8; exact crossing criterion and strong inverse condition
Sheared inertial suspensions (Takada et al., 2020)	Common heated steady state	Competition of viscous heating, drag, and collisional cooling; normal and anomalous inverse Mpemba plus mixed processes
Binary molecular suspensions (González et al., 2020)	Bath temperature $D[p(t);T_b]$9	Species-dependent drag and nonlinear coupling of mixture temperature to partial temperatures
Radiative VO$$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$0-SiC nanostructures (Herz, 21 Jun 2026)	Radiative heating to hot substrate	Hysteresis, latent-heat-enhanced effective heat capacity, and near-field coupling

In anisotropically driven inelastic Maxwell gases, the exact reduced dynamics are written for the total and difference energies,

$$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$1

with a linear relaxation matrix whose off-diagonal element

$$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$2

couples anisotropy to total energy. The inverse effect under heating is governed by the same exact crossing condition as the direct effect,

$$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$3

and the strong inverse effect arises when the slow-mode overlap $$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$4 vanishes (Biswas et al., 2021).

In sheared inertial suspensions, the heating counterpart appears in both normal and anomalous forms. The temperature balance contains viscous heating, drag to the bath, and collisional dissipation,

$$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$5

and the different initial sheared or unsheared preparations alter the initial stress and hence the early heating rate. The paper reports both normal inverse Mpemba and anomalous inverse Mpemba, as well as mixed processes in which one trajectory both heats and cools during relaxation (Takada et al., 2020).

Binary molecular suspensions provide another route. There the mixture temperature

$$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$6

is nonlinearly coupled to the partial temperatures through species-dependent drag coefficients $$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$7. The necessary condition for crossover depends on the relative ordering of initial $$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$8 and $$D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m,$$9, and all Mpemba variants disappear when $D[p^c(0);T_b] > D[p^h(0);T_b].$0, because the mixture temperature decouples from the hidden variable $D[p^c(0);T_b] > D[p^h(0);T_b].$1 (González et al., 2020).

A distinct macroscopic realization is radiative. In the hysteretic VO$D[p^c(0);T_b] > D[p^h(0);T_b].$2 nanoparticle near a SiC substrate, the temperature obeys

$D[p^c(0);T_b] > D[p^h(0);T_b].$3

and the onset condition derived in the paper is

$D[p^c(0);T_b] > D[p^h(0);T_b].$4

Here the latent-heat peak in $D[p^c(0);T_b] > D[p^h(0);T_b].$5 acts as a thermal buffer, and the same framework yields both ordinary and inverse radiative Mpemba effects; a passive version also appears when the hysteretic memory is stored externally in the substrate reflection rather than the relaxing particle itself (Herz, 21 Jun 2026).

Polymer crystallization offers a more limited connection. In polybutene-1, the study argues that both Mpemba and inverse Mpemba effects can be framed within nonequilibrium thermodynamics of configurational entropy and relaxation, but it reports only the standard Mpemba effect experimentally and explicitly states that no inverse Mpemba experiment was demonstrated in that system (Liu et al., 2022).

5. Quantum, prethermal, and strong inverse effects

The inverse Mpemba effect is not confined to classical thermalization. In driven open quantum systems, the relevant object is a Liouvillian spectrum around a final NESS. The single trapped-ion qubit realization uses the GKSL master equation

$D[p^c(0);T_b] > D[p^h(0);T_b].$6

with $D[p^c(0);T_b] > D[p^h(0);T_b].$7 and $D[p^c(0);T_b] > D[p^h(0);T_b].$8. In Bloch coordinates, the relaxation around the final NESS is

$D[p^c(0);T_b] > D[p^h(0);T_b].$9

and the inverse effect is governed by the nonmonotonicity of the slow-mode overlap $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$0 (Shapira et al., 2024).

This system also realizes the strong inverse Mpemba effect. The slowest mode can be canceled exactly at

$$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$1

which requires

$$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$2

Experimentally, the effect was observed robustly for $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$3. For $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$4, a cold state $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$5 and a hot one $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$6 crossed at approximately $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$7, and for $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$8 the crossing occurred at $$\tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\},$$9. The strong-effect point at $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$0 was consistent with a measured zero of the slow-mode amplitude (Shapira et al., 2024).

The quantum-mechanical aspect is specific. The strong inverse effect in this qubit requires sufficient coherence: in the extreme dephasing limit $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$1, the dynamics reduce to incoherent population relaxation, $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$2 becomes monotone, and neither inverse nor strong inverse behavior is possible (Shapira et al., 2024). The general quantum review places this within the same spectral paradigm as classical Markov processes, but with Liouvillian eigenoperators and NESS targets replacing detailed-balance eigenmodes (Teza et al., 3 Feb 2025).

A different extension is prethermal rather than dissipative. In the periodically driven isolated ladder of spinless fermions, the ultimate steady state is infinite temperature, but relaxation first approaches a long-lived prethermal plateau. The prethermal inverse Mpemba criterion is formulated by

$\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$3

with inverse behavior when

$\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$4

In the reported numerics, the coldest initial state $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$5 reaches a higher prethermal plateau faster than hotter initial states such as $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$6, while subsystem diagnostics like $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$7 and entanglement entropy show analogous crossings (Sugimoto et al., 7 Jul 2025).

Strong inverse effects also arise in many-spin models with equilibrium phase-structure constraints. In the antiferromagnetic Ising model with a reentrant phase transition, the pair-approximation analysis shows that when the final state lies in the paramagnetic phase, the slowest mode is purely staggered,

$\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$8

Hence any initial paramagnetic state has zero overlap with that slow mode, whereas an antiferromagnetic initial state excites it. Reentrance then makes it possible to realize strong direct and strong inverse Mpemba effects by choosing one initial state from a paramagnetic lobe and the other from the intermediate antiferromagnetic phase (Blom et al., 30 Apr 2026).

6. Protocol engineering, macroscopic formulations, and limitations

A separate strand of work treats inverse Mpemba not only as a diagnostic of spontaneous relaxation but also as a target of protocol design. In the inverse-engineering study of cooling and heating laws, the baseline Newtonian equation

$\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$9

does not generate anomalous ordering when $T_c$00 is constant. The paper realizes inverse Mpemba phenomenologically by making the relaxation rate depend on the initial state,

$T_c$01

with $T_c$02, so lower initial temperature implies larger rate and faster heating. The corresponding inverse-engineered bath protocol is

$T_c$03

The same paper analyzes memory, nonlinear cooling functions, and delay terms, and emphasizes that backward-engineered protocols can fail to exist or can be non-unique under physical constraints (Löwen, 13 Apr 2026).

Delay alone can also generate exact inverse-Mpemba windows. In the Descartes protocol based on a time-delayed Newton law,

$T_c$04

the heating problem is expressed in terms of the distance-to-hot variable

$T_c$05

For instantaneous quenches, the inverse effect exists if and only if

$T_c$06

equivalently

$T_c$07

where $T_c$08, $T_c$09 is the waiting time, and $T_c$10 is the quasi-exponential solution of the delay equation. At fixed delay $T_c$11, the maximal inverse Mpemba magnitude occurs at $T_c$12 (Santos, 3 Feb 2026).

At a more macroscopic level, the cumulative-heat formulation derives a generalized Newton law directly from linear irreversible thermodynamics:

$T_c$13

with $T_c$14 and

$T_c$15

In this framework, heating corresponds to $T_c$16, and $T_c$17 induces the inverse Mpemba effect because the effective rate

$T_c$18

increases more strongly for the colder trajectory, which absorbs more heat sooner. The coefficient $T_c$19 shifts the asymptotic target and predicts incomplete thermalization when $T_c$20 is finite (Lin et al., 20 Mar 2026).

Despite this breadth, the literature also delineates clear limitations. The original sufficient conditions assume finite-state Markovianity, ergodicity, and detailed balance, and departures from these assumptions can alter relaxation pathways and obscure spectral ordering (Lu et al., 2016). The review literature emphasizes that the choice of distance functional matters operationally, even though the slowest-mode amplitude controls the asymptotic speedup across standard monotone metrics (Teza et al., 3 Feb 2025). Several works also stress that inverse heating is often weaker than direct cooling because very hot targets can suppress metastability and reduce spectral separation, making crossings harder to resolve experimentally (Shapira et al., 2024, Teza et al., 3 Feb 2025).

Across these formulations, a unifying conclusion recurs. The inverse Mpemba effect is not a statement about anomalous equilibrium thermodynamics; it is a statement about nonequilibrium state-space geometry. Whether in a detailed-balance master equation, a Liouvillian NESS, a Floquet prethermal plateau, a hysteretic radiative nanostructure, or a macroscopic cumulative-heat model, anomalous heating appears when the colder initial preparation is less burdened by the slowest relaxation channel than a warmer one.