Strong Mpemba Point in Relaxation Dynamics

• Strong Mpemba point is defined as the precise initial-state parameter where the overlap with the slowest relaxation mode cancels, leading to accelerated thermalization.
• It manifests in various systems including overdamped Langevin dynamics, granular gases, classical spin models, and open quantum systems governed by Lindblad dynamics.
• The phenomenon leverages spectral geometry and symmetry, offering practical routes to engineer multistage cooling transitions and control dominant relaxation timescales.

Searching arXiv for papers on strong Mpemba points and related quantum/classical formulations. arXiv search query: "strong Mpemba point" A strong Mpemba point is a special value of an initial-state control parameter—most commonly temperature, but also bias, mixing angle, or another preparation coordinate—at which the overlap of the initial deviation from stationarity with the slowest relaxation mode vanishes exactly. The asymptotic dynamics then bypasses the slowest channel and is governed by faster modes, so a state initially farther from equilibrium can relax exponentially faster than a closer state. This notion was formulated first in Markovian relaxation theory and has since been developed in overdamped Langevin systems, granular gases, classical spin models, and open quantum systems governed by Lindblad dynamics (Klich et al., 2017, Zhang et al., 2024, Wei et al., 20 May 2026).

1. Definition and formal criterion

The generic Mpemba effect refers to anomalous relaxation in which two initial states in a family $\rho(x)$ or $p(x)$, both evolving toward the same steady state, are ordered oppositely at late times compared with their initial distances from stationarity. In the classical and quantum formulations summarized in recent work, the effect is most naturally expressed through modal decompositions of the relaxation generator (Wei et al., 20 May 2026, Bagui et al., 2 Dec 2025).

For open quantum dynamics with Liouvillian superoperator $\mathcal{L}$, right modes $v_k$, left modes $l_k$, and steady state $\rho_{\rm ss}$, one writes

$$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$

The slowest nontrivial mode has eigenvalue(s) $\lambda_{\rm slow}$ with the smallest magnitude of $\operatorname{Re}\lambda_k$ among $k\ge2$. A weak Mpemba point is characterized by a small but nonzero slow-mode amplitude, so faster subleading modes dominate only for a transient. A strong Mpemba point $p(x)$0 is defined by exact cancellation,

$p(x)$1

so the asymptotic relaxation is governed by faster accessible modes (Wei et al., 20 May 2026).

The same structure appears in classical detailed-balance dynamics. If

$p(x)$2

then the strong condition is $p(x)$3, where $p(x)$4 is the slowest nonstationary decay rate in the conventional ordering used in classical Markov chains (Klich et al., 2017). In parameterized quantum families $p(x)$5, the same condition is written as $p(x)$6, with $p(x)$7 the left eigenmatrix of the slowest nonzero Liouvillian mode (Bagui et al., 2 Dec 2025).

The practical significance of the strong point is that the dominant late-time scale changes discontinuously. If generic initial states relax on $p(x)$8, then a strong point relaxes on the faster scale set by the next accessible mode, such as $p(x)$9 in open quantum systems or $\mathcal{L}$0 in classical notation (Zhang et al., 2024, Klich et al., 2017).

2. Spectral geometry, topology, and measure-independent formulations

In classical detailed-balance Markov processes, the strong Mpemba point is not merely an isolated kinematic curiosity but part of a broader spectral geometry. The function $\mathcal{L}$1 can have multiple zeros as the initial temperature is varied, and the number of such zeros defines the Mpemba index. The parity of this index is a topological property: it is fixed by endpoint signs and is stable under continuous deformations that do not alter the relevant spectral ordering (Klich et al., 2017).

This topological viewpoint is tied to the equilibrium locus in probability space. Along the curve of initial Gibbs states, the strong points are the crossings of that curve with the hyperplane orthogonal to the slow mode. In mean-field antiferromagnetic Ising dynamics, this structure is further connected to thermal overshoot: trajectories can approach the bath equilibrium from opposite sides, implying by continuity the existence of a trajectory aligned with the fast eigendirection, which is precisely the strong Mpemba point (Klich et al., 2017).

A different refinement concerns the dependence of the Mpemba effect on the chosen figure of merit. The thermomajorization formulation replaces any single distance by the order relation $\mathcal{L}$2 with respect to the target Gibbs state $\mathcal{L}$3. In that framework, the thermomajorization Mpemba effect is equivalent to the occurrence of finite-time crossovers for all monotone measures, and the strong point is defined spectrally by $\mathcal{L}$4 in the nondegenerate case or by vanishing projection onto the degenerate slow subspace in the degenerate case (Vu et al., 2024).

This measure-independent formulation sharpens a common ambiguity. The conventional inequality $\mathcal{L}$5 is necessary but not sufficient for thermomajorization Mpemba behavior at long times. By contrast, the strong condition removes the slow mode altogether and thereby survives changes of monotone more robustly (Vu et al., 2024).

3. Classical stochastic and nonequilibrium realizations

The strong Mpemba point has concrete realizations across several classical nonequilibrium models. In overdamped Langevin dynamics on a double-well potential, the small-diffusion analysis shows that the strong effect occurs when the equilibrium probability of occupying a well at the initial temperature matches that at the bath temperature. In the paper’s notation this is the leading-order condition $\mathcal{L}$6, which is the small-diffusion form of the spectral constraint $\mathcal{L}$7 (Walker et al., 2022).

For piecewise-linear double-well potentials, an exact Fokker–Planck spectral treatment gives the same general criterion in operator form: $\mathcal{L}$8 The paper does not report explicit nontrivial values of $\mathcal{L}$9, but it supplies the exact overlap integral needed to locate them numerically for any choice of slopes, minima positions, and boundaries (Biswas et al., 2023).

Recent work on one-dimensional overdamped Langevin dynamics identifies boundaries, rather than metastability alone, as the central structural feature behind Mpemba behavior. In the low-temperature regime, the derivative of the first nontrivial left eigenmode acts as a Dirac delta peak, and the strong point is determined by the cancellation of the associated cumulative-probability contribution. This produces explicit scaling of the Mpemba temperature with wall position and allows the engineering of multistage Mpemba transitions with multiple strong points (Liu et al., 2 Jun 2026).

Granular gases provide another explicit setting. In anisotropically driven hard-disk gases, linearization near the stationary state yields two coupled modes for $v_k$0, and the strong Mpemba manifold is the line

$v_k$1

written explicitly in the paper as

$v_k$2

Any stationary initial state on this line relaxes purely with the faster rate $v_k$3 (Biswas et al., 2021).

In driven bi-dispersed granular Maxwell gases, the strong manifold likewise becomes a line in the space of initial total and differential energies,

$v_k$4

with the slow-mode coefficient $v_k$5 vanishing exactly on that line. By contrast, the mono-dispersed model does not exhibit a strong Mpemba point among steady-state initial conditions, because its steady states reduce the dynamics effectively to a single scalar variable (Biswas et al., 2020).

A further classical realization appears in the antiferromagnetic Ising model with a reentrant phase transition. In the paramagnetic final phase, the slowest linearized mode is purely staggered,

$v_k$6

so the slow-mode amplitude is

$v_k$7

Hence all initial temperatures in the paramagnetic phase satisfy $v_k$8 and are strong Mpemba points when quenched to a paramagnetic $v_k$9 (Blom et al., 30 Apr 2026).

4. Open quantum systems and Liouvillian strong points

In Markovian open quantum systems, the strong Mpemba point is formulated directly in Liouville space. For Lindblad dynamics

$l_k$0

one expands

$l_k$1

If $l_k$2 for the slowest nonzero mode, the long-time relaxation is governed by $l_k$3 rather than $l_k$4 (Bagui et al., 2 Dec 2025, Zhang et al., 2024).

This formulation has been realized experimentally in a single trapped-ion qutrit. There, a pure initial state $l_k$5 was prepared such that

$l_k$6

and the relaxation proceeded with rate $l_k$7 rather than $l_k$8. The same experiment identified a Liouvillian exceptional point through coalescence of $l_k$9 and $\rho_{\rm ss}$0 and of the corresponding eigenmodes, and found that strong Mpemba behavior exists only on the real side of that exceptional point (Zhang et al., 2024).

The strong-point criterion also underlies recent observable-based detection schemes. If an observable $\rho_{\rm ss}$1 satisfies $\rho_{\rm ss}$2, then the slow-mode amplitude in the observable dynamics,

$\rho_{\rm ss}$3

vanishes if and only if the state is at a strong Mpemba point. This permits detection of strong quantum Mpemba behavior without full state tomography, provided the steady state and the initial preparations are known (Bagui et al., 2 Dec 2025).

In Davies maps and related thermalizing semigroups, the same spectral condition can be understood as the elimination of the slow coherence or population sector, depending on the model. Recent work also emphasizes decoherence-free subspaces as a mechanism for producing large separations between slow and fast decay channels, and in collective-decay models the fast rate can scale linearly with system size, yielding what that paper calls an extreme version of the effect (Saliba et al., 15 Dec 2025).

5. Mechanisms that create or protect strong Mpemba points

Several distinct mechanisms can generate strong Mpemba points by enforcing exact or near-exact orthogonality to the slowest mode.

A particularly explicit mechanism is symmetry-filtered mode accessibility. In the long-range XXZ chain with local dephasing, the operator

$\rho_{\rm ss}$4

is an exact Liouvillian eigenmode at the $\rho_{\rm ss}$5-symmetric point $\rho_{\rm ss}$6, with

$\rho_{\rm ss}$7

For $\rho_{\rm ss}$8, highly symmetric initial states—most prominently the $\rho_{\rm ss}$9-symmetric ground state—have zero overlap with spatially nonuniform slow modes and project exclusively onto the exact $$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$0 channel. In that setting the ground state at $$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$1 is a strong Mpemba point, with

$$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$2

while finite-temperature states remain contaminated by slower sectors (Wei et al., 20 May 2026).

Quantum criticality offers a different route. In dissipative XXZ and $$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$3–$$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$4 spin chains with bulk dephasing, the overlap with the slowest Liouvillian mode vanishes for extreme-temperature initial states at the critical points $$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$5, and along the isotropic $$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$6–$$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$7 line with $$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$8. The zero-temperature ground state at $$\delta\rho(t)\equiv \rho(t)-\rho_{\rm ss} =\sum_{k\ge2} c_k\, e^{\lambda_k t}\, v_k, \qquad c_k=\operatorname{Tr}(l_k\,\rho(0)).$$9 and the highest-energy state at $\lambda_{\rm slow}$0 act as strong Mpemba points because the slow-mode coefficient $\lambda_{\rm slow}$1 vanishes there, whereas finite-temperature states retain slow-mode weight (Wei et al., 26 Aug 2025).

Bath engineering can also induce strong points. In a qubit coupled to squeezed thermal reservoirs, the jump operators are Bogoliubov-rotated,

$\lambda_{\rm slow}$2

and the paper reports no quantum Mpemba effect at $\lambda_{\rm slow}$3 within its protocol, but finds weak and strong regimes for $\lambda_{\rm slow}$4, with large Mpemba parameter values and explicit suppression of the slowest mode for suitably tuned squeezing phase and drive parameters (Furtado et al., 2024).

A broader emergent viewpoint appears in work on collective environments and anisotropic relaxation. For a shared environment, the mean-field Bloch dynamics has a slow radial mode and a faster collective angular mode, and the strong condition reduces geometrically to $\lambda_{\rm slow}$5. With local anisotropic relaxation, it becomes $\lambda_{\rm slow}$6. In both cases the strong point is an initial state lying on the manifold orthogonal to the slow direction (Das, 10 Dec 2025).

In dissipative SYK models coupled to non-Markovian SYK baths, explicit strong points were not tabulated, but the paper states that the observed Mpemba crossings and multi-mode relaxation imply that a strong point $\lambda_{\rm slow}$7 can be located numerically by fitting the slowest amplitude and solving $\lambda_{\rm slow}$8. The same work reports that no Mpemba crossings and no strong Mpemba behavior were seen in the Lindblad SYK model within the parameter regime explored (Wang et al., 2024).

6. Detection, ambiguities, and open questions

The practical identification of a strong Mpemba point depends on both spectral access and the choice of diagnostic. In many-body quantum systems, full tomography is often prohibitive, which motivates the use of “good observables” satisfying $\lambda_{\rm slow}$9 so that disappearance of the slow exponential in $\operatorname{Re}\lambda_k$0 directly diagnoses $\operatorname{Re}\lambda_k$1 (Bagui et al., 2 Dec 2025).

At the same time, several works emphasize that overtaking in a chosen distance can be metric-dependent even when the spectral strong-point condition is not. In Davies-map analyses, the non-equilibrium free energy can fail to show overtaking at $\operatorname{Re}\lambda_k$2 even though the trace distance still does, whereas the spectral condition $\operatorname{Re}\lambda_k$3 is independent of the metric (Saliba et al., 15 Dec 2025). The thermomajorization approach was introduced precisely to remove this ambiguity by replacing a single monotone with an order relation that unifies all monotone measures (Vu et al., 2024).

Degeneracies and non-diagonalizability complicate the definition. In degenerate slow subspaces one must cancel the full slow-sector projection rather than a single scalar amplitude, and at Liouvillian exceptional points the correct description involves Jordan structure and polynomial prefactors rather than pure exponentials (Zhang et al., 2024, Bagui et al., 2 Dec 2025).

Several open problems recur across the literature. One is robustness away from idealized symmetry points, for example under anisotropy $\operatorname{Re}\lambda_k$4, altered couplings, inhomogeneous fields, or modified dissipators, all of which restore slow-mode overlap in the long-range XXZ mechanism (Wei et al., 20 May 2026). Another is the scaling of slowest decay rates and strong-point structure near quantum criticality, where current evidence is largely numerical (Wei et al., 26 Aug 2025). Non-Markovianity is a further frontier: in SYK systems it appears to enable anomalous crossings absent in Lindblad reductions, but a systematic strong-point theory remains incomplete (Wang et al., 2024).

Across these settings, the central content of the strong Mpemba point remains stable. It is the exact zero of the slowest-mode overlap along a controllable initial-state manifold. Everything else—the presence of symmetry, reentrance, criticality, walls, squeezed correlations, collective dissipation, or carefully chosen observables—serves either to create that zero, protect it, or make it experimentally visible (Klich et al., 2017, Wei et al., 20 May 2026, Bagui et al., 2 Dec 2025).