The Mpemba effect likes to hit a wall

Abstract: The historical Mpemba effect involves a first-order phase transition. This has prompted the experimental realization of microscopic proxies in the form of a colloidal particle trapped in an asymmetric double well, for which the Mpemba effect has indeed been observed. We establish that the existence of the one-dimensional Mpemba effect for a polynomial potential is driven solely by the presence of a hard enough boundary, irrespective of the potential's double-well shape. We then show that the physics of the underlying Mpemba effect is governed not only by single-well physics but also by the high-temperature initial regime.

• The paper shows that the Mpemba effect emerges only when a sufficiently hard boundary is present, enforcing a crossover of probability between potential wells.
• It employs an analytical treatment using the second eigenmode of the Fokker-Planck operator to reveal a scaling law for the critical inverse temperature.
• Numerical analysis with quartic, sextic, and piecewise potentials confirms that boundary manipulation, rather than intrinsic asymmetry, predominantly affects the relaxation dynamics.

Boundary-Driven Origin of the Mpemba Effect in One-Dimensional Potentials

Introduction

The paper "The Mpemba effect likes to hit a wall" (2604.01543) undertakes a systematic theoretical analysis of the Mpemba effect—the phenomenon wherein a system initially prepared at a higher temperature relaxes to equilibrium faster than a system begun at a lower temperature after both are quenched into the same environment. Contrary to the common interpretation that attributes the effect in double-well or metastable systems to free-energy landscapes or activation barriers, this work makes the bold claim that, in one-dimensional overdamped dynamics, the essential requirement for the Mpemba effect is the existence of a sufficiently hard boundary (wall) on the shallow side of the confining potential, with the effect disappearing in its absence. Thus, the effect is ultimately a boundary artifact rather than a direct reflection of "metastability" or inner potential structure.

Physical Framework and Analytical Methodology

The analysis is rooted in overdamped Langevin dynamics for a particle in a one-dimensional potential $V(x)$, possibly with hard walls at $x = -L_-$ (shallow side) and $x = L_+$ (deep side): $\frac{d x}{d t} = -V'(x) + \sqrt{2T} \eta,$ with detailed-balance Fokker-Planck evolution. Considering initial conditions sampled from Boltzmann distributions at temperature $T_i$ (distinct from the bath temperature $T$ for relaxation), the relaxation to equilibrium is characterized via the spectral decomposition of the evolution operator.

The theoretical focus is on the second eigenmode $a_2$ of the Fokker-Planck operator, which controls the leading non-trivial relaxation process. The Mpemba effect is defined by the non-monotonicity of $a_2(T_i)$ as a function of the initial temperature: there exists $T_\mathcal{M}$ where $a_2$ vanishes, marking a switch from faster to slower relaxation. Analytical treatment proceeds by deriving the $x = -L_-$0-dependence of $x = -L_-$1 and examining how boundary placement (i.e., the positions of $x = -L_-$2) influences the Mpemba signature.

Key Theoretical Results

The main result is the identification of a hard wall on the shallow side ($x = -L_-$3) as a necessary condition for observing the Mpemba effect in generic asymmetric double-well (or generally confining) polynomial potentials. Specifically:

• When the shallow-side wall at $x = -L_-$4 is sent to infinity, the effect disappears ($x = -L_-$5, or $x = -L_-$6 loses extremality).
• Walls (or strongly diverging branches) on the deep side, when sufficiently distant from the relevant well, do not induce the effect, while hard boundaries too close suppress it.
• The explicit scaling for the critical inverse temperature is

$x = -L_-$7

for large $x = -L_-$8 and a confining monomial tail $x = -L_-$9.

• The populations and internal energies in the left and right branches must cross as a function of $x = L_+$0 for the effect to occur, a condition enforced by the finite boundary on the shallow side.

This results in the somewhat counterintuitive and contradictory claim (relative to standard intuition) that the intricate structure of the double-well (i.e., barrier height, asymmetry, or metastability) is of minor relevance to the existence of the Mpemba effect in 1D; instead, the effect is generically observed when the probability in the potential is forced (by the wall) to shift from one well to another as $x = L_+$1 increases.

Numerical Analysis and Validation

The authors numerically solve the discretized Fokker-Planck evolution for quartic, sextic, and piecewise quadratic potentials, systematically varying $x = L_+$2 (the left wall location). The empirical findings:

• The scaling $x = L_+$3 holds robustly, even away from strictly asymptotic regimes.
• For soft boundaries or walls at infinity, the critical temperature diverges and the effect vanishes.
• When the right wall is brought sufficiently close to the deep well, the effect is also suppressed, confirming the asymmetric boundary requirement.

A further demonstration shows that even non-analytic or piecewise potentials with asymmetrically diverging tails (without true hard walls) can display the effect, so long as their asymptotics enforce a probability crossover as initial temperature increases.

Physical and Conceptual Implications

The strong conclusion is that the experimentally observed Mpemba effect in 1D systems—e.g., colloidal particles in optical tweezers, as in [kumar2020exponentially]—is “wall-driven” and does not require the presence of a metastable state or high energy barrier per se. This has conceptual and practical implications:

• Theoretically, this work suggests that “quicker cooling of hotter systems” in experimental or numerical studies using potential engineering may arise from poorly controlled boundaries (finite support of optical traps, simulation boxes, or non-physical constraints), not generic relaxation dynamics.
• The findings call for a careful reevaluation of “Mpemba effects” in low-dimensional and single-particle systems and cast doubt on their use as direct analogues for macroscopic first-order transitions or glassy relaxation.
• Experimentally, the boundary shape at high energies (e.g., steepness, divergence rate, or finite support) should be directly probed and varied to confirm the wall-driven mechanism.
• The analysis offers a recipe for creating or destroying the Mpemba effect in model systems via boundary manipulation, which bears on the design of relaxation acceleration protocols.

Broader Context and Future Directions

This boundary-driven perspective contrasts with alternative approaches that attribute the Mpemba effect in many-body, quantum, or higher-dimensional systems to spectral properties, memory effects, or non-equilibrium initial conditions (see, e.g., [TEZA2026, Biswas23a, Holtzman22, Carollo21, Ohga2024]). The current manuscript’s analysis is restricted to high-temperature initial states, one-dimensional geometry, and detailed-balance (i.e., equilibrium) initializations.

Open avenues include:

• Extension to systems with symmetric double-well potentials, where the $x = L_+$4 mode may not suffice to capture relaxation anomalies.
• Investigation of the effect in higher dimensions, multi-particle/coupled systems, or in the quantum regime, where boundary influences are less direct or may couple to nontrivial system modes [takadahayakawa].
• Quantitative experimental tests probing the steepness or shape of optical or mechanical potentials at high energies, explicitly linking practical constraints (trap size, steepness) to anomalous relaxation signatures.

Conclusion

This work establishes that, in 1D overdamped relaxation for particles in double-well or generic polynomial potentials, the appearance of the Mpemba effect is fundamentally a boundary-driven phenomenon, requiring a hard or steep wall on the shallow side of the potential. The effect is destroyed in the absence of such a boundary, regardless of the inner landscape. This finding re-frames the discussion around observed Mpemba phenomena in experimental and theoretical studies on low-dimensional systems, highlighting the central role of boundaries and contesting explanations based on metastability or potential asymmetry alone. Future research should systematically probe wall-driven versus intrinsic relaxation anomalies in complex systems.