Mpemba Effect in an Expanding Lieb-Liniger Bose gas in a hard wall box

Abstract: The Mpemba effect, broadly understood as the counterintuitive phenomenon in which a system initially farther from equilibrium relaxes faster than a system closer to equilibrium, has been widely studied in classical stochastic systems and, more recently, in quantum settings. However, its manifestation is strongly dependent on the choice of observable and the dynamical constraints of the system. In this work, we investigate the emergence of a Mpemba-type effect in the density redistribution dynamics of a strongly interacting one-dimensional Bose gas in the Tonks-Girardeau regime undergoing a sudden box expansion from length L_0 to L. By defining a physically motivated distance function based on the difference of densities between spatial regions, we provide evidence that -the relaxation dynamics of the ground and excited symmetry sectors exhibit a clear crossing in time, indicating a reversal in relaxation ordering. We emphasize that the Mpemba effect is not a universal law but rather an observable-dependent phenomenon that arises under specific dynamical conditions. In particular, we show that the interplay between initial state structure, integrability, and spatial redistribution leads to distinct relaxation pathways that enable the effect. Our results clarify common misconceptions linking the Mpemba effect to Newton's law of cooling and highlight the conditions under which such anomalous relaxation behavior can emerge in integrable quantum systems.

• The paper demonstrates the Mpemba effect via density-based observables in an expanding Lieb-Liniger Bose gas, with excited states overtaking ground states after a crossover time.
• It employs ab initio path-integral Monte Carlo simulations to accurately track unitary dynamics and preserve many-body correlations in the Tonks-Girardeau regime.
• The findings highlight that anomalous relaxation is strongly observable-dependent, offering new insights into nonequilibrium phenomena in integrable quantum systems.

Mpemba Effect in the Lieb-Liniger Bose Gas: Observable-Dependent Anomalous Relaxation under Expansion Quenches

Introduction

The paper "Mpemba Effect in an Expanding Lieb-Liniger Bose gas in a hard wall box" (2604.05408) provides a systematic analysis of anomalous relaxation dynamics, specifically Mpemba-type effects, in the nonequilibrium expansion of a strongly interacting one-dimensional Bose gas. The work situates itself at the interface of quantum quench dynamics, integrable systems, and relaxation theory, analyzing conditions under which a quantum analog of the classical Mpemba effect manifests in the redistribution of spatial density following a sudden change of trap geometry.

System and Methodology

The system consists of $N$ bosons with repulsive contact interactions, governed by the Lieb-Liniger Hamiltonian, initially confined within a 1D hard-wall box of length $L_0$. At $t=0$, a quantum quench is implemented by instantaneously expanding the box to a larger length $L$, without altering the interaction strength. The model operates in the strongly interacting regime (Tonks-Girardeau limit), but retains full many-body correlations due to the use of an ab initio path-integral Monte Carlo (PIMC) method based on the Feynman-Kac representation.

Key features of the methodology include:

• Explicit sampling of many-body wavefunctions: No mapping to non-interacting systems is performed; strong interactions are fully retained.
• Evaluation of time-evolved densities: The evolution under the post-quench Hamiltonian is tracked for both ground and excited initial symmetry sectors.
• Observable selection: The dynamics is probed via a physically motivated density-based distance function, defined through regionally averaged densities in the initial and expanded spatial domains.

Distance Functions and the Mpemba Criterion

Unlike relaxation in classical or open quantum systems, unitary dynamics in integrable quantum systems requires careful definition of distance from equilibrium. The observable of focus, $D(t) = |\rho_{L_0}(t) - \rho_L(t)|$, quantifies the imbalance in particle density between the original and expanded regions. A further distance to stationarity $\mathcal{D}(t)$ is defined as the deviation of $D(t)$ from its saturated long-time value.

The principal signature of a Mpemba-type effect is a reversal in the ordering of relaxation among initial states. Specifically, starting from $$\mathcal{D}_{\mathrm{exc}}(0) > \mathcal{D}_{\mathrm{gnd}}(0)$$ (excited state further from stationarity), the anomalous behavior is evidenced by the existence of a time $t_c$ where $$\mathcal{D}_{\mathrm{exc}}(t_c) = \mathcal{D}_{\mathrm{gnd}}(t_c)$$ and $L_0$0 for $L_0$1; that is, the initially further state relaxes faster.

Numerical Evidence for Mpemba-Type Relaxation

The PIMC computations reveal:

• Clear crossing in relaxation trajectories: The difference function $L_0$2 changes sign at a well-defined time and remains negative, indicating persistent reversal of relaxation order.
• Robustness to numerical parameters: Variation of time step or sampling resolution does not qualitatively affect the observed crossing.
• Observable dependence: The effect is not present for arbitrary observables but is sharply resolved in the density-based regionally averaged quantity.

The physical origin is tied to the projection of initial symmetry sectors onto post-quench modes. Ground and excited states display distinct dephasing and mode-filling properties due to their differing initial spatial profiles and momentum distributions. Ground states, being more localized pre-quench, undergo rapid initial redistribution, whereas excited states, with broader profiles, exhibit more gradual but overtaking relaxation trajectories. The system approaches a stationary state associated with long-time dephased configurations rather than canonical thermal equilibrium, consistent with integrable dynamics.

Implications and Theoretical Context

This study provides concrete evidence that the Mpemba effect in isolated quantum many-body systems is highly non-universal and strictly observable-dependent. The results reinforce and expand upon previous findings in both classical [5–13] and quantum integrable and non-integrable systems [3, 16–20, 22–30]:

• No contradiction with thermodynamics: The effect does not violate relaxation principles—rather, it emerges from nontrivial mode structure and the interplay of initial condition projections.
• Role of integrability and initial state structure: The system's conserved quantities and the spatial properties of the initial state underpin the possibility of anomalous relaxation pathways.
• Specificity of the phenomenon: Absence of universal Mpemba behavior is emphasized; the effect may be absent for other observables such as local densities or global energies.

Prospects and Open Questions

The research opens several avenues for both theoretical and experimental exploration:

• Interaction dependence: Extending the analysis to varying interaction strengths, including the crossover from integrable to non-integrable regimes, would clarify the robustness of Mpemba-type dynamics.
• Observable and system size scaling: Mapping the dependency of the effect on observable choice and on extensive system parameters is critical for comprehensive classification.
• Extensions to higher dimensions and different geometries: Investigating similar quenches in higher-dimensional or lattice systems may reveal new aspects of anomalous relaxation.
• Experimental realizations: Ultracold atom platforms are poised for direct tests of such nonmonotonic relaxation in real time and with controllable initial states.

Conclusion

The study establishes the emergence of Mpemba-type anomalous relaxation in an expanding Lieb-Liniger Bose gas probed via density redistribution, using controlled many-body simulations without approximations. The effect arises from specific dynamical and observable circumstances, tightly linked to the structure of initial states and the integrable modal landscape of the post-quench system. The findings clearly delineate the observable-dependent and non-universal nature of Mpemba-type effects in closed quantum systems and set the stage for future theoretical and experimental investigations into the phenomenology of nonequilibrium relaxation.