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

This paper demonstrates that a Mpemba-type effect, where a system initially farther from equilibrium relaxes faster than one closer to it, emerges in the density redistribution dynamics of a strongly interacting one-dimensional Bose gas undergoing sudden box expansion, highlighting that this phenomenon is an observable-dependent consequence of specific dynamical conditions rather than a universal law.

The Gist

The Big Idea: The "Mpemba Effect" in a Quantum World

You might have heard of the Mpemba effect from a strange fact about water: sometimes, hot water freezes faster than cold water. It sounds impossible, right? But it happens because the "hot" water has a different internal structure that allows it to cool down more efficiently under specific conditions.

This paper takes that weird idea and asks: "Does this happen in the quantum world?"

Specifically, the authors looked at a line of tiny, bouncing particles (bosons) trapped in a box. They wanted to see if a "hotter" (more excited) version of this gas could settle down faster than a "cooler" (calmer) version after the box suddenly got bigger.

The Setup: The Quantum Balloon

Imagine a crowded dance floor (the box) where everyone is holding hands tightly (strong interactions).

1. The Start: The dancers are in a small room ($L_0$). Some are dancing in a perfect, synchronized line (the Ground State). Others are dancing wildly and chaotically (the Excited State).
2. The Quench: Suddenly, the walls of the room are knocked down, and the dancers can run into a huge, empty warehouse ($L$).
3. The Question: As they spread out to fill the new space, who gets to the "calm, evenly spread out" state first?
   • Intuition says: The calm dancers (Ground State) should settle down faster because they are already organized.
   • The Mpemba Effect says: Maybe the chaotic dancers (Excited State) actually settle down faster because their wild energy helps them explore the new space more quickly.

The Experiment: Measuring the "Messiness"

To figure out who wins, the scientists needed a way to measure how "messy" the system was. They didn't just look at the whole room; they split the warehouse into two zones:

• Zone A: The original small room.
• Zone B: The new, empty part of the warehouse.

They measured the density difference.

• If the particles are all stuck in Zone A, the difference is huge (very messy).
• If the particles are evenly spread between Zone A and Zone B, the difference is zero (perfectly calm).

They tracked this "messiness score" over time for both the calm dancers and the wild dancers.

The Discovery: The Great Overtaking

Here is what they found, which is the core of the paper:

1. The Start: At the very beginning, the Wild Dancers (Excited State) were much messier than the Calm Dancers (Ground State). They were further away from the goal.
2. The Race: As time went on, both groups started spreading out.
3. The Twist: Suddenly, the Wild Dancers started catching up and passed the Calm Dancers. For a while, the Wild Dancers were actually closer to being perfectly calm than the Calm Dancers were.

This is the Mpemba Effect: The system that started "worse" (further from equilibrium) ended up relaxing faster than the one that started "better."

Why Did This Happen? (The Secret Sauce)

The paper explains that this isn't magic; it's about how they move.

• The Calm Dancers (Ground State): They are very organized. When the wall opens, they move slowly and carefully. They get stuck in a "local" rhythm. They move fast at first but then get bogged down in a slow, steady shuffle.
• The Wild Dancers (Excited State): They are chaotic. When the wall opens, their wild energy allows them to bounce around the new space very aggressively. They explore the new territory so quickly that they find the "evenly spread out" pattern faster than the careful dancers.

It's like two people trying to clean a messy room:

• Person A (Calm): Starts by picking up one sock, then one shoe. Very methodical.
• Person B (Wild): Starts by throwing everything into a pile and then sweeping the whole floor at once.
• Result: Even though Person B started with a bigger mess, their aggressive method got the room clean faster than Person A's slow method.

The Important Warning: It's Not a Universal Law

The authors are very careful to say: "Don't expect this to happen every time."

The Mpemba effect here only happened because:

1. They chose a specific way to measure "messiness" (looking at the density difference between two specific zones).
2. The particles were interacting strongly (holding hands).
3. The system was "integrable" (meaning it follows strict rules of motion without random friction).

If they had measured something else (like the total energy), the effect might not have shown up at all. It's like saying, "If you measure a race by who runs the fastest first 10 meters, the tortoise might win. But if you measure by who finishes the whole marathon, the hare wins." The result depends entirely on what you are measuring.

The Bottom Line

This paper proves that in the quantum world, starting "farther away" from a goal doesn't always mean you will arrive later. Sometimes, the chaotic, high-energy start provides a shortcut to stability that the calm, organized start misses.

It's a reminder that nature is full of counter-intuitive surprises, and how we choose to look at a problem (our "observable") changes the story we tell.

Technical Summary

1. Problem Statement and Motivation

The Mpemba effect is a counterintuitive phenomenon where a system initially farther from equilibrium relaxes faster than a system initially closer to equilibrium. While extensively studied in classical stochastic systems and recently in open quantum systems (governed by Lindblad equations), its manifestation in closed, integrable quantum systems remains less understood.

In closed quantum systems, relaxation occurs via dephasing following a quantum quench, leading to a stationary state (Generalized Gibbs Ensemble) rather than thermal equilibrium. A critical challenge in identifying the Mpemba effect in these systems is the lack of a unique definition for "distance from equilibrium." Different observables can yield qualitatively different relaxation behaviors.

The specific problem addressed:
Does a Mpemba-type effect emerge in the density redistribution dynamics of a strongly interacting one-dimensional Bose gas (Lieb-Liniger model) undergoing a sudden box expansion? The authors aim to determine if an excited state can relax faster than the ground state under specific observable definitions, and to clarify the role of integrability and interactions in this process.

2. Methodology

Physical Model

• System: $N$ bosons in 1D interacting via a repulsive delta potential ($g > 0$).
• Regime: Strongly interacting, approaching the Tonks-Girardeau (TG) regime.
• Hamiltonian:
  $$H = -\sum_{i=1}^N \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_i^2} + g\sum_{i<j}\delta(x_i - x_j) + V_L(x)$$
  where $V_L(x)$ is a hard-wall box potential of length $L$.
• Quench Protocol:
  1. The system is prepared in the ground state (or an excited state) within a box of length $L_0$.
  2. At $t=0$, the box is suddenly expanded to length $L$ ($L > L_0$).
  3. The interaction strength $g$ remains active throughout the evolution.
  4. The system evolves unitarily under the new Hamiltonian.

Numerical Approach

• Technique: Path-Integral Monte Carlo (PIMC) based on the generalized Feynman-Kac representation.
• Advantage: Unlike methods that map interacting bosons to non-interacting fermions (valid only in the strict TG limit), this approach is ab initio and retains full many-body interactions.
• Implementation:
  • The Schrödinger equation is mapped to a diffusion process in imaginary time.
  • A trial wavefunction ($\phi_T$) and trial energy ($\lambda_T$) are used to guide the stochastic sampling (importance sampling) to improve convergence.
  • The many-body spatial density $\rho(x, t)$ is calculated by sampling stochastic trajectories.

Observable and Distance Function

To quantify relaxation, the authors define a physically motivated distance function based on spatial density redistribution:

1. Partitioning: The expanded box is divided into the "Initial region" $[0, L_0]$ and the "Expanded region" $[L_0, L]$.
2. Average Densities:
   $$\rho_{L_0}(t) = \frac{1}{L_0}\int_0^{L_0} \rho(x,t)dx, \quad \rho_{L}(t) = \frac{1}{L-L_0}\int_{L_0}^L \rho(x,t)dx$$
3. Imbalance Observable: $D(t) = |\rho_{L_0}(t) - \rho_{L}(t)|$.
4. Distance from Stationary State:
   $$\mathcal{D}(t) = |D(t) - D(t_{final})|$$
   where $t_{final}$ is a sufficiently large time where the system has saturated.

Mpemba Criterion

The effect is identified if:

1. Initially, the excited state is farther from equilibrium: $\mathcal{D}_{exc}(0) > \mathcal{D}_{gnd}(0)$.
2. At a later time $t_c$, the ordering reverses: $\mathcal{D}_{exc}(t) < \mathcal{D}_{gnd}(t)$ for $t > t_c$.
3. This is quantified by the difference function $\Delta(t) = \mathcal{D}_{gnd}(t) - \mathcal{D}_{exc}(t)$, which must cross zero from positive to negative and remain negative.

3. Key Results

• Observation of Crossing: Numerical simulations reveal a clear crossing in the relaxation trajectories of the ground state and an excited state. The excited state, initially having a larger density imbalance, relaxes faster than the ground state after a specific crossing time $t_c$.
• Robustness:
  • Time Step Independence: The crossing behavior persists across different numerical time steps ($dt$), confirming it is not a discretization artifact.
  • Single Crossing: The difference function $\Delta(t)$ exhibits a single zero crossing and remains negative thereafter, satisfying the robust Mpemba criterion.
• Observable Dependence: The effect is highly sensitive to the definition of the observable. The crossing arises specifically from the spatial redistribution of particles between the initial and expanded regions. Other observables (e.g., total energy or local densities at specific points) may not exhibit this behavior.
• Role of Interactions: The effect occurs in a system with finite but strong interactions, computed via a fully interacting framework. This suggests the Mpemba effect is not restricted to exactly solvable (non-interacting or strict TG) limits but can emerge in strongly correlated systems.

4. Physical Interpretation

The authors attribute the Mpemba effect to the interplay between initial state structure and the spectrum of post-quench modes:

• Ground State: Characterized by strong spatial correlations and sharp confinement. It undergoes a rapid initial redistribution followed by a slower relaxation phase.
• Excited State: Possesses a broader momentum distribution. It exhibits a more gradual but sustained relaxation.
• Mechanism: The distinct relaxation pathways lead to a transient reversal in the ordering of the distance functions. This is a result of coherent many-body dynamics and dephasing, not dissipative mechanisms.

5. Significance and Conclusions

• Non-Universality: The paper reinforces the view that the Mpemba effect is not a universal law of relaxation. It is an observable-dependent phenomenon that arises only under specific dynamical conditions (choice of observable, initial state structure, and integrability).
• Integrable Systems: It demonstrates that Mpemba-type behavior can emerge in closed, integrable quantum systems governed by dephasing, challenging the notion that such effects require dissipation or thermalization.
• Methodological Contribution: The use of Path-Integral Monte Carlo provides a rigorous ab initio framework for studying non-equilibrium dynamics in interacting quantum gases without relying on mappings to free fermions.
• Clarification of Misconceptions: The work explicitly decouples the Mpemba effect from Newton's law of cooling, highlighting that it arises from the projection of initial conditions onto relaxation modes rather than simple thermal gradients.

Future Outlook: The authors suggest further systematic exploration of the effect's dependence on interaction strength, system size, and the definition of observables to establish general criteria for its emergence in quantum systems.