Metrological Mpemba Effect: Quantitative Analysis

• The paper establishes the derivation of maximum extractable work via the Kullback–Leibler divergence, defining a strong Mpemba condition in double-well systems.
• The metrological Mpemba effect is characterized by two-stage relaxation dynamics, where rapid intra-well equilibration followed by slow inter-well transitions yields non-monotonic behavior.
• Experimental protocols, including precision sensor calibration and optimized finite-rate quenches, are designed to harness the effect for enhanced quantum thermometry and many-body system assessments.

The Metrological Mpemba Effect denotes the rigorous, quantifiable manifestation of the Mpemba phenomenon—the counterintuitive observation that, under certain conditions, a system initially prepared at a higher temperature relaxes to thermal equilibrium with a cold bath faster than a system started closer to the final temperature. In the metrological context, this effect is recast in terms of measurable quantities such as extractable work, mean first-passage times, or information-theoretic metrics, providing a robust framework for both experimental validation and the development of optimal control protocols across classical, stochastic, and quantum systems.

1. Theoretical Basis: Nonequilibrium Thermodynamics and Extractable Work

The metrological interpretation of the Mpemba effect is anchored in the two-stage relaxation dynamics of systems with separated timescales, typically realized in double-well potentials with strong barrier separation. In such configurations, a Brownian particle in a potential $U(x)$ with two minima (metastable and global-stable) exhibits rapid intra-well relaxation (Stage 1), establishing a local equilibrium at the bath temperature $T_b$ but retaining the initial population imbalance between wells. The subsequent slow inter-well equilibration (Stage 2) redistributes populations via rare, activated transitions.

The post-Stage 1 state $\rho^{\rm leq}(x)$ is formed by matching the $T_b$-Gibbs profiles within each well, but with well populations $a_L(\beta)$ and $a_R(\beta)$ inherited from the initial state at temperature $T_0=\beta^{-1}$:

$$\rho^{\rm leq}_{\beta,\beta_b}(x) = \begin{cases} a_L(\beta) \frac{g_{\beta_b}(x)}{a_L^*} & x < 0 \ a_R(\beta) \frac{g_{\beta_b}(x)}{a_R^*} & x > 0 \end{cases}$$

The maximum extractable work in an isothermal process driving the system from this state to global equilibrium is given by the Kullback–Leibler divergence between the local equilibrium and bath equilibrium distributions:

$$W_{\rm ex}(\beta, \beta_b) = k_B T_b \left[ a_L(\beta) \ln \frac{a_L(\beta)}{a_L^*} + a_R(\beta) \ln \frac{a_R(\beta)}{a_R^*} \right]$$

The non-monotonic dependence of $W_{\rm ex}(T_0)$ on $T_0$ is the signature of the metastable Mpemba effect, with points of vanishing extractable work corresponding to the “strong Mpemba” condition, where rapid global equilibration occurs immediately after Stage 1 (Chétrite et al., 2021).

2. Metrological Conditions: Strong and Weak Mpemba Effect

The rigorous criteria for the Mpemba effect, both strong and weak, can be formulated in terms of observables, as established via the small-diffusion limit of overdamped Langevin dynamics in the double-well:

• Strong Mpemba Effect: The amplitude $a_2$ of the slowest relaxation mode vanishes if and only if the initial well occupancy matches the bath equilibrium:

$\Pi_L(T) = \Pi_L(T_b)$

This establishes an operational experimental criterion: by observing the initial populations in each well, the possibility of a strong Mpemba crossing can be directly determined.

• Weak Mpemba Effect: The overlap $a_2(T)$ as a function of $T$ exhibits a non-monotonic extremum, determined to leading order by

$W^{(0)}(T) = 0$

where $W^{(0)}(T)$ involves averages of the potential $U(x)$ within each well and global averages, weighted by initial and bath equilibrium populations.

• Mean First-Passage Times (MFPT): All kinetic terms can be characterized via MFPTs measured in controlled experiments. For a colloidal particle, MFPTs from each well and equilibrium occupations provide all inputs required to validate the metrological Mpemba criteria (Walker et al., 2022).

3. Quantum Metrological Generalization

In quantum systems, the Mpemba effect is recast as a resource for enhanced temperature estimation. The metrological Mpemba effect in quantum thermometry is defined by the transient dominance of the quantum Fisher information (QFI) for temperature, achieved by initializing a probe in a nonequilibrium (“hotter”) state. Using explicit models (two-level and Λ-level probes under amplitude-damping dynamics), it is shown that:

$$\mathcal F_T[\rho_{\rm hot}(t^*)] > \mathcal F_T[\rho_{\rm cold}(t^*)] \geq \mathcal F_T[\rho_{\rm eq}(T)]$$

for a finite crossover time $t^*$, indicating that nonequilibrium initialization can outperform equilibrium and even “colder” strategies for temperature sensing. The theoretical underpinning involves decomposing the Lindblad dynamics into modes, with the QFI enhancement arising from state-dependent relaxation rates and their positive temperature-derivatives (Chattopadhyay et al., 8 Jan 2026).

4. Experimental and Protocol Design Considerations

Metrological implementation of the Mpemba effect demands precision in both preparation and measurement:

• Sensor Calibration and Reproducibility: State-of-the-art thermometry, e.g., NIST-traceable platinum resistance thermometers with sub-0.05 °C expanded uncertainty, is essential. Systematic uncertainties (drift, pressure variation, mass loss) are minimized, and data are averaged over several runs to ensure statistical significance (Wang et al., 2011).
• Protocol Optimization: To harness rapid equilibration, initial temperatures should be chosen just beyond the extractable work maximum $W(T_0)$, minimizing Stage 2 relaxation time. For maximal work extraction, the system should be initialized near the $W(T_0)$ peak. The possibility of two distinct initial temperatures yielding identical extractable work motivates bistable thermodynamic engine protocols.
• Finite-Rate Quenches: Analysis under time-delayed Newtonian cooling confirms that a “genuine” Mpemba inversion disappears if the quench duration $\sigma$ is finite and both samples are exposed to identical baths, although for $\sigma \ll 1$ the effect can persist within experimental tolerance (Santos, 19 Feb 2025).
• Classical Liquids: Experimental evidence, as in controlled vacuum-pump setups for water, quantifies the crossover temperature $T_c$ at which cooling curves intersect with sub-decikelvin precision, demonstrating that reproducible Mpemba phenomena can be established under well-constrained boundary conditions (Wang et al., 2011).

5. Application to Collective and Cooperative Systems

The metrological framework extends to many-body and cooperative systems, including magnetic phase transitions. Monte Carlo investigations in $q$-state Potts and Ising models establish that higher-temperature initial states can reach ordered equilibrium more rapidly than colder ones when quenched below $T_c$. The time to reach a reference order parameter scales as a super-universal power law with respect to the initial correlation length $\xi$:

$t_c \propto \xi^\kappa, \qquad \kappa \approx 0.9$

The “Mpemba number” $M$—the log-ratio of crossing times to log-ratio of initial correlation lengths—emerges as a universal metric for quantifying the strength of the Mpemba effect across diverse systems (Chatterjee et al., 2023).

System/Class	Metrological Observable	Mpemba Criterion/Formulation
Double-well/Colloid	Extractable work $W_{\rm ex}(T_0)$, MFPT, $\Pi_{L,R}(T)$	$W_{\rm ex}$ non-monotonic in $T_0$; $\Pi_L(T) = \Pi_L(T_b)$ for strong effect
Quantum probe	Quantum Fisher Information $\mathcal{F}_T$	$$\mathcal{F}_T^{\rm hot}(t^*) > \mathcal{F}_T^{\rm cold}(t^*)$$ at $t^*$
Magnetic system	Crossing time $t_c$ of order parameter/energy, correlation length $\xi$	$t_c \propto \xi^{\,\kappa}$ with $\kappa \approx 0.9$

6. Practical Implications and Future Directions

The metrological Mpemba effect provides a blueprint for optimizing thermal protocols in nanoscale engines, quantum sensors, and cooperative many-body systems:

• Protocol Design: Calculation of $W_{\rm ex}(T_0)$ using only equilibrium distributions at $T_0$ and $T_b$ enables the tailoring of fast-relaxing, efficient switching or cooling cycles in colloidal and nano-mechanical devices (Chétrite et al., 2021).
• Ultra-Fast Quantum Thermometry: Leveraging Mpemba-type dynamics in quantum probes realizes ultrafast, high-precision temperature measurements unattainable in equilibrium-based strategies (Chattopadhyay et al., 8 Jan 2026).
• Benchmarking and Universality: The dimensionless Mpemba number $M$ supports cross-system comparison, enabling rigorous assessment of the Mpemba effect’s prevalence and strength beyond the specifics of microscopic mechanisms (Chatterjee et al., 2023).
• Limits and Caveats: The effect is strictly bounded by the precise matching of initial and bath conditions and can be nullified or modified by finite-rate quenches, system size, or explicit protocol details (Santos, 19 Feb 2025).

The metrological Mpemba effect thus constitutes the cornerstone of quantitative, reproducible exploration and exploitation of anomalous non-monotonic relaxation phenomena across scales and platforms.