Mpemba effect as an entropy-production race:Finite-amplitude bounds on crossing times in Markovian relaxation
Abstract
Two canonical preparations evolving under a common final Markov generator can exhibit the Mpemba effect because they excite different relaxation modes. We bound the first crossing of their relative entropies by treating relaxation as an entropy-production race. The exact balance −Ḋ=ėp implies that a meeting occurs when the accumulated \emph{difference} of normalized entropy-production rates, ∫0t(rH−rC) ds with rα=ėpα/Dα, equals the initial logarithmic deficit ln[DH(0)/DC(0)]. Global entropy-decay inequalities yield a lower crossing-time bound when the entropy-decay constant is bounded from below and the hot rate from above; we supply both certificates. A nonzero cubic moment of a slowest eigenfunction is sufficient for the modified logarithmic Sobolev constant to lie strictly below the spectral gap. Combining established relative-entropy comparison inequalities with Markov contraction and the actual mode amplitudes gives finite-amplitude crossing bounds without a small-deviation expansion. For an exactly strong, reversible three-state preparation these give 0.1659≤ tM≤0.1674 in model time units, independently certified by outward-rounded interval arithmetic. This interval is narrower than 1% and contains tM=0.1662209. The conservative global-rate comparison used here gives only tM≥0.042. The hotter preparation starts $6.12$ times farther from equilibrium, yet it reaches D=10−5 at t=0.233, compared with t=0.844 for the colder one. Mean energy crosses later than relative entropy. For a fixed driven generator, nonadiabatic entropy production governs the same balance. These results concern finite-state relaxation in a specified distance and do not predict water-freezing times.
Related
- Two-loop amplitude for the associated production of a top-anti-top pair and a W boson at hadron colliders
- Amplitude-level NLO predictions for exclusive J/ψ+γ production in ultraperipheral collisions
- Information Entropy as a Sensitive Diagnostic of Relaxation Dynamics in a Quasiperiodic Mott Insulator
- Field-Driven Coupled Magnon--Phonon--Electron Relaxation in Magnetite Using Steepest-Entropy-Ascent Quantum Thermodynamic Formalism
- Wasserstein-2 gradient flows and the geometry of entropy production in classical and quantum stochastic thermodynamics
BibTeX
@misc{taye2026mpemba,
title = {{Mpemba effect as an entropy-production race:Finite-amplitude bounds on crossing times in Markovian relaxation}},
author = {Mesfin Taye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40704},
eprint = {2610.40704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40704},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mpemba effect as an entropy-production race:Finite-amplitude bounds on crossing times in Markovian relaxation
AU - Mesfin Taye
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.40704
UR - https://arxiv.org/abs/2610.40704
AB - Two canonical preparations evolving under a common final Markov generator can exhibit the Mpemba effect because they excite different relaxation modes. We bound the first crossing of their relative entropies by treating relaxation as an entropy-production race. The exact balance −Ḋ=ė_p implies that a meeting occurs when the accumulated \emph{difference} of normalized entropy-production rates, ∫_0^t(r_H−r_C) ds with r_α=ė_p^α/D_α, equals the initial logarithmic deficit ln[D_H(0)/D_C(0)]. Global entropy-decay inequalities yield a lower crossing-time bound when the entropy-decay constant is bounded from below and the hot rate from above; we supply both certificates. A nonzero cubic moment of a slowest eigenfunction is sufficient for the modified logarithmic Sobolev constant to lie strictly below the spectral gap. Combining established relative-entropy comparison inequalities with Markov contraction and the actual mode amplitudes gives finite-amplitude crossing bounds without a small-deviation expansion. For an exactly strong, reversible three-state preparation these give 0.1659≤ t_M≤0.1674 in model time units, independently certified by outward-rounded interval arithmetic. This interval is narrower than 1% and contains t_M=0.1662209. The conservative global-rate comparison used here gives only t_M≥0.042. The hotter preparation starts $6.12$ times farther from equilibrium, yet it reaches D=10^−5 at t=0.233, compared with t=0.844 for the colder one. Mean energy crosses later than relative entropy. For a fixed driven generator, nonadiabatic entropy production governs the same balance. These results concern finite-state relaxation in a specified distance and do not predict water-freezing times.
ER -