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 $-\dot D=\dot e_p$ implies that a meeting occurs when the accumulated \emph{difference} of normalized entropy-production rates, $\int_0^t(r_H-r_C)\,\mathrm ds$ with $r_α=\dot e_p^α/D_α$, 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 $0.1659\le t_M\le0.1674$ 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 $t_M\ge0.042$. 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.