Martingale-induced local invariance in Progressive Quenching

TL;DR

This paper reveals how martingale properties induce local invariance in the probability distribution evolution during progressive quenching in a coupled Ising system, linking path weight invariance to system memory and distribution limits.

Contribution

It demonstrates that martingale-induced local invariance governs the path weight and distribution evolution in progressive quenching, providing new insights into system memory and distribution limits.

Findings

Martingale property implies local invariance of path weight.

Distribution evolution maintains a Boltzmann-like structure.

Recycled quenching distribution converges to the stage distribution.

Abstract

Progressive quenching (PQ) is a stochastic process during which one fixes, one after another, the degrees of freedom of a globally coupled Ising spin system while letting it thermalize through a heat bath. It has previously been shown that during PQ, the mean equilibrium spin value follows a martingale process and this process can characterize the memory of the system. In the present study, we find that the aforementioned martingale implies a local invariance of the path weight for the total quenched magnetization, the Markovian process whose increment is the spin that is fixed last. Consequently, PQ lets the probability distribution for the total quenched magnetization evolve while keeping the Boltzmann-like factor, or a canonical structure, under constraint, which consists of a path-independent potential and a path-counting entropy. Moreover, when the PQ starts from full equilibrium, the probability distribution at each stage of PQ is found to be the limit distribution of what we call recycled quenching, the process in which a randomly chosen quenched spin is unquenched after a single step of PQ. The local invariance is a consequence of the martingale property, and not an application of known theorems for the martingale process.