Martingale-induced local invariance in progressive quenching

Area of Science:

• Statistical Mechanics
• Complex Systems
• Computational Physics

Background:

• Progressive quenching (PQ) is a stochastic process involving sequential fixing of degrees of freedom in globally coupled Ising spin systems.
• Previous work established that the mean equilibrium spin value during PQ follows a martingale process, characterizing system memory.

Purpose of the Study:

• To investigate the implications of the martingale property in PQ on the dynamics of quenched magnetization.
• To explore the evolution of probability distributions under PQ and its connection to canonical statistical mechanics.

Main Methods:

• Analysis of the martingale process associated with the total quenched magnetization in PQ.
• Derivation of local invariance from the martingale property.
• Comparison of PQ with a novel process termed 'recycled quenching'.

Main Results:

• The martingale property in PQ implies a local invariance of the path weight for total quenched magnetization.
• PQ allows the probability distribution of total quenched magnetization to evolve while preserving a constrained canonical structure (path-independent potential and path-counting entropy).
• When PQ starts from equilibrium, its distributions match the limit distributions of recycled quenching.

Conclusions:

• The local invariance is a direct consequence of the martingale property in PQ, independent of other martingale theorems.
• PQ provides a framework for understanding how stochastic processes can maintain aspects of canonical equilibrium during non-equilibrium evolution.
• Recycled quenching offers a complementary perspective on the probability distributions generated by PQ.