Microscopic realizations of the Trap Model

TL;DR

This paper demonstrates that Monte Carlo methods applied to Number Partitioning and Diophantine approximations serve as microscopic models of the Trap Model, revealing that response and correlation functions follow fluctuation-dissipation relations even during aging.

Contribution

It introduces microscopic realizations of the Trap Model through optimization problems, providing new insights into its physical interpretation and broader applicability.

Findings

Response and correlation functions obey fluctuation-dissipation theorem during aging.

Microscopic models offer a new perspective on the physics of the Trap Model.

Relevance to optimization problems with non-linear cost functions.

Abstract

Monte Carlo optimizations of Number Partitioning and of Diophantine approximations are microscopic realizations of `Trap Model' dynamics. This offers a fresh look at the physics behind this model, and points at other situations in which it may apply. Our results strongly suggest that in any such realization of the Trap Model, the response and correlation functions of smooth observables obey the fluctuation-dissipation theorem even in the aging regime. Our discussion for the Number Partitioning problem may be relevant for the class of optimization problems whose cost function does not scale linearly with the size, and are thus awkward from the statistical mechanic point of view.