Long time limit of equilibrium glassy dynamics and replica calculation

TL;DR

This paper demonstrates that the long-time limit of equilibrium glassy dynamics can be computed using a replica approach with one-step replica symmetry breaking, establishing a formal equivalence between static replica calculations and dynamic relaxation behavior.

Contribution

It establishes a rigorous connection between the long-time dynamic self-energy and the static self-energy in a replicated system with one-step RSB, proving the equivalence to all orders in perturbation for scalar theories.

Findings

Long-time limit of dynamic self-energy derived from static replica self-energy.

Dyson equation in replicated system yields bifurcation equation for ergodicity breaking.

Proves equivalence of replica formalism and dynamic relaxation in scalar theories.

Abstract

It is shown that the limit $t-t'\to\infty$ of the equilibrium dynamic self-energy can be computed from the $n\to 1$ limit of the static self-energy of a $n$-times replicated system with one step replica symmetry breaking structure. It is also shown that the Dyson equation of the replicated system leads in the $n\to 1$ limit to the bifurcation equation for the glass ergodicity breaking parameter computed from dynamics. The equivalence of the replica formalism to the long time limit of the equilibrium relaxation dynamics is proved to all orders in perturbation for a scalar theory.