Dynamical TAP equations for non-equilibrium Ising spin glasses

TL;DR

This paper derives dynamical TAP equations for non-equilibrium Ising spin glasses with both synchronous and asynchronous updates, accommodating asymmetric couplings and time-dependent external fields, and shows they become exact in large systems.

Contribution

It introduces a generating functional approach to derive dynamical TAP equations for non-equilibrium Ising models with general couplings and dynamics.

Findings

TAP equations are self consistent for magnetizations over time.

Numerical simulations indicate TAP equations become exact for large systems.

The framework applies to both synchronous and asynchronous dynamics.

Abstract

We derive and study dynamical TAP equations for Ising spin glasses obeying both synchronous and asynchronous dynamics using a generating functional approach. The system can have an asymmetric coupling matrix, and the external fields can be time-dependent. In the synchronously updated model, the TAP equations take the form of self consistent equations for magnetizations at time $t+1$, given the magnetizations at time $t$. In the asynchronously updated model, the TAP equations determine the time derivatives of the magnetizations at each time, again via self consistent equations, given the current values of the magnetizations. Numerical simulations suggest that the TAP equations become exact for large systems.