Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function

TL;DR

This paper generalizes the fluctuation dissipation theorem for Ising spins out of equilibrium, introduces an efficient numerical method for response function computation, and applies it to analyze the dynamics of Ising models.

Contribution

It provides a new off-equilibrium fluctuation dissipation relation for Ising spins and a novel algorithm for linear response calculation without perturbations.

Findings

The structure of the response function is consistent across different dynamics.

The integrated response function follows a specific scaling law with exponent approximately 0.26.

The new method yields highly accurate data for Ising chain and 2D models.

Abstract

We derive for Ising spins an off-equilibrium generalization of the fluctuation dissipation theorem, which is formally identical to the one previously obtained for soft spins with Langevin dynamics [L.F.Cugliandolo, J.Kurchan and G.Parisi, J.Phys.I France \textbf{4}, 1641 (1994)]. The result is quite general and holds both for dynamics with conserved and non conserved order parameter. On the basis of this fluctuation dissipation relation, we construct an efficient numerical algorithm for the computation of the linear response function without imposing the perturbing field, which is alternative to those of Chatelain [J.Phys. A \textbf{36}, 10739 (2003)] and Ricci-Tersenghi [Phys.Rev.E {\bf 68}, 065104(R) (2003)]. As applications of the new algorithm, we present very accurate data for the linear response function of the Ising chain, with conserved and non conserved order parameter dynamics, finding that in both cases the structure is the same with a very simple physical interpretation. We also compute the integrated response function of the two dimensional Ising model, confirming that it obeys scaling $\chi (t,t_w)\simeq t_w^{-a}f(t/t_w)$, with $a =0.26\pm 0.01$, as previously found with a different method.