Violation of the Fluctuation Dissipation Theorem during Domain Growth in the Long-range Ising Model

TL;DR

This work investigates aging and fluctuation-dissipation theorem (FDT) violations during domain growth in a two-dimensional long-range Ising model (LRIM) with nonconserved (Glauber) and conserved (Kawasaki) dynamics. Using the generalized FDT framework and an off-equilibrium method to compute the integrated response $\chi(t,t_w)$, it characterizes two-time quantities $C(t,t_w)$, $R(t,t_w)$, and the effective temperature $T_{\rm eff}(t,t_w)$ across interaction ranges set by $\sigma$ with Ewald summation for LR interactions. Key findings include strong FDT violations with $T_{\rm eff}(t,t_w) \to \infty$ as $t\to\infty$, aging scaling of $\chi(t,t_w)$ and $T_{\rm eff}(t,t_w)$ that depends on $\sigma$ and dynamics, and a clear separation between SU in spatial morphologies (via $C_{\rm sp}(r,t)$) and non-SU behavior in two-time observables. Bray–Rutenberg growth exponents are recovered with $\sigma$-dependent crossover between GD and KD, while the susceptibility exponent $a$ varies smoothly with $\sigma$ (no kink at $\sigma=1$), illustrating distinct effects of interaction range and conservation on aging in LR systems.

Abstract

The celebrated {\it fluctuation dissipation theorem} (FDT) does not apply to nonequilibrium systems. In this context, Cugliandolo and Kurchan [Phys. Rev. Lett. {\bf 71}, 173 (1993)] introduced a generalized FDT which interprets the nonequilibrium evolution as a composition of {\it time sectors} corresponding to different {\it effective temperatures}. We use this framework to study domain growth in the $d=2$ long-range Ising model (LRIM) with nonconserved kinetics ({\it Glauber spin-flip}) and conserved kinetics ({\it Kawasaki spin-exchange}). We study the dynamical scaling and super-universal (SU) scaling of various two-time quantities, e.g., autocorrelation function, response function, effective temperature, etc. In particular, we investigate how the interaction range and conservation laws affect these characteristic features of domain growth.

Figures (7)

• Figure 1: Evolution snapshots for the LRIM with GD after a quench at $t=0$. The values of $\sigma$ and $t$ are as shown. The lateral system size is $L = 1024$. The up and down spins are marked blue and green, respectively.
• Figure 2: Evolution snapshots for the LRIM with KD. The values of $\sigma$ and $t$ are as shown. The lateral system size is $L = 256$. The up and down spins are marked blue and green, respectively.
• Figure 3: The upper row plots $C_{\rm sp}(r,t)$ vs. $r/\ell(t)$ for (a) GD at $t=128$, (b) KD at $t = 5120$. The lower row plots $C(t,t_w)$ vs. $\ell(t)/\ell(t_w)$ on a log-log scale for (c) GD with $t_w=32$, (d) KD with $t_w = 1024$. In all frames, we superpose data for $\sigma = 0.6,0.8,1.6,\infty$.
• Figure 4: Plot of $T \chi(t,t_w)$ vs. $C(t,t_w)$ for the LRIM. We plot data for 3 values of $t_w$ (as indicated) for (a) GD with $\sigma=0.6$, (b) GD with $\sigma=1.6$, (c) GD with $\sigma=\infty$, (d) KD with $\sigma=0.6$, (e) KD with $\sigma=1.6$, (f) KD with $\sigma=\infty$. The straight line in each frame denotes the FDT line.
• Figure 5: Plots of $T \chi(t,t_w)$ vs. $t_w$ on a log-log scale for (a) GD with $\sigma = 0.6$, (b) KD with $\sigma = 0.6$. The data sets correspond to fixed values of $t/t_w$, as indicated. The best-fit lines yield estimates of $a$. Scaling plots of $t_w^a T \chi(t,t_w)$ vs. $t/t_w$ with 3 values of $t_w$ for (c) GD with $\sigma = 0.6$, (d) KD with $\sigma = 0.6$. Scaling plots of $t_w^{-a} T_{\rm eff} (t,t_w)$ vs. $t/t_w$ with 3 values of $t_w$ for (e) GD with $\sigma = 0.6$, (f) KD with $\sigma = 0.6$.