Large deviations of the finite-time magnetization of the Curie-Weiss random-field Ising model

We study the large deviations of the magnetization at some finite time in the Curie-Weiss random field Ising model with parallel updating. While relaxation dynamics in an infinite-time horizon gives rise to unique dynamical trajectories [specified by initial conditions and governed by first-order dynamics of the form $m_{t+1}=f\left(m_t\right)]$, we observe that the introduction of a finite-time horizon and the specification of terminal conditions can generate a host of metastable solutions obeying second-order dynamics. We show that these solutions are governed by a Newtonian-like dynamics in discrete time which permits solutions in terms of both the first-order relaxation (“forward”) dynamics and the backward dynamics $m_{t+1}=f^{-1}\left(m_t\right)$. Our approach allows us to classify trajectories for a given final magnetization as stable or metastable according to the value of the rate function associated with them. We find that in analogy to the Freidlin-Wentzell description of the stochastic dynamics of escape from metastable states, the dominant trajectories may switch between the two types (forward and backward) of first-order dynamics. Additionally, we show how to compute rate functions when uncertainty in the quenched disorder is introduced.

Figure 1

Peretto rate functions in the paramagnetic ((a), $\beta =0.67$, $h=0.2$) and ferromagnetic ((b), $\beta =2.5$, $h=0.4$) parts of the phase diagram.

Figure 2

Phase portraits of trajectories described by Eq. (35) [(a) and (c)] and maps of the relaxation dynamics $m_{t+1}=f\left(m_t\right)$ together with the diagonal $m_{t+1}=m_t$ [(b) and (d)] for the parameter settings: (a, b) $\beta =2.5,h=0.4$ (ferromagnetic phase), (c, d) $\beta =2.5,h=0.485$ (ferromagnetic phase with metastable state at $m=0$).

Figure 3

Phase portraits of trajectories described by Eq. (35) [(a) and (c)] and maps of the relaxation dynamics $m_{t+1}=f\left(m_t\right)$ together with the diagonal $m_{t+1}=m_t$ [(b) and (d)] for the parameter settings: (a, b) $\beta =2.5,h=0.6$ (paramagnetic phase) (c, d) $\beta =1.1,h=0.1$ (ferromagnetic phase near a second-order phase transition).

Figure 4

Position-dependent potential (solid line) and path-dependent potential (circles) with $T=20$ for the magnetic regime at $\beta =2.5,h=0.4$, and $r_0=0$.

Figure 5

Position-dependent potential (solid line) and initial total energy for $r_0=0.3$ (dashed) and $r_0=0.7$ (dash-dotted) for the ferromagnetic regime at $\beta =2.5$ and $h=0.4$. The level of the highest potential peak is fixed at 0.

Figure 6

(a) Peretto rate function (dashed) and finite-time rate function ($T=50$, dots) for the ferromagnetic phase at $\beta =2.5$, $h=0.4,r_0=0.3$. (b) Peretto rate function (solid line) and finite-time rate function ($T=20$, circles) for the paramagnetic phase at $\beta =1$, $h=0.4,r_0=0.3$.

Figure 7

(a) Dominant trajectories as a function of time for $\beta =2.5,h=0.4,r_0=0.3$. (b) dominant trajectories (circles) and limit curves $m_{t+1}=f\left(m_t\right)$ (solid line) and $m_{t+1}=f^{-1}\left(m_t\right)$ (dashed line) for $\beta =2.5,h=0.4$.

Figure 8

Quasi-Peretto solutions for $\beta =2.5$, $h=0.4$, $r_0=0.3$ (a) as a function of time. (b) in the $(m_t,m_{t+1})$ plane (circles) along with limit curves $m_{t+1}=f\left(m_t\right)$ (dashed line) and $m_{t+1}=f^{-1}\left(m_t\right)$ (solid line) for $\beta =2.5,h=0.4$. (c) Rate function associated with the quasi-Peretto trajectories (circles) and Peretto rate function (solid line).

Figure 9

Finite-time ($T=50$) theoretical (solid red line) and empirical (circles) rate functions for $\beta =2.5$, $h=0.4,r_0=0.0$.

Figure 10

Rate functions for $\beta =2.5$, $h=0.4$, $r_0=0.3$, and $T=50$ with atypical disorder (circles) and without (red dotted line).