Non-equilibrium dynamics in Ising like models with biased initial condition

TL;DR

This paper studies the non-equilibrium dynamics of Ising-like models with biased initial conditions, analyzing fixed points, exit probabilities, and scaling behaviors through mean field and numerical simulations.

Contribution

It introduces a detailed stability analysis of fixed points and explores the behavior of exit probabilities and scaling functions in Ising-like models with varying coordination numbers.

Findings

Existence of a coordination number-dependent exponent in fixed point stability.

In mean field, the exit probability equals initial fraction for z=2.

Scaling collapse of exit probability using a universal scaling function.

Abstract

We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number $z$ in the Ising model. For the generalised voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of $z$ are also obtained. A related study is the behaviour of the exit probability $E(x_0)$, defined as the probability that a configuration ends up with all spins up starting with $x_0$ fraction of up spins. An interesting result is $E(x_0) = x_0$ in the mean field approximation when $z=2$, which is consistent with the conserved magnetisation in the system. For larger values of $z$, $E(x_0)$ shows the usual finite size dependent non linear behaviour both in the mean field model and in Ising model with nearest neighbour interaction on different two dimensional lattices. For such a behaviour, a data collapse of $E(x_0)$ is obtained using $y = \frac{(x_0 - x_c)}{x_c}L^{1/\nu}$ as the scaling variable and $f(y)=\frac{1+\tanh(\lambda y)}{2}$ appears as the scaling function. The universality of the exponent and the scaling factor is investigated.