Dynamics of the directed Ising chain

TL;DR

This paper extends Glauber's analysis of the Ising chain to asymmetric dynamics, revealing a transition in correlation behavior and fluctuation-dissipation ratio, with explicit solutions for key quantities in stationary and zero-temperature regimes.

Contribution

It introduces a model with asymmetric influence in the Ising chain, providing explicit solutions and analyzing the transition in correlation and response functions.

Findings

Transition in correlation function behavior at critical asymmetry

Fluctuation-dissipation ratio decreases from 1 to 0 at critical point

Asymmetric dynamics suppress criticality at zero temperature

Abstract

The study by Glauber of the time-dependent statistics of the Ising chain is extended to the case where each spin is influenced unequally by its nearest neighbours. The asymmetry of the dynamics implies the failure of the detailed balance condition. The functional form of the rate at which an individual spin changes its state is constrained by the global balance condition with respect to the equilibrium measure of the Ising chain. The local magnetization, the equal-time and two-time correlation functions and the linear response to an external magnetic field obey linear equations which are solved explicitly. The behaviour of these quantities and the relation between the correlation and response functions are analyzed both in the stationary state and in the zero-temperature scaling regime. In the stationary state, a transition between two behaviours of the correlation function occurs when the amplitude of the asymmetry crosses a critical value, with the consequence that the limit fluctuation-dissipation ratio decays continuously from the value 1, for the equilibrium state in the absence of asymmetry, to 0 for this critical value. At zero temperature, under asymmetric dynamics, the system loses its critical character, yet keeping many of the characteristic features of a coarsening system.