Transfer operator analysis of the parallel dynamics of disordered Ising chains

Abstract

We study the synchronous stochastic dynamics of the random field and random bond Ising chain. For this model the generating functional analysis method of De Dominicis leads to a formalism with transfer operators, similar to transfer matrices in equilibrium studies, but with dynamical paths of spins and (conjugate) fields as arguments, as opposed to replicated spins. In the thermodynamic limit the macroscopic dynamics is captured by the dominant eigenspace of the transfer operator, leading to a relatively simple and transparent set of equations that are easy to solve numerically. Our results are supported excellently by numerical simulations.

Keywords:

• disordered Ising chains
• parallel dynamics
• transfer operator

Acknowledgements

KT was supported by a Grant-in-Aid (no. 18079006) and Program for Promoting Internationalisation of University Education from MEXT, Japan.

Notes

Notes

1. if the spectrum of M has continuous parts, the eigenvalue sum becomes an integral.

2. We interpret (49) as the fixed-point condition of a map (where denotes the operator in the right-hand side of (49)), and iterate this map until no longer decreases, upon which ε is reduced and the process repeated until a solution of (49) is found. The complexity of solving (49) numerically scales with the dimension of Φ, i.e. as .