Effect of disorder on relaxation in the one-dimensional Glauber model

Abstract

We study the long-time relaxation of magnetization in a disordered linear chain of Ising spins from an initially aligned state. The coupling constants are ferromagnetic and nearest-neighbor only, taking values J₀ and J₁ with probabilitiesp and 1-p, respectively. The time evolution of the system is governed by the Glauber master equation. It is shown that for large times t, the magnetization M(t) varies as [exp(-λ₀ t]Φ(t), where λ₀ is a function of the stronger bond strength J₀ only, and Φ(t) decreases slower than an exponential. For very long times, we find that ln Φ(t) varies as -t ^1/3. For low enough temperatures, there is an intermediate time regime when ln Φ(t) varies as -t ^½. The results can be extended to more general probability distributions of ferromagnetic coupling constants, assuming that M(t) can only increase if any bond in the chain is strengthened. If the coupling constants have a continuous distribution in which the probability density varies as a power law near some maximum value J₀, we find that ln Φ(t) varies as -t ^1/3(ln t)^⅔ for large times.