Scaling forms for relaxation times of the fiber bundle model

Abstract

Using extensive numerical analysis of the fiber bundle model with equal load sharing dynamics we studied the finite-size scaling forms of the relaxation times against the deviations of applied load per fiber from the critical point. Our most crucial result is we have not found any ln(N) dependence of the average relaxation time <T(σ,N)> in the precritical state. The other results are as follows: (i) The critical load σ(c)(N) for the bundle of size N approaches its asymptotic value σ(c)(∞) as σ(c)(N)=σ(c)(∞)+AN(-1/ν). (ii) Right at the critical point the average relaxation time <T(σ(c)(N),N)> scales with the bundle size N as <T(σ(c)(N),N)>~N(η) and this behavior remains valid within a small window of size |Δσ|~N(-ζ) around the critical point. (iii) When 1/N<|Δσ|<100N(-ζ) the finite-size scaling takes the form <T(σ,N)>/N(η)~G[{σ(c)(N)-σ}N(ζ)] so in the limit of N→∞ one has <T(σ)>~(σ-σ(c))(-τ). The high precision of our numerical estimates led us to verify that ν=3/2, conjecture that η=1/3, ζ=2/3, and, therefore, τ=1/2.