Scaling forms for Relaxation Times of the Fiber Bundle model
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 $\langle T(σ,N) \rangle$ in the precritical state. The other results are: (i) The critical load $σ_c(N)$ for the bundle of size $N$ approaches its asymptotic value $σ_c(\infty)$ as $σ_c(N) = σ_c(\infty) + AN^{-1/ν}$. (ii) Right at the critical point the average relaxation time $\langle T(σ_c(N),N) \rangle$ scales with the bundle size $N$ as: $\langle T(σ_c(N),N) \rangle \sim N^η$ and this behavior remains valid within a small window of size $|Δσ| \sim N^{-ζ}$ around the critical point. (iii) When $1/N < |Δσ| < 100N^{-ζ}$ the finite-size scaling takes the form: $\langle T(σ,N) \rangle / N^η \sim {\cal G}[\{σ_c(N)-σ\}N^ζ]$ so that in the limit of $N \to \infty$ one has $\langle T(σ) \rangle \sim (σ- σ_c)^{-τ}$. 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 $τ= 1/2$.