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 $\langle T(\sigma,N) \rangle$ in the precritical state. The other results are: (i) The critical load $\sigma_c(N)$ for the bundle of size $N$ approaches its asymptotic value $\sigma_c(\infty)$ as $\sigma_c(N) = \sigma_c(\infty) + AN^{-1/\nu}$. (ii) Right at the critical point the average relaxation time $\langle T(\sigma_c(N),N) \rangle$ scales with the bundle size $N$ as: $\langle T(\sigma_c(N),N) \rangle \sim N^{\eta}$ and this behavior remains valid within a small window of size $|\Delta \sigma| \sim N^{-\zeta}$ around the critical point. (iii) When $1/N < |\Delta \sigma| < 100N^{-\zeta}$ the finite-size scaling takes the form: $\langle T(\sigma,N) \rangle / N^{\eta} \sim {\cal G}[{\sigma_c(N)-\sigma}N^{\zeta}]$ so that in the limit of $N \to \infty$ one has $\langle T(\sigma) \rangle \sim (\sigma - \sigma_c)^{-\tau}$. The high precision of our numerical estimates led us to verify that $\nu = 3/2$, conjecture that $\eta = 1/3$, $\zeta = 2/3$ and therefore $\tau = 1/2$.