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 $⟨T(\sigma ,N)⟩$ 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 ${\sigma }_c(N)={\sigma }_c(\infty )+A{N}^{-1/\nu }$. (ii) Right at the critical point the average relaxation time $⟨T({\sigma }_c(N),N)⟩$ scales with the bundle size $N$ as: $⟨T({\sigma }_c(N),N)⟩\sim N^{\eta }$ 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: $⟨T(\sigma ,N)⟩/N^{\eta }\sim \mathcal{G}[\{{\sigma }_c(N)-\sigma \}{N}^{\zeta }]$ so that in the limit of $N\to \infty$ one has $⟨T(\sigma )⟩\sim (\sigma -{\sigma }_c{)}^{-\tau }$. 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 $\tau =1/2$.

8 pages, 11 figures

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