Statistics of a Large Number of Renewals in Equilibrium and Non-Equilibrium Renewal Processes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909331689046016 |
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| author | Wang, Wanli Burov, Stanislav |
| author_facet | Wang, Wanli Burov, Stanislav |
| contents | The renewal process is a key statistical model for describing a wide range of stochastic systems in Physics. This work investigates the behavior of the probability distribution of the number of renewals in renewal processes in the short-time limit, with a focus on cases where the number of renewals is large. We find that the specific details of the sojourn time distribution $ϕ(τ)$ in this limit can significantly modify the behavior in the large-number-of-renewals regime. We explore both non-equilibrium and equilibrium renewal processes, deriving results for various forms of $ϕ(τ)$. Using saddle point approximations, we analyze cases where $ϕ(τ)$ follows a power-series expansion, includes a cutoff, or exhibits non-analytic behavior near $τ= 0$. Additionally, we show how the short-time properties of $ϕ(τ)$ shape the decay of the number of renewals in equilibrium compared to non-equilibrium renewal processes. The probability of the number of renewals plays a crucial role in determining rare event behaviors, such as Laplace tails. The results obtained here are expected to help advance the development of a theoretical framework for rare events in transport processes in complex systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Statistics of a Large Number of Renewals in Equilibrium and Non-Equilibrium Renewal Processes Wang, Wanli Burov, Stanislav Statistical Mechanics The renewal process is a key statistical model for describing a wide range of stochastic systems in Physics. This work investigates the behavior of the probability distribution of the number of renewals in renewal processes in the short-time limit, with a focus on cases where the number of renewals is large. We find that the specific details of the sojourn time distribution $ϕ(τ)$ in this limit can significantly modify the behavior in the large-number-of-renewals regime. We explore both non-equilibrium and equilibrium renewal processes, deriving results for various forms of $ϕ(τ)$. Using saddle point approximations, we analyze cases where $ϕ(τ)$ follows a power-series expansion, includes a cutoff, or exhibits non-analytic behavior near $τ= 0$. Additionally, we show how the short-time properties of $ϕ(τ)$ shape the decay of the number of renewals in equilibrium compared to non-equilibrium renewal processes. The probability of the number of renewals plays a crucial role in determining rare event behaviors, such as Laplace tails. The results obtained here are expected to help advance the development of a theoretical framework for rare events in transport processes in complex systems. |
| title | Statistics of a Large Number of Renewals in Equilibrium and Non-Equilibrium Renewal Processes |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2409.19966 |