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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2309.02896 |
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| _version_ | 1866914755999956992 |
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| author | Chakraborty, Tanmoy Pradhan, Punyabrata |
| author_facet | Chakraborty, Tanmoy Pradhan, Punyabrata |
| contents | We investigate steady-state current fluctuations in two models of run-and-tumble particles (RTPs) on a ring of $L$ sites, for \textit{arbitrary} tumbling rate $γ=τ_p^{-1}$ and density $ρ$; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called long-ranged lattice gas (LLG). We show that, in the limit of $L$ large, the fluctuation of cumulative current $Q_i(T, L)$ across $i$th bond in a time interval $T \gg 1/D$ grows first {\it subdiffusively} and then {\it diffusively} (linearly) with $T$, where $D$ is the bulk diffusion coefficient. Remarkably, regardless of the model details, the scaled bond-current fluctuations $D \langle Q_i^2(T, L) \rangle/2 χL \equiv {\cal W}(y)$ as a function of scaled variable $y=DT/L^2$ collapse onto a {\it universal} scaling curve ${\cal W}(y)$, where $χ(ρ,γ)$ is the collective particle {\it mobility}. In the limit of small density and tumbling rate $ρ, γ\rightarrow 0$ with $ψ=ρ/γ$ fixed, there exists a scaling law: The scaled mobility $γ^{a} χ(ρ, γ)/χ^{(0)} \equiv {\cal H} (ψ)$ as a function of $ψ$ collapse onto a scaling curve ${\cal H}(ψ)$, where $a=1$ and $2$ in models I and II, respectively, and $χ^{(0)}$ is the mobility in the limiting case of symmetric simple exclusion process (SSEP). For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, ${\cal W}(y)$ and ${\cal H}(ψ)$. We also calculate spatial correlation functions for the current, and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length $ξ\sim τ_p^{1/2}$ diverging with persistence time $τ_p \gg 1$. Overall our theory is in excellent agreement with simulations and complements the findings of Ref. {\it arXiv:2209.11995}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02896 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Time-dependent properties of run-and-tumble particles. II.: Current fluctuations Chakraborty, Tanmoy Pradhan, Punyabrata Statistical Mechanics We investigate steady-state current fluctuations in two models of run-and-tumble particles (RTPs) on a ring of $L$ sites, for \textit{arbitrary} tumbling rate $γ=τ_p^{-1}$ and density $ρ$; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called long-ranged lattice gas (LLG). We show that, in the limit of $L$ large, the fluctuation of cumulative current $Q_i(T, L)$ across $i$th bond in a time interval $T \gg 1/D$ grows first {\it subdiffusively} and then {\it diffusively} (linearly) with $T$, where $D$ is the bulk diffusion coefficient. Remarkably, regardless of the model details, the scaled bond-current fluctuations $D \langle Q_i^2(T, L) \rangle/2 χL \equiv {\cal W}(y)$ as a function of scaled variable $y=DT/L^2$ collapse onto a {\it universal} scaling curve ${\cal W}(y)$, where $χ(ρ,γ)$ is the collective particle {\it mobility}. In the limit of small density and tumbling rate $ρ, γ\rightarrow 0$ with $ψ=ρ/γ$ fixed, there exists a scaling law: The scaled mobility $γ^{a} χ(ρ, γ)/χ^{(0)} \equiv {\cal H} (ψ)$ as a function of $ψ$ collapse onto a scaling curve ${\cal H}(ψ)$, where $a=1$ and $2$ in models I and II, respectively, and $χ^{(0)}$ is the mobility in the limiting case of symmetric simple exclusion process (SSEP). For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, ${\cal W}(y)$ and ${\cal H}(ψ)$. We also calculate spatial correlation functions for the current, and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length $ξ\sim τ_p^{1/2}$ diverging with persistence time $τ_p \gg 1$. Overall our theory is in excellent agreement with simulations and complements the findings of Ref. {\it arXiv:2209.11995}. |
| title | Time-dependent properties of run-and-tumble particles. II.: Current fluctuations |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2309.02896 |