Aging properties of the voter model with long-range interactions
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| Format: | Preprint |
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2024
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| _version_ | 1866913375892537344 |
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| author | Corberi, Federico Smaldone, Luca |
| author_facet | Corberi, Federico Smaldone, Luca |
| contents | We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent $S_i=\pm 1$ positioned at a lattice vertex $i$, copies the state of another one located at a distance $r$, selected randomly with a probability $P(r) \propto r^{-α}$. Employing both analytical and numerical methods, we compute the two-time correlation function $G(r;t,s)$ ($t\ge s$) between the state of a variable $S_i$ at time $s$ and that of another one, at distance $r$, at time $t$. At time $t$, the memory of an agent of its former state at time $s$, expressed by the {\it autocorrelation function} $A(t,s)=G(r=0;t,s)$, decays algebraically for $α>1$ as $[L(t)/L(s)]^{-λ}$, where $L$ is a time-increasing coherence length and $λ$ is the Fisher-Huse exponent. We find $λ=1$ for $α>2$, and $λ=1/(α-1)$ for $1<α\le 2$. For $α\le 1$, instead, there is an exponential decay, as in mean-field. Then, at variance with what is known for the related Ising model, here we find that $λ$ increases upon decreasing $α$. The space-dependent correlation $G(r;t,s)$ obeys a scaling symmetry $G(r;t,s)=g[r/L(s);L(t)/L(s)]$ for $α>2$. Similarly, for $1<α\le 2$ one has $G(r;t,s)=g[r/{\cal L}(t);{\cal L}(t)/{\cal L}(s)] $, where now the length ${\cal L}$ regulating two-time correlations differs from the coherence length as ${\cal L}\propto L^δ$, with $δ=1+2(2-α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Aging properties of the voter model with long-range interactions Corberi, Federico Smaldone, Luca Statistical Mechanics We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent $S_i=\pm 1$ positioned at a lattice vertex $i$, copies the state of another one located at a distance $r$, selected randomly with a probability $P(r) \propto r^{-α}$. Employing both analytical and numerical methods, we compute the two-time correlation function $G(r;t,s)$ ($t\ge s$) between the state of a variable $S_i$ at time $s$ and that of another one, at distance $r$, at time $t$. At time $t$, the memory of an agent of its former state at time $s$, expressed by the {\it autocorrelation function} $A(t,s)=G(r=0;t,s)$, decays algebraically for $α>1$ as $[L(t)/L(s)]^{-λ}$, where $L$ is a time-increasing coherence length and $λ$ is the Fisher-Huse exponent. We find $λ=1$ for $α>2$, and $λ=1/(α-1)$ for $1<α\le 2$. For $α\le 1$, instead, there is an exponential decay, as in mean-field. Then, at variance with what is known for the related Ising model, here we find that $λ$ increases upon decreasing $α$. The space-dependent correlation $G(r;t,s)$ obeys a scaling symmetry $G(r;t,s)=g[r/L(s);L(t)/L(s)]$ for $α>2$. Similarly, for $1<α\le 2$ one has $G(r;t,s)=g[r/{\cal L}(t);{\cal L}(t)/{\cal L}(s)] $, where now the length ${\cal L}$ regulating two-time correlations differs from the coherence length as ${\cal L}\propto L^δ$, with $δ=1+2(2-α)$. |
| title | Aging properties of the voter model with long-range interactions |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2402.11079 |