Position distribution in a generalized run-and-tumble process
| hal.structure.identifier | Laboratoire Ondes et Matière d'Aquitaine [LOMA] | |
| dc.contributor.author | DEAN, David S. | |
| hal.structure.identifier | Laboratoire de Physique Théorique et Modèles Statistiques [LPTMS] | |
| dc.contributor.author | MAJUMDAR, Satya N. | |
| hal.structure.identifier | Laboratoire de Physique Théorique et Modélisation [LPTM - UMR 8089] | |
| dc.contributor.author | SCHAWE, Hendrik | |
| dc.date.issued | 2021-01 | |
| dc.identifier.issn | 2470-0045 | |
| dc.description.abstractEn | We study a class of stochastic processes of the type $\frac{d^n x}{dt^n}= v_0\, \sigma(t)$ where $n>0$ is a positive integer and $\sigma(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to the other with a constant rate $\gamma$. For $n=1$, it reduces to the standard run and tumble process for active particles in one dimension. This process can be analytically continued to any $n>0$ including non-integer values. We compute exactly the mean squared displacement at time $t$ for all $n>0$ and show that at late times while it grows as $\sim t^{2n-1}$ for $n>1/2$, it approaches a constant for $n<1/2$. In the marginal case $n=1/2$, it grows very slowly with time as $\sim \ln t$. Thus the process undergoes a {\em localisation} transition at $n=1/2$. We also show that the position distribution $p_n(x,t)$ remains time-dependent even at late times for $n\ge 1/2$, but approaches a stationary time-independent form for $n<1/2$. The tails of the position distribution at late times exhibit a large deviation form, $p_n(x,t)\sim \exp\left[-\gamma\, t\, \Phi_n\left(\frac{x}{x^*(t)}\right)\right]$, where $x^*(t)= v_0\, t^n/\Gamma(n+1)$. We compute the rate function $\Phi_n(z)$ analytically for all $n>0$ and also numerically using importance sampling methods, finding excellent agreement between them. For three special values $n=1$, $n=2$ and $n=1/2$ we compute the exact cumulant generating function of the position distribution at all times $t$. | |
| dc.language.iso | en | |
| dc.publisher | American Physical Society (APS) | |
| dc.title.en | Position distribution in a generalized run-and-tumble process | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1103/PhysRevE.103.012130 | |
| dc.subject.hal | Physique [physics] | |
| dc.identifier.arxiv | 2009.01487 | |
| bordeaux.journal | Physical Review E | |
| bordeaux.volume | 103 | |
| bordeaux.issue | 1 | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-03223889 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03223889v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Physical%20Review%20E&rft.date=2021-01&rft.volume=103&rft.issue=1&rft.eissn=2470-0045&rft.issn=2470-0045&rft.au=DEAN,%20David%20S.&MAJUMDAR,%20Satya%20N.&SCHAWE,%20Hendrik&rft.genre=article |
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