Position distribution in a generalised run and tumble process

David S. Dean; Satya N. Majumdar; Hendrik Schawe

doi:10.1103/PhysRevE.103.012130

paper

Position distribution in a generalised run and tumble process

arXiv:2009.01487 · doi:10.1103/PhysRevE.103.012130

Abstract

We study a class of stochastic processes of the type $\frac{d ^{n} x}{d t ^{n}} = v_{0} σ (t)$ where $n > 0$ is a positive integer and $σ (t) = \pm 1$ represents an `active' telegraphic noise that flips from one state to the other with a constant rate $γ$ . 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 $\sim t^{2 n - 1}$ 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 $n \geq 1/2$ , 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, $p_{n} (x, t) \sim exp [- γ t Φ_{n} (\frac{x}{x ^{*} ( t )})]$ , where $x^{*} (t) = v_{0} t^{n} /Γ (n + 1)$ . We compute the rate function $Φ_{n} (z)$ 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 $t$ .

24 pages, 4 Figures

Position distribution in a generalised run and tumble process

Abstract

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