Position distribution in a generalised run and tumble process [article]

David S. Dean, Satya N. Majumdar, Hendrik Schawe
2020 arXiv   pre-print
We study a class of stochastic processes of the type d^n x/dt^n= v_0 σ(t) where n>0 is a positive integer and σ(t)=± 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
more » ... mes while it grows as ∼ 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 ∼ln t. Thus the process undergoes a 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≥ 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)∼[-γ t Φ_n(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.
arXiv:2009.01487v1 fatcat:zmubrrls75dt7bmu45lws4dggu