Ratcheting by Stochastic Resetting With Fat‐Tailed Time Distributions

We investigated both numerically and analytically the drift of a Brownian particle in a ratchet potential under stochastic resetting with fat-tailed distributions. As a study case we chose a Pareto time distribution with tail index β. We observed that for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo><</mml:mo> <mml:mi>β</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn></mml:mrow> <mml:annotation>${1/2\char60 \beta \char60 1}$</mml:annotation> </mml:semantics> </mml:math> rectification occurs even if for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>β</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn></mml:mrow> <mml:annotation>${\beta \char60 1}$</mml:annotation> </mml:semantics> </mml:math> the mean resetting time is infinite. However, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>β</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn></mml:mrow> <mml:annotation>${\beta \le 1/2}$</mml:annotation> </mml:semantics> </mml:math> rectification is completely suppressed. For low noise levels, the drift speed attains a maximum for β immediately above 1, that is for finite but large mean resetting times. In correspondence with such an optimal drift the particle diffusion over the ratchet potential turns from normal to superdiffusive, a property also related to the fat tails of the resetting time distribution.