Run-and-tumble motion in a harmonic potential: field theory and entropy production

Abstract

Run-and-tumble (RnT) motion is an example of active motility where particles move at constant speed and change direction at random times. In this work we study RnT motion with diffusion in a harmonic potential in one dimension via a path integral approach. We derive a Doi-Peliti field theory and use it to calculate the entropy production and other observables in closed form. All our results are exact.

Field Theory of Free Run and Tumble Particles in d Dimensions

“…It is this competition that drives the rich phenomenology of RnT particles in a harmonic potential with spring-constant k [8], as the deviation from the equilibrium behaviour vanishes with v/(Dk).…”

“…. ; 0) is in fact the kernel of the Fokker-Planck equation [8,14,15], re-written to match the form of a master equation,…”

“…Doi-Peliti field theory [4,5] is a class of field theories to deal with non-equilibrium statistical mechanics [6]. Observables in this field theory are calculated from the master or Fokker-Planck equation [7,8], applying second quantisation and the path integral, in a straight forward manner. The aim in the following is to apply Doi-Peliti field theory to the one and higher dimensional RnT process, to calculate the particle density subject to initial conditions, the propagators and to use it to determine the mean squared displacement (MSD) as well as the entropy production.…”

Entropy production in exactly solvable systems

“…However, in interacting many-particle systems, such a description is not available in general. Instead, we may choose to use the Doi-Peliti formalism [50][51][52][53][54][55][56][57][58] to describe the system, since it provides a systematic approach based on the microscopic dynamics and which retains the particle entity.…”

Non-equilibrium thermodynamics of diffusion in fluctuating potentials

“…Introducing fluctuations into what would otherwise be time-independent model parameters is a recurrent theme in non-equilibrium physics. Indeed, think for example of Run-and-Tumble (RnT) and Active Ornstein-Uhlenbeck (AOUPs) particles, whose self-propulsion velocity is described by a telegraph process and an OU process, respectively [18,19]. Fluctuating interactions are a generic feature of living systems and can have striking consequences including clustering in populations of bacteria interacting via type IV pili [20][21][22], arrested coalescence in cellular aggregates [23] and fluidization of embryonic tissues [24].…”