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Run, Tumble and Paint

Emir Sezik, Callum Britton, Alex Touma, Gunnar Pruessner

Abstract

The visit probability, quantifying whether a particle has reached a given point for the first time by a specified time, provides access to various extreme value statistics and serves as a fundamental tool for characterising active matter models. However, previous studies have largely neglected how the visit probability depends on the internal degree of freedom driving the active particle. To address this, we calculate the "state-dependent'' visit probability for a Run-and-Tumble particle, that is the probability that the particle first passes through $x$ before time $t$, keeping track of its internal state during first passage. This process may be thought of as the particle "painting'' the positions it passes through for the time in the colour of its self-propulsion state. We perform this calculation in one dimension using Doi-Peliti field theory, by extending the tracer mechanism from previous works to incorporate such "polar deposition'' and demonstrate that state-dependent visit probabilities can be elegantly captured within this field-theoretic framework. We further derive the total volume covered by a right- (or left-) moving Run-and-Tumble particle and compare our results with known expressions for Brownian motion.

Run, Tumble and Paint

Abstract

The visit probability, quantifying whether a particle has reached a given point for the first time by a specified time, provides access to various extreme value statistics and serves as a fundamental tool for characterising active matter models. However, previous studies have largely neglected how the visit probability depends on the internal degree of freedom driving the active particle. To address this, we calculate the "state-dependent'' visit probability for a Run-and-Tumble particle, that is the probability that the particle first passes through before time , keeping track of its internal state during first passage. This process may be thought of as the particle "painting'' the positions it passes through for the time in the colour of its self-propulsion state. We perform this calculation in one dimension using Doi-Peliti field theory, by extending the tracer mechanism from previous works to incorporate such "polar deposition'' and demonstrate that state-dependent visit probabilities can be elegantly captured within this field-theoretic framework. We further derive the total volume covered by a right- (or left-) moving Run-and-Tumble particle and compare our results with known expressions for Brownian motion.

Paper Structure

This paper contains 11 sections, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Field Theory
  4. Observables
  5. Conclusion
  6. Scaling of longest excursion
  7. Limiting cases
  8. $\nu_0\to0, \alpha\to0$
  9. $D_{x}\to0^{+}, \alpha\to0$
  10. $\nu_0\to0$, $\alpha >0$
  11. $D_{x}\to0^{+}$, $\alpha > 0$

Figures (2)

  • Figure 1: The tracer mechanism --- Cartoon: (a) The RnT particle, shown as a ball, is initialised as a right mover (red). (b) Particle evolves in time, travelling on average to the right and leaving a trail of right (red) tracers, or painting the real line with a colour corresponding to its orientation. (c) Particle tumbles to become a left mover (blue) and the jitter of its diffusive motion leaves a "puddle" of blue to the right of the red stretch. (d) Particle travels further to the left and leaves left (blue) tracers specifically where the particle has not already traversed. The particle doesn't paint over the existing red tracers. (e) Numerical realisation of the process, initialised as a right mover with $\alpha = 10.0;\, D_{x} = 1.0;\, \nu_0 = 1.0;\, T=100$.
  • Figure 2: Comparison with simulations of (a) the asymptotic expression for the total volume covered by an RnT particle, Eq. \ref{['eq:total_Volume_covered']} and (b) the volume covered on the positive half line in each state as a function of Péclet, Eq. \ref{['eq:asymptotic_volume_explored_half_line']} with a fixed large time, $t \gg 1/\alpha$, with $1/\alpha$ the tumbling timescale. For convenience, the vertical axis is scaled by $\sqrt{D_{x} t}$ to elucidate the dependence on the Péclet number, $\text{Pe}$. The error bars associated with the numerical results are shown, but are smaller than the symbol size. For simplicity, we simulate the system on a lattice in discrete time and count all the distinct lattice sites visited during time $t \gg 1/\alpha$ to obtain our observables. Simulations were run for $10^{5}$ time steps, with ensemble averaging over $10^{4}$, for $D_x=1.0$; $\alpha=0.5$; $\nu_0\in\{0.5, 1.0,\dots,10.0\}$.