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A General Approach to the Shape Transition of Run-and-Tumble Particles: The 1D PDMP Framework for Invariant Measure Regularity

Leo Hahn

TL;DR

This work presents a general 1D PDMP framework to study shape transitions in run-and-tumble particles under confining potentials, characterizing when the invariant-density of the system is continuous or diverges without requiring explicit invariant measures. The authors prove regularity of the invariant density on noncritical intervals, and derive precise criteria for continuity vs divergence at simply and multiply critical points where vector fields vanish, via a combination of Fokker-Planck reformulations and a linear-system-based criterion for auxiliary integrals E^c. They show that the vector fields need only be $C^r$ to guarantee $C^r$ densities, even with position-dependent jump rates or resetting, and apply the theory to natural models like a two-velocity power-law process and a six-velocity harmonic process, revealing nuanced regularity patterns including interior singularities in multi-vanishing-point scenarios. The results unify and extend Bakhtin’s regularity theory for PDMPs, provide practical tools to assess shape transitions in RTPs, and offer insights into how near- vs far-from-equilibrium behavior manifests in invariant-measure regularity. Overall, the paper delivers a robust, general framework for understanding how nonequilibrium features of RTPs shape the invariant measure through local dynamics and switching structure, with implications for higher-velocity or interacting RTPs and resetting mechanisms.

Abstract

Run-and-tumble particles (RTPs) have emerged as a paradigmatic example for studying nonequilibrium phenomena in statistical mechanics. The invariant measure of a wide class of RTPs subjected to a potential possesses a density that is continuous at high tumble rates but exhibits divergences at low ones. This key feature, known as shape transition, constitutes a qualitative indicator of the relative closeness (continuous density) or strong deviation (diverging density) from the equilibrium setting. Furthermore, the points at which the density diverges correspond to the configurations where the system spends most of its time in the low tumble rate regime. Building on and extending existing results concerning the regularity of the invariant measure of one-dimensional dynamical systems with random switching, we show how to characterize the shape transition even in situations where the invariant measure cannot be computed explicitly. Our analysis confirms shape transition as a robust, general feature of RTPs subjected to a potential. We also refine the regularity theory for the invariant measure of one-dimensional dynamical systems with random switching.

A General Approach to the Shape Transition of Run-and-Tumble Particles: The 1D PDMP Framework for Invariant Measure Regularity

TL;DR

This work presents a general 1D PDMP framework to study shape transitions in run-and-tumble particles under confining potentials, characterizing when the invariant-density of the system is continuous or diverges without requiring explicit invariant measures. The authors prove regularity of the invariant density on noncritical intervals, and derive precise criteria for continuity vs divergence at simply and multiply critical points where vector fields vanish, via a combination of Fokker-Planck reformulations and a linear-system-based criterion for auxiliary integrals E^c. They show that the vector fields need only be to guarantee densities, even with position-dependent jump rates or resetting, and apply the theory to natural models like a two-velocity power-law process and a six-velocity harmonic process, revealing nuanced regularity patterns including interior singularities in multi-vanishing-point scenarios. The results unify and extend Bakhtin’s regularity theory for PDMPs, provide practical tools to assess shape transitions in RTPs, and offer insights into how near- vs far-from-equilibrium behavior manifests in invariant-measure regularity. Overall, the paper delivers a robust, general framework for understanding how nonequilibrium features of RTPs shape the invariant measure through local dynamics and switching structure, with implications for higher-velocity or interacting RTPs and resetting mechanisms.

Abstract

Run-and-tumble particles (RTPs) have emerged as a paradigmatic example for studying nonequilibrium phenomena in statistical mechanics. The invariant measure of a wide class of RTPs subjected to a potential possesses a density that is continuous at high tumble rates but exhibits divergences at low ones. This key feature, known as shape transition, constitutes a qualitative indicator of the relative closeness (continuous density) or strong deviation (diverging density) from the equilibrium setting. Furthermore, the points at which the density diverges correspond to the configurations where the system spends most of its time in the low tumble rate regime. Building on and extending existing results concerning the regularity of the invariant measure of one-dimensional dynamical systems with random switching, we show how to characterize the shape transition even in situations where the invariant measure cannot be computed explicitly. Our analysis confirms shape transition as a robust, general feature of RTPs subjected to a potential. We also refine the regularity theory for the invariant measure of one-dimensional dynamical systems with random switching.

Paper Structure

This paper contains 16 sections, 15 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Run-and-tumble particles as PDMPs
  3. Main results
  4. Regularity on noncritical intervals
  5. Continuity at simply critical points
  6. Continuity at multiply critical points
  7. Regularity on noncritical intervals
  8. Continuity at multiply critical points
  9. Proof of Theorem \ref{['thm:continuity_critical_point']} (i)
  10. Proof of Theorem \ref{['thm:continuity_critical_point']} (ii)
  11. Proof of Theorem \ref{['thm:continuity_critical_point']} (iii)
  12. Explicit computation of Ecσ
  13. Applications to the power-law and harmonic processes
  14. Acknowledgments
  15. Data availability statement
  16. ...and 1 more sections

Key Result

Proposition 3

The process $X_t$ is a homogeneous strong Markov process. A bounded measurable function $f : \mathbb R \to \mathbb R$ is in the domain $D(\mathcal{L})$ of its extended generator $\mathcal{L}$ if and only if and in that case

Figures (2)

  • Figure 1: Markov jump process followed by the single-particle velocities
  • Figure 2: Markov jump process followed by the relative velocity

Theorems & Definitions (47)

  • Definition 1: Local characteristics
  • Definition 2: One-dimensional piecewise-deterministic Markov process
  • Proposition 3: Extended generator
  • Remark 4
  • Definition 5: Two-particle instantaneous power-law process
  • Definition 6: Two-particle finite harmonic process
  • Definition 7: Single-particle resetting process evans18
  • Definition 8
  • Definition 9
  • Theorem 1
  • ...and 37 more