Run-and-tumble particles in a harmonic external field: constructing the solution of the Fokker-Planck equation from the equations of motion

Aoran Sun; Fangfu Ye; Jeff Z. Y. Chen

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arXiv · 2610.24961

Run-and-tumble particles in a harmonic external field: constructing the solution of the Fokker-Planck equation from the equations of motion

Aoran Sun·Fangfu Ye·Jeff Z. Y. Chen

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Abstract

A method for integrating the Langevin equations of motion for run-and-tumble particles is introduced to compute the joint distribution of their position and active velocity, with and without a harmonic external force field. The key step is the construction of the moments of the distribution and the correlation functions, conditional on the particles being in a specific state at the final time. For one-dimensional run-and-tumble particles in a harmonic external potential, the exact steady-state distributions for the tumble state, as well as for the run state conditional on a specified velocity direction, are obtained. This enables us to demonstrate their equivalence to the exact solutions of the corresponding Fokker-Planck equations.

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Aoran Sun, Fangfu Ye, Jeff Z. Y. Chen. 2026-10-18. Run-and-tumble particles in a harmonic external field: constructing the solution of the Fokker-Planck equation from the equations of motion. https://arxiv.org/abs/2610.24961

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