RESEARCH ARTICLE

Effective diffusion of a tracer in active bath: A path-integral approach

Hefei National Research Center for Physical Sciences at the Microscale & Key Laboratory of Precision and Intelligent Chemistry, University of Science and Technology of China, Hefei 230026, China

^* Corresponding authors (emails: fengmk@ustc.edu.cn (Mengkai Feng); hzhlj@ustc.edu.cn (Zhonghuai Hou))

Received: 30 October 2023
Revised: 3 January 2024
Accepted: 5 February 2024

Abstract

We investigate the effective diffusion of a tracer immersed in an active particle bath consisting of self-propelled particles. Utilising the Dean’s method developed for the equilibrium bath and extending it to the nonequilibrium situation, we derive a generalized Langevin equation (GLE) for the tracer particle. The complex interactions between the tracer and bath particles are shown as a memory kernel term and two colored noise terms. To obtain the effective diffusivity of the tracer, we use path integral technique to calculate all necessary correlation functions. Calculations show the effective diffusion decreases with the persistent time of active force, and has rich behavior with the number density of bath particles, depending on different activities. All theoretical results regarding the dependence of such diffusivity on bath parameters have been confirmed by direct computer simulation.