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We present an analysis of the stationary distributions of run-and-tumble particles trapped in external potentials in terms of a thermophoretic potential, that emerges when trapped active motion is mapped to trapped passive Brownian motion…

Statistical Mechanics · Physics 2019-02-06 Francisco J. Sevilla , Alejandro V. Arzola , Enrique Puga Cital

Motivated by various recent experimental findings, we propose a dynamical model of intermittently self-propelled particles: active particles that recurrently switch between two modes of motion, namely an active run-state and a turn state,…

Soft Condensed Matter · Physics 2025-10-30 Agniva Datta , Carsten Beta , Robert Großmann

Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for…

Probability · Mathematics 2008-01-03 Rudolf Gorenflo , Entsar A. A. Abdel-Rehim

We study the transport of self-propelled particles in dynamic complex environments. To obtain exact results, we introduce a model of run-and-tumble particles (RTPs) moving in discrete time on a $d$-dimensional cubic lattice in the presence…

Statistical Mechanics · Physics 2018-07-12 Thibault Bertrand , Yongfeng Zhao , Olivier Bénichou , Julien Tailleur , Raphaël Voituriez

A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing…

Classical Analysis and ODEs · Mathematics 2007-05-23 J. B. Sanders , N. M. Temme

We consider a simple model for active random walk with general temporal correlations, and investigate the shape of the probability distribution function of the displacement during a short time interval. We find that under certain conditions…

Statistical Mechanics · Physics 2020-01-06 Eial Teomy , Yael Roichman , Yair Shokef

We consider a random walk model in a one-dimensional environment, formed by several zones of finite width with the fixed transition probabilities. It is also assumed that the transitions to the left and right neighboring points have unequal…

Statistical Mechanics · Physics 2017-08-18 A. V. Nazarenko , V. Blavatska

We study the motion of a one-dimensional run-and-tumble particle with three discrete internal states in the presence of a harmonic trap of stiffness $\mu.$ The three internal states, corresponding to positive, negative and zero velocities…

Statistical Mechanics · Physics 2020-02-19 Urna Basu , Satya N. Majumdar , Alberto Rosso , Sanjib Sabhapandit , Gregory Schehr

A non--linear diffusion equation is derived by taking into account hopping rates depending on the occupation of next neighbouring sites. There appears additonal repulsive and attractive forces leading to a changed local mobiltiy. The…

Statistical Mechanics · Physics 2009-10-31 S. Artz , M. Schulz , S. Trimper

A particle subject to successive, random displacements is said to execute a random walk (in position or some other coordinate). The mathematical properties of random walks have been very thoroughly investigated, and the model is used in…

Statistical Mechanics · Physics 2007-05-23 M. Wilkinson , B. Mehlig

We study the dynamics of a one-dimensional run and tumble particle subjected to confining potentials of the type $V(x) = \alpha \, |x|^p$, with $p>0$. The noise that drives the particle dynamics is telegraphic and alternates between $\pm 1$…

Statistical Mechanics · Physics 2022-06-22 Abhishek Dhar , Anupam Kundu , Satya N. Majumdar , Sanjib Sabhapandit , Grégory Schehr

We consider a walker that at each step keeps the same direction with a probabilitythat depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent…

Probability · Mathematics 2015-09-15 Peggy Cénac , Basile De Loynes , Arnaud Le Ny , Yoann Offret

A random walk scheme, consisting of alternating phases of regular Brownian motion and L\'evy walks, is proposed as a model for run-and-tumble bacterial motion. Within the continuous-time random walk approach we obtain the long-time and…

Biological Physics · Physics 2017-01-26 Felix Thiel , Lutz Schimansky-Geier , Igor M. Sokolov

We study the relaxation dynamics of a run and tumble particle in a one-dimensional piecewise linear potential $U(x)=b|x|$, from delta-function initial conditions at $x=0$ to steady state. In addition to experiencing active telegraphic…

Statistical Mechanics · Physics 2025-06-04 R. K. Singh , Oded Farago

Random walk has wide applications in many fields, such as machine learning, biology, physics, and chemistry. Random walk can be discrete or continuous in time and space. Asymmetric random walk could be described by drift-diffusion equation.…

Statistical Mechanics · Physics 2024-03-01 Guoxing Lin , Shaokun Zheng

We consider an overdamped run-and-tumble particle in two dimensions, with self propulsion in an orientation that stochastically rotates by 90 degrees at a constant rate, clockwise or counter-clockwise with equal probabilities. In addition,…

Statistical Mechanics · Physics 2022-11-18 Naftali R. Smith , Pierre Le Doussal , Satya N. Majumdar , Gregory Schehr

We investigate the nonlocal behavior of passive tracer dispersion with random stopping at various sites in fluids. This kind of dispersion processes is modeled by an integral partial differential equation, i.e., an advection-diffusion…

Dynamical Systems · Mathematics 2025-10-20 Jinqiao Duan , James R. Brannan , H. Gao

We study the problem of a run and tumble particle in a harmonic trap, with a finite run and tumble time, by a direct integration of the equation of motion. An exact 1D steady state distribution, diagram laws and a programmable Volterra…

Statistical Mechanics · Physics 2025-04-17 Aoran Sun , Fangfu Ye , Rudolf Podgornik

We study a class of stochastic processes of the type $\frac{d^n x}{dt^n}= v_0\, \sigma(t)$ where $n>0$ is a positive integer and $\sigma(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to the other with a…

Statistical Mechanics · Physics 2021-01-27 David S. Dean , Satya N. Majumdar , Hendrik Schawe

The motion of self-propelled particles is modeled as a persistent random walk. An analytical framework is developed that allows the derivation of exact expressions for the time evolution of arbitrary moments of the persistent walk's…

Soft Condensed Matter · Physics 2015-07-28 Zeinab Sadjadi , M. Reza Shaebani , Heiko Rieger , Ludger Santen