Long-time position distribution of an active Brownian particle in two dimensions

Basu, Urna; Majumdar, Satya N.; Rosso, Alberto; Schehr, Grégory

doi:10.1103/physreve.100.062116

Cited by 72 publications

(83 citation statements)

References 64 publications

(105 reference statements)

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“…Moreover, the probability law of configurations of a system under resetting is related to the probability law of the ordinary system (without resetting) by an integral equation, obtained from a renewal argument [20,29,[33][34][35][36]. In particular, the probability law at steady state of the system under resetting can be worked out from the Laplace transform of the probability law at steady state of the ordinary system.…”

Section: Introductionmentioning

confidence: 99%

Non-conserving zero-range processes with extensive rates under resetting

Grange

2020

J. Phys. Commun.

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We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and vanish with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the meanfield geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model can be mapped to population dynamics (if the creation rates are reproductive fitnesses in a haploid population, and the hopping rate is the mutation rate). It can also be mapped to a Bianconi-Barabási model of a growing network with random rewiring of links (if creation rates are the rates of acquisition of links by nodes, and the hopping rate is the rewiring rate). The steady state has recently been worked out and gives rise to occupation numbers that reproduce Kingman's house-of-cards model of selection and mutation. In this paper we solve the master equation using a functional method, which yields integral equations satisfied by the occupation numbers. The occupation numbers are shown to forget initial conditions at an exponential rate that decreases linearly with the fitness level. Moreover, they can be computed exactly in the Laplace domain, which allows to obtain the steady state of the system under resetting. The result modifies the house-of-cards result by simply adding a skewed version of the initial conditions, and by adding the resetting rate to the hopping rate.

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Section: Introductionmentioning

confidence: 99%

Non-conserving zero-range processes with extensive rates under resetting

Grange

2020

J. Phys. Commun.

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“…which reflects the bimodality of the density distribution 55, [71][72][73][74][75] (see also Refs. 54,76 for experimental studies) as a distinct feature compared to the Gaussian shape of the AOUP solution.…”

Section: Parental Active Modelmentioning

confidence: 99%

The Parental Active Model: a unifying stochastic description of self-propulsion

Caprini,

Sprenger,

Löwen

et al. 2022

Preprint

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We propose a new overarching model for self-propelled particles that flexibly generates a full family of "descendants". The general dynamics introduced in this paper, which we denote as "parental" active model (PAM), unifies two special cases commonly used to describe active matter, namely active Brownian particles (ABPs) and active Ornstein-Uhlenbeck particles (AOUPs). We thereby document the existence of a deep and close stochastic relationship between them, resulting in the subtle balance between fluctuations in the magnitude and direction of the self-propulsion velocity. Besides illustrating the relation between these two common models, the PAM can generate additional offspring, interpolating between ABP and AOUP dynamics, that could provide more suitable models for a large class of living and inanimate active matter systems, possessing characteristic distributions of their self-propulsion velocity. Our general model is evaluated in the presence of a harmonic external confinement. For this reference example, we present a two-state phase diagram which sheds light on the transition in the shape of the positional density distribution, from a unimodal Gaussian for AOUPs to a Mexican-hat-like profile for ABPs.

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“…In this context, studies of simple models have been crucial. They displayed several ballistic-diffusive crossovers, non-Boltzmann steady-state, localization away from potential minima, and associated re-entrant transition for steadystate properties of trapped particles [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Self-propulsion with speed and orientation fluctuation: exact computation of moments and dynamical bistabilities in displacement

Shee,

Chaudhuri

2021

Preprint

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We consider the influence of active speed fluctuations on the dynamics of a d-dimensional active Brownian particle performing a persistent stochastic motion. We use the Laplace transform of the Fokker-Planck equation to obtain exact expressions for time-dependent dynamical moments. Our results agree with direct numerical simulations and show several dynamical crossovers determined by the activity, persistence, and speed fluctuation. The persistence in the motion leads to anisotropy, with the parallel component of displacement fluctuation showing sub-diffusive behavior and non-monotonic variation. The kurtosis remains positive at short times determined by the speed fluctuation, crossing over to a negative minimum at intermediate times governed by the persistence before vanishing asymptotically. The probability distribution of particle displacement obtained from numerical simulations in two-dimension shows two crossovers between contracted and expanded trajectories via two bimodal distributions at intervening times. While the speed fluctuation dominates the first crossover, the second crossover is controlled by persistence like in the worm-like chain model of semiflexible polymers.

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Long-time position distribution of an active Brownian particle in two dimensions

Cited by 72 publications

References 64 publications

Non-conserving zero-range processes with extensive rates under resetting

Non-conserving zero-range processes with extensive rates under resetting

The Parental Active Model: a unifying stochastic description of self-propulsion

Self-propulsion with speed and orientation fluctuation: exact computation of moments and dynamical bistabilities in displacement

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