Analytic Solution of an Active Brownian Particle in a Harmonic Well

Abstract

We provide an analytical solution for the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle trapped in an isotropic harmonic potential. Using the passive Brownian particle as basis states we show that the Fokker-Planck operator becomes lower diagonal, implying that the eigenvalues are unaffected by the activity. The propagator is then expressed as a combination of the equilibrium eigenstates with weights obeying exact iterative relations. We show that for the low-order correlation functions, such as the positional autocorrelation function, the recursion terminates at finite order in the Péclet number allowing us to generate exact compact expressions and derive the velocity autocorrelation function and the time-dependent diffusion coefficient. The nonmonotonic behavior of latter quantities serves as a fingerprint of the non-equilibrium dynamics.

11pages, 3 figures

References in corpus (18)

• Novel type of phase transition in a system of self-driven particles
• The hydrodynamics of swimming microorganisms
• Self-motile colloidal particles: from directed propulsion to random walk
• Physics of Microswimmers - Single Particle Motion and Collective Behavior
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Sedimentation, trapping, and rectification of dilute bacteria
• Dynamics of a Brownian circle swimmer
• Dynamic clustering and chemotactic collapse of self-phoretic active particles
• How does a flexible chain of active particles swell?
• Activated escape of a self-propelled particle from a metastable state
• Intermediate scattering function of an anisotropic active Brownian particle
• Active escape dynamics: the effect of persistence on barrier crossing
• Active Brownian particles: Entropy production and fluctuation-response
• Active Brownian filaments with hydrodynamic interactions: conformations and dynamics
• Intermediate scattering function of an anisotropic Brownian circle swimmer
• Time-Dependent Fluctuations and Superdiffusivity in the Driven Lattice Lorentz Gas
• Optimal navigation strategy of active Brownian particles in target-search problems

Cited by in corpus (7)

• Exact position distribution of a harmonically-confined run-and-tumble particle in two dimensions
• Adaptive active Brownian particles searching for targets of unknown positions
• Transition-path sampling for Run-and-Tumble particles
• Two-Dimensional Active Brownian Particles Crossing a Parabolic Barrier: Transition-Path Times, Survival Probability, and First-Passage Time
• Free chiral self-propelled robots compared to active Brownian circle swimmers
• When velocity autocorrelations mirror force autocorrelations: Exact noise-cancellation in interacting Brownian systems
• Density-Independent transient caging in the high-density phase of motility-induced phase separation