18 research outputs found
The Fokker-Planck equation, and stationary densities
The most general local Markovian stochastic model is investigated, for which
it is known that the evolution equation is the Fokker-Planck equation. Special
cases are investigated where uncorrelated initial states remain uncorrelated.
Finally, stochastic one-dimensional fields with local interactions are studied
that have kink-solutions.Comment: 10 page
Collective motion of active Brownian particles in one dimension
We analyze a model of active Brownian particles with non-linear friction and
velocity coupling in one spatial dimension. The model exhibits two modes of
motion observed in biological swarms: A disordered phase with vanishing mean
velocity and an ordered phase with finite mean velocity. Starting from the
microscopic Langevin equations, we derive mean-field equations of the
collective dynamics. We identify the fixed points of the mean-field equations
corresponding to the two modes and analyze their stability with respect to the
model parameters. Finally, we compare our analytical findings with numerical
simulations of the microscopic model.Comment: submitted to Eur. Phys J. Special Topic
Active Brownian Particles. From Individual to Collective Stochastic Dynamics
We review theoretical models of individual motility as well as collective
dynamics and pattern formation of active particles. We focus on simple models
of active dynamics with a particular emphasis on nonlinear and stochastic
dynamics of such self-propelled entities in the framework of statistical
mechanics. Examples of such active units in complex physico-chemical and
biological systems are chemically powered nano-rods, localized patterns in
reaction-diffusion system, motile cells or macroscopic animals. Based on the
description of individual motion of point-like active particles by stochastic
differential equations, we discuss different velocity-dependent friction
functions, the impact of various types of fluctuations and calculate
characteristic observables such as stationary velocity distributions or
diffusion coefficients. Finally, we consider not only the free and confined
individual active dynamics but also different types of interaction between
active particles. The resulting collective dynamical behavior of large
assemblies and aggregates of active units is discussed and an overview over
some recent results on spatiotemporal pattern formation in such systems is
given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte
Harmonic oscillators driven by colored noise: crossovers, resonances and spectra
We study second-order properties of linear oscillators driven by exponentially correlated noise. We focus our attention on dynamical exponents and crossovers and also on resonance phenomena that appear when the driving noise is dichotomous. We also obtain the power spectrum and show its different behaviors according to the color of the noise