31 research outputs found
Theory of Second and Higher Order Stochastic Processes
This paper presents a general approach to linear stochastic processes driven
by various random noises. Mathematically, such processes are described by
linear stochastic differential equations of arbitrary order (the simplest
non-trivial example is , where is not a Gaussian white
noise). The stochastic process is discretized into time-steps, all possible
realizations are summed up and the continuum limit is taken. This procedure
often yields closed form formulas for the joint probability distributions.
Completely worked out examples include all Gaussian random forces and a large
class of Markovian (non-Gaussian) forces. This approach is also useful for
deriving Fokker-Planck equations for the probability distribution functions.
This is worked out for Gaussian noises and for the Markovian dichotomous noise.Comment: 35 pages, PlainTex, accepted for publication in Phys Rev. E
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
Langevin equation for the squeezing of light by means of a parametric oscillator
We show that the Langevin equation for a nonlinear-optical system may be obtained directly from the Heisenberg equation of motion for the annihilation operators, provided a certain linearization procedure is valid. We apply the technique to the parametric oscillator used to generate squeezed light and compare our results to those obtained from Fokker-Planck-type equations. We argue that, only when the Wigner, as opposed to the P or Q, representation of quantum optics is used, do we get a correct description of the underlying stochastic process. We show how the linearization procedure may be carried out to describe the operation of the parametric oscillator both below threshold, where a squeezed vacuum state results, and above threshold, where we find a squeezed coherent state. In the region of the threshold a heuristic extension of the method leads to a possible description of the system by means of a nonlinear Langevin equation
Macroscopic quantum fluctuations in noise-sustained optical patterns
We investigate quantum effects in pattern formation for a degenerate optical parametric oscillator with walk-off. This device has a convective regime in which macroscopic patterns are both initiated and sustained by quantum noise. Familiar methods based on linearization about a pseudoclassical field fail in this regime and new approaches are required. We employ a method in which the pump field is treated as a c-number variable but is driven by the c-number representation of the quantum subharmonic signal field. This allows us to include the effects of the fluctuations in the signal on the pump, which in turn act back on the signal. We find that the nonclassical effects, in the form of squeezing, survive just above the threshold of the convective regime. Further, above threshold, the macroscopic quantum noise suppresses these effects
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
Mean first-passage times for systems driven by gamma and McFadden dichotomous noise
We consider mean first-passage times (MFPTs) for systems driven by non-Markov gamma and McFadden dichotomous noises. A simplified derivation is given of the underlying integral equations and the theory for ordinary renewal processes is extended to modified and equilibrium renewal processes. The exact results are compared with the MFPT for Markov dichotomous noise and with the results of Monte Carlo simulations
Mean exit times for free inertial stochastic processes
We study the mean exit time of a free inertial random process from a region in space. The acceleration alternatively takes the values +[ital a] and [minus][ital a] for random periods of time governed by a common distribution [psi]([ital t]). The mean exit time satisfies an integral equation that reduces to a partial differential equation if the random acceleration is Markovian. Some qualitative features of the behavior of the system are discussed and checked by simulations. Among these features, the most striking is the discontinuity of the mean exit time as a function of the initial conditions
Second-order processes driven by dichotomous noise
We study free second-order processes driven by dichotomous noise. We obtain an exact differential equation for the marginal density p(x,t) of the position. It is also found that both the velocity ¿(t) and the position X(t) are Gaussian random variables for large t
Free inertial processes driven by Gaussian noise: Probability distributions, anomalous diffusion and fractal behavior
We study the motion of an unbound particle under the influence of a random force modeled as Gaussian colored noise with an arbitrary correlation function. We derive exact equations for the joint and marginal probability density functions and find the associated solutions. We analyze in detail anomalous diffusion behaviors along with the fractal structure of the trajectories of the particle and explore possible connections between dynamical exponents of the variance and the fractal dimension of the trajectories