T.D. Frank

Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.

Generalized multivariate Fokker-Planck equations derived from kinetic transport theory and linear nonequilibrium thermodynamics

Abstract We study many particle systems in the context of mean field forces, concentration-dependent diffusion coefficients, generalized equilibrium distributions, and quantum statistics. Using kinetic transport theory and linear nonequilibrium thermodynamics we derive for these systems a generalized multivariate Fokker–Planck equation. It is shown that this Fokker–Planck equation describes relaxation processes, has stationary maximum entropy distributions, can have multiple stationary solutions and stationary solutions that differ from Boltzmann distributions.

Measuring the interaction law of dissipative solitons

We investigate the interaction of self-organized solitary current filaments (dissipative solitons) under the influence of noise in a planar dc gas-discharge system with high-ohmic barrier. These localized structures exhibit interesting properties such as propagation and scattering as well as the transient formation of molecules and clusters. To quantitatively examine these interesting properties, methods of stochastic data analysis are developed to determine interaction laws from experimentally recorded data series. It turns out that in most cases, the sign of the resulting interaction functions alternates with the distance of the dissipative solitons. This phenomenon can be connected to experimentally detectable oscillatorily decaying tails of the filaments, which is in agreement with theoretical predictions.

Classical Langevin equations for the free electron gas and blackbody radiation

Among others, Uhling and Uhlenbeck, Kaniadakis and Quarati and Kadanoff have suggested to describe the evolution of quantum systems exhibiting Fermi–Dirac and Bose–Einstein statistics by means of classical but nonlinear evolution equations for density measures such as generalized Boltzmann equations and nonlinear Fokker–Planck equations. We use this approach in order to derive classical Langevin equations for quantum systems and apply the Langevin equations thus obtained to two fundamental quantum systems, namely, the free electron gas and blackbody radiation.

Drift bifurcation detection for dissipative solitons

We report on the experimental detection of a drift bifurcation for dissipative solitons, which we observe in the form of current filaments in a planar semiconductor–gas-discharge system. By introducing a new stochastic data analysis technique we find that due to a change of system parameters the dissipative solitons undergo a transition from purely noise-driven objects with Brownian motion to particles with a dynamically stabilized finite velocity.

On reducible nonlinear time-delayed stochastic systems : fluctuation-dissipation relations, transitions to bistability, and secondary transitions to non-stationarity

We show the conditions under which nonlinear time-delayed dynamical systems with multiplicative noise sources can be transformed into linear time-delayed systems with additive noise sources. We show that, for such reducible systems, analytical expressions for stationary distributions can be obtained. We demonstrate that fluctuation-dissipation relations of reducible systems become trivial and we show that reducible systems may exhibit delay- and noise-induced transitions to bistability and secondary transitions to non-stationarity. Our general findings are exemplified for three models: a Gompertz model, a Hongler model and a model involving a 1 - x 2 noise amplitude.

A note on the Markov property of stochastic processes described by nonlinear Fokker–Planck equations

We study the Markov property of processes described by generalized Fokker–Planck equations that are nonlinear with respect to probability densities such as mean field Fokker–Planck equations and Fokker–Planck equations related to generalized thermostatistics. We show that their transient solutions describe non-Markov processes. In contrast, stationary solutions can describe Markov processes. As a result, nonlinear Fokker–Planck equations can be used to model transient non-Markov processes that converge to stationary Markov processes.

Variability, Covariation, and Invariance with Respect to Coordinate Systems in Motor Control: Reply to Smeets and Louw (2007).

In their comment on the tolerance-noise covariation (TNC) method for decomposing variability by H. Müller and D. Sternad (2003, 2004b), J. B. J. Smeets and S. Louw show that covariation (C), as defined within the TNC method, is not invariant with respect to coordinate transformations and contend that it is, therefore, meaningless. Although the observation is correct, their interpretation is misleading in the following ways: (a) They equate covariation C with the known statistical quantity covariance and noise (N) with standard deviations. The two quantities C and N are conceptually different statistical measures. (b) Dependency on the reference frame is not only a feature of C but of all 3 components. However, such dependency is ubiquitous in motor control. (c) As the frame of reference in biological systems is poorly understood, the TNC method may afford evaluation of different coordinates for control.

Modelling the stochastic single particle dynamics of relativistic fermions and bosons using nonlinear drift-diffusion equations

Recently, there is a considerable interest in quantum kinetic models of fermions and bosons given by generalised Boltzmann equations and their corresponding diffusion approximations: nonlinear drift-diffusion equations. In this context, we propose a model that describes the dynamical properties of relativistic fermions and bosons consistent with the Fermi-Dirac and Bose-Einstein statistics. In particular, we discuss the first- and second-order statistics of quantum particle trajectories in energy space.

Stationary distributions of stochastic processes described by a linear neutral delay differential equation

Stationary distributions of processes are derived that involve a time delay and are defined by a linear stochastic neutral delay differential equation. The distributions are Gaussian distributions. The variances of the Gaussian distributions are either monotonically increasing or decreasing functions of the time delays. The variances become infinite when fixed points of corresponding deterministic processes become unstable.