Nonlinear Fokker Research Articles

Collective behavior of coupled mesoscopic cylinders

AbstractWe address the objective of the generation of finite magnetic flux out of unbiased thermal current fluctuations in a collection of identical mesoscopic cylinders which are coupled via mutual inductances. The influence of thermal Nyquist fluctuations are described in terms of a set of Langevin equations or a corresponding Fokker–Planck equation, respectively. In the limit of infinitely many constituents, the steady‐state of the system is determined by an effective, nonlinear Fokker–Planck equation. The system exhibits in this thermodynamic limit a second‐order phase transition: the average flux through each cylinder changes continuously from zero to non‐zero value and the phase diagram depicts a critical line. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Well-Posedness of a Multiscale Model for Concentrated Suspensions

In a previous work [E. Cancès, I. Catto, and Y. Gati, SIAM J. Math. Anal., 37 (2005), pp. 60--82], three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles which are subjected to a given time-dependent shear rate. In the present work we extend the model to a more physically relevant situation where the shear rate actually depends on the macroscopic velocity of the fluid. As a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker--Planck-type equation with nonhomogeneous, nonlocal, and possibly degenerate coefficients. We show the existence and the uniqueness of the global-in-time weak solutionto such a system.

Estimating the nonextensivity of systems from experimental data: a nonlinear diffusion equation approach

We consider nonextensive systems that are related to the nonextensive entropy proposed by Tsallis and can be described by means of the nonlinear porous medium equation and the nonlinear Fokker–Planck equation proposed by Plastino and Plastino. We show how to determine the degree of nonextensivity of these systems from experimental data. Both transient and stationary cases are addressed.

Fluctuation–dissipation theorems for nonlinear Fokker–Planck equations of the Desai–Zwanzig type and Vlasov–Fokker–Planck equations

For many-body systems described by nonlinear Fokker–Planck equations and Vlasov–Fokker–Planck equations we determine the linear system response to small external driving forces. Thus, we derive fluctuation–dissipation theorems that relate dissipative properties of the perturbed systems to the second-order fluctuations of the unperturbed systems.

Dynamic mean field models: H-theorem for stochastic processes and basins of attraction of stationary processes

Stochastic processes of dynamic mean field models are studied in terms of nonlinear Fokker–Planck equations. We show that H-theorems for single time-point distributions can be used to derive H-theorems for hierarchies of joint distributions. In doing so, we prove the convergence of transient stochastic processes to stationary ones. For multistable systems we furthermore determine the basins of attraction of these stationary processes. Our results are illustrated by means of a mean field model exhibiting bistability.

Stability analysis of nonequilibrium mean field models by means of self-consistency equations

We consider systems that operator far from thermal equilibrium and can be described by means of nonlinear Fokker–Planck equations using mean field theory. We determine the stability of stationary states by means of Prigogine's Lyapunov functional and self-consistency equations.

Anomalous diffusion and anisotropic nonlinear Fokker–Planck equation

We analyse a N-dimensional anisotropic nonlinear Fokker–Planck equation by considering stationary and time-dependent solutions. The stationary solutions are obtained for very general situations, including those when the diffusion coefficients are spatial dependents. Time-dependent solutions are found in the absence of external force and with constant diffusion coefficients. When restricted to the bi-dimensional case, our investigation about time-dependent solutions focuses on situations where the diffusion coefficient are D x ∝| x| − θ x and D y ∝| y| − θ y with θ x,θ y∈ R . In general, we verify an anomalous behavior induced in a given direction due to the other directions.

Stability analysis of nonequilibrium mean field models by means of self-consistency equations

We consider systems that operator far from thermal equilibrium and can be described by means of nonlinear Fokker–Planck equations using mean field theory. We determine the stability of stationary states by means of Prigogine's Lyapunov functional and self-consistency equations.

Complete description of a generalized Ornstein?Uhlenbeck process related to the nonextensive Gaussian entropy

We consider a generalized Ornstein–Uhlenbeck process described by a nonlinear Fokker–Planck equation related to the one-parametric, nonextensive Gaussian entropy, which is a special case of the two-parametric Sharma–Mittal entropy. We derive the entire hierarchy of distribution functions for that process and, in doing so, derive a complete description of a stochastic process related to a nonextensive entropy measure.

Complete description of a generalized Ornstein–Uhlenbeck process related to the nonextensive Gaussian entropy

We consider a generalized Ornstein–Uhlenbeck process described by a nonlinear Fokker–Planck equation related to the one-parametric, nonextensive Gaussian entropy, which is a special case of the two-parametric Sharma–Mittal entropy. We derive the entire hierarchy of distribution functions for that process and, in doing so, derive a complete description of a stochastic process related to a nonextensive entropy measure.

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.

A procedure for obtaining general nonlinear Fokker–Planck equations

A procedure for deriving general nonlinear Fokker–Planck equations (FPEs) directly from the master equation is presented. The nonlinear effects are introduced in the transition probabilities, which present a dependence on the probabilities for finding the system in a given state. It is shown that the FPEs, obtained from master equations describing transitions among discrete and continuous sets of states, are identical. Within such a procedure, we construct nonlinear FPEs that appear to be very general. Our general FPEs recover, as particular cases, nonlinear FPEs investigated previously by many authors, introduced on a purely phenomenological basis, and they lead to the possibility of more complete and complex diffusive equations.

Influence of the non-linearity of the collision operator on ion cyclotron resonance heating

The distribution function of ions heated in the ion cyclotron range of frequencies is obtained by solving the Fokker–Planck equation. For non-minority heating scenarios, the full non-linear Coulomb collision operator must be used in this equation. A method of resolution accounting for the complexity of this operator is presented. We consider a plasma immersed in a homogeneous magnetic field. The adopted method of resolution is based on an expansion of the distribution function in a series of Legendre polynomials. Results of the corresponding code non-linear Fokker–Planck in two-dimensional velocity space (NLFP-2D) are discussed for experimentally relevant JET-like parameters. The convergence of the Legendre expansion has been tested and the importance of the non-linear effects has been shown. Three different regimes, characterized by different modes of indirect ion heating and depending on the absorbed RF power density as well as the minority concentration, were identified. The NLFP-2D code has been used to study the validity of the Maxwellian approximation of the self-collision operator. One finds that this model is only correct in a very limited range of parameters, and leads mostly to an overestimation of the indirect ion heating.

Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations

Abstract We discuss two fundamental aspects of Fokker–Planck equations that are nonlinear with respect to probability densities. First, we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions. Second, we demonstrate that nonlinear Fokker–Planck equations can be interpreted as linear Fokker–Planck equations describing nonlinear families of Markov diffusion processes. We exploit this finding in order to derive complete hierarchies of probability densities from nonlinear Fokker–Planck equations.

The optimal evolution of the free energy of interacting gases and its applications

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived. To cite this article: M. Agueh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Controlling the collective motion of interacting particles: analytical study via the nonlinear Fokker–Planck equation

We propose a nonlinear Fokker–Planck equation for the description of stochastic transport in systems of short-range interacting particles. We develop a perturbation scheme, valid for high frequencies, for particles on an asymmetric potential driven by a time-oscillating temperature. For a particular type of asymmetric potential, the net DC current shows two current inversions when increasing either the particle density or the interaction strength.

Erratum to: “Anomalous diffusion: nonlinear fractional Fokker–Planck equation”

Erratum to: “Anomalous diffusion: nonlinear fractional Fokker–Planck equation”

Tomography of solitons

We develop the tomographic representation of wavefunctions which are solutions of the generalized nonlinear Schrodinger equation (NLSE) and show its connection with the Weyl–Wigner map. The generalized NLSE is presented in the form of a nonlinear Fokker–Planck-type equation for the standard probability distribution function (tomogram). In particular, this theory is applied to solitons, where tomograms for envelope bright solitons of a family of modified NLSEs are presented and numerically evaluated. Examples of symplectic tomography and Fresnel tomography of linear and nonlinear signals are discussed.

Self-organized criticality within fractional Lorenz scheme

The theory of a flux steady-state related to avalanche formation is presented for the simplest model of a sand pile within the framework of the Lorenz approach. The stationary values of sand velocity and sand pile slope are derived as functions of a control parameter (driven sand pile slope). The additive noise of above values are introduced for building a phase diagram, where the noise intensities determine both avalanche and non-avalanche domains, as well as mixed one. Corresponding to the SOC regime, the last domain is crucial to affect of the noise intensity of the vertical component of sand velocity and especially sand pile slope. To address to a self-similar behavior, a fractional feedback is used as an efficient ingredient of the modified Lorenz system. In the spirit of Edwards paradigm, an effective thermodynamics is introduced to determine a distribution over an avalanche ensemble with negative temperature. Steady-state behavior of the moving grains number, as well as non-extensive values of entropy and energy is studied in detail. The power law distribution over the avalanche size is described within a fractional Lorenz scheme, where the energy noise plays a crucial role. This distribution is shown to be a solution of both fractional and nonlinear Fokker–Planck equation. As a result, we obtain new relations between the exponent of the size distribution, fractal dimension of phase space, characteristic exponent of multiplicative noise, number of governing equations, dynamical exponents and non-extensivity parameter.

Derivative pricing with non-linear Fokker–Planck dynamics

We examine how the Black–Scholes derivative pricing formula is modified when the underlying security obeys non-extensive statistics and Fokker–Planck dynamics. An unusual feature of such securities is that the volatility in the underlying Ito–Langevin equation depends implicitly on the actual market rate of return. This complicates most approaches to valuation. Here we show that progress is possible using variations of the Cox–Ross valuation technique.