Fokker-Planck Type Equation Research Articles

WELL-POSEDNESS OF FOKKER–PLANCK TYPE EQUATIONS ON THE WIENER SPACE

We consider Fokker–Planck type equations on the abstract Wiener space. Under the assumptions that the coefficients have a certain Sobolev regularity and they, together with their gradients and divergences, are exponentially integrable, we establish the existence of solutions to these equations, based on the estimates for solutions to Fokker–Planck equations in the finite-dimensional case. Moreover, the solution is unique if it belongs to the first-order Sobolev space.

Information entropy production for steady-state electrochemical systems

In this work, we develop non-equilibrium information theory formalism for the investigation of the steady-state heterogeneous electron transfer. In order to analyze the information transfer, we introduce the concept of information potential and information affinity. These new functions allow us to describe the transformation from the initial to final state by means of Fokker–Planck type equation. Accordingly, the transformation can be viewed as a diffusion process within the phase space. Also, we are able to determine the information entropy production. We show that the Shannon information entropy production is equal to zero for a reversible system. However, the change in the relative entropy or Kullback–Leibler measure of information distance is positive for an irreversible system. In all cases, we find that a non-negative information entropy production is always observed during the experiments. Here, we demonstrate the self-consistency of the non-equilibrium formalism in the information theory for studying steady-state electrochemical systems.

Phi-entropy inequalities for diffusion semigroups

We obtain and study new Φ-entropy inequalities for diffusion semigroups, with Poincaré or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker–Planck type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The Γ 2 criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.

Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders

We propose a mathematical model for opinion formation in a society that is built of two groups, one group of ‘ordinary’ people and one group of ‘strong opinion leaders’. Our approach is based on an opinion formation model introduced in Toscani (Toscani 2006 Commun. Math. Sci. 4 , 481–496) and borrows ideas from the kinetic theory of mixtures of rarefied gases. Starting from microscopic interactions among individuals, we arrive at a macroscopic description of the opinion formation process that is characterized by a system of Fokker–Planck-type equations. We discuss the steady states of this system, extend it to incorporate emergence and decline of opinion leaders and present numerical results.

Fermi–Dirac–Fokker–Planck equation: Well-posedness & long-time asymptotics

A Fokker–Planck type equation for interacting particles with exclusion principle is analyzed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in L 1 are characterized by the Fermi–Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropy-dissipation arguments for initial data controlled by Fermi–Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi–Dirac equilibrium is shown to converge to zero without decay rate.

Non-Markovian renormalization of kinetic coefficients for drift-type turbulence in magnetized plasmas

The problem of derivation of the kinetic equations for inhomogeneous plasma in an external magnetic field is considered. The Fokker–Planck-type equations with the non-Markovian kinetic coefficients are proposed. In the time-local limit (small correlation times with respect to the distribution function relaxation time) the relations obtained recover the results known from the appropriate quasilinear theory and the Dupree–Weinstock theory of plasma turbulence. Kinetic calculations of the dielectric response function are also performed with regard to the influence of turbulent fields on particle motion. The equations proposed are used to describe zonal flow generation and to estimate the diffusion coefficient for saturated turbulence.

Analytical and numerical study on the nonequilibrium relaxation by the simplified Fokker–Planck equation

Nonequilibrium relaxation by the simplified Fokker–Planck equation is studied by calculating the relaxation of Grad’s moments. From calculated relaxation of moments, we propose two simplified Fokker–Planck-type equations. For the quantitative validation of nonequilibrium relaxation by using the simplified Fokker–Planck-type equations, we solve a rarefied shock layer problem. To study the stronger diffusion of the distribution function than the simplified Fokker–Planck-type equation, we consider diffusion of the distribution function due to the gain term of the Boltzmann equation, because the Boltzmann equation can be described by a partial differential equation with higher-order terms than the simplified Fokker–Planck equation. The effect of the gain term of the Boltzmann equation is discussed based on comparisons with our proposed Bhatnagar–Gross–Krook equation with velocity-dependent collision frequency.

A Boltzmann-like equation for choice formation

We describe here a possible approach to the formation of choice in a society bymethods borrowed from the kinetic theory of rarefied gases. It is shown thatthe evolution of the continuous density of opinions obeys a linear Boltzmannequation where the background density represents the fixed distribution ofpossible choices. The binary interactions between individuals are in generalnon-local, and take into account both the compromise propensity and theself-thinking. In particular regimes, the linear Boltzmann equation is welldescribed by a Fokker-Planck type equation, for which in some cases the steadystates (distribution of choices) can be obtained in analytical form. ThisFokker-Planck type equation generalizes analogous one obtained by mean fieldapproximation of the voter model in [27]. Numerical examples illustratethe influence of different model parameters in the description both of theshape of the distribution of choices, and in its mean value.

Boltzmann and Fokker-Planck Equations Modelling Opinion Formation in the Presence of Strong Leaders

We propose a mathematical model for opinion formation in a society which is built of two groups, one group of 'ordinary' people and one group of 'strong opinion leaders'. Our approach is based on an opinion formation model introduced in Toscani (2006) and borrows ideas from the kinetic theory of mixtures of rarefied gases. Starting from microscopic interactions among individuals, we arrive at a macroscopic description of the opinion formation process which is characterized by a system of Fokker-Planck type equations. We discuss the steady states of this system, extend it to incorporate emergence and decline of opinion leaders, and present numerical results.

International and Domestic Trading and Wealth Distribution

We introduce and discuss a kinetic model for wealth distribution in a simple market economy which is built of a number of countries or social groups. Our approach is based on the model with risky investments introduced by Cordier, Pareschi and Toscani and borrows ideas from the kinetic theory of mixtures of rarefied gases. Wealth is exchanged by individuals inside these countries (domestic trade) as well as in between different countries (international trade). Under a suitable scaling we derive a system of Fokker-Planck type equations and discuss its extension to a two-dimensional model with distributed trading propensity. Theoretical and numerical results for two groups show that the wealth distribution develops a bimodal (and in general, a polymodal) shape.

Existence and Uniqueness of Solutions to Fokker–Planck Type Equations with Irregular Coefficients

We study the existence and the uniqueness of the solution to a class of Fokker–Planck type equations with irregular coefficients, more precisely with coefficients in Sobolev spaces W 1, p . Our arguments are based upon the DiPerna–Lions theory of renormalized solutions to linear transport equations and related equations [5]. The present work extends the results of our previous article [14], where only the simpler case of a Fokker–Planck equation with constant diffusion matrix was addressed. The consequences of the present results on the well-posedness of the associated stochastic differential equations are only outlined here. They will be more thoroughly examined in a forthcoming work [15].

Theoretical study of sub-to-picosecond laser pulse interaction with dielectrics, semiconductors and semiconductor heterostructures

A summary of theoretical investigations of short pulse (τL ≤ 1 ps) laser interaction with dielectrics, semiconductors and semiconductor heterostructures is presented. The time-dependent kinetic Fokker-Planck type equation for excited conduction electrons is used to describe this interaction in SiO2 and is systematically derived for GaAs. The energy spectra of the electron distribution function and the time dependence of the electron density are presented to illustrate the role of the different physical processes responsible for the conduction electron dynamics. Some possible types of laser induced damage in dielectrics and semiconductors including optical, electrical and structural damage are explored. In addition, the photon-absorption-induced intrasubband and intersubband phonon scattering of conduction electrons in GaAs/AlGaAs quantum wells are predicted by using the Boltzmann equation in the presence of a normally incident mid-IR pulsed laser field.

Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior

This paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results.

International and domestic trading and wealth distribution

We introduce and discuss a kinetic model for wealth distribution in a simple market economy which is built of a number of countries or social groups. Our approach is based on the model with risky investments introduced by Cordier, Pareschi, and Toscani in On a kinetic model for a simple market economy, and borrows ideas from the kinetic theory of mixtures of rarefied gases. Wealth is exchanged by individuals inside these countries (domestic trade) as well as in between different countries (international trade). Under a suitable scaling we derive a system of Fokker-Planck type equations and discuss its extension to a two-dimensional model with distributed trading propensity. Theoretical and numerical results for two groups show that the wealth distribution develops a bimodal (and in general, a polymodal) shape.

Fokker-Planck-type equations for a simple gas and for a semirelativistic Brownian motion from a relativistic kinetic theory

A covariant Fokker-Planck-type equation for a simple gas and an equation for the Brownian motion are derived from a relativistic kinetic theory based on the Boltzmann equation. For the simple gas the dynamic friction four-vector and the diffusion tensor are identified and written in terms of integrals which take into account the collision processes. In the case of Brownian motion, the Brownian particles are considered as nonrelativistic, whereas the background gas behaves as a relativistic gas. A general expression for the semirelativistic viscous friction coefficient is obtained and the particular case of constant differential cross section is analyzed for which the nonrelativistic and ultrarelativistic limiting cases are calculated.

Variation of fiber orientation in turbulent flow inside a planar contraction with different shapes

In this study, we have investigated the influence of shape of planar contractions on the orientation distribution of stiff fibers suspended in turbulent flow. To do this, we have employed a model for the orientational diffusion coefficient based on the data obtained by high-speed imaging of suspension flow at the centerline of a contraction with flat walls. This orientational diffusion coefficient depends only on the contraction ratio and turbulence intensity. Our measurements show that the turbulence intensity decays exponentially independent of the contraction angle. This implies that the turbulence variation in the contraction is independent of the shape, consistent with the results by the rapid distortion theory and the experimental results of axisymmetric contractions. In order to determine the orientation anisotropy, we have solved a Fokker–Planck type equation governing the orientation distribution of fibers in turbulent flow. Although the turbulence variation and the orientational diffusion are independent of the contraction shape, the results show that the variation of the orientation anisotropy is dependent on shape. This can be explained by the variation of the rotational Péclet number, Pe r, inside the contractions. This quantity is a measure of the importance of the mean rate of the strain relative to the orientational diffusion. We have shown that when Pe r < 10 turbulence can significantly influence the evolution of the orientation anisotropy. Since in contractions with identical inlet conditions the streamwise position where Pe r = 10 depends on the shape, the orientation anisotropy is dependent on the variation of rate of strain in a given contraction. We demonstrate the shape effect by considering contraction with flat walls as well as three contractions with different mean rate of strain variation.

Asymptotic behavior of a fractional Fokker–Planck-type equation

It is proved that the asymptotic shape of the solution for a wide class of fractional Fokker–Planck-type equations with coefficients depending on coordinate and time is a stretched Gaussian for the initial condition being pulse function in the homogeneous and heterogeneous fractal structures, whose mean square displacement behaves like 〈 ( Δ x ) 2 ( t ) 〉 ∼ t γ and 〈 ( Δ x ) 2 ( t ) 〉 ∼ x - θ t γ ( 0 < γ < 1 ,- ∞ < θ < + ∞ ) , respectively.

Potential symmetry and invariant solutions of Fokker-Planck equation in cylindrical coordinates related to magnetic field diffusion in magnetohydrodynamics including the Hall current

Lie groups involving potential symmetries are applied in connection with the system of magnetohydrodynamic equations for incompressible matter with Ohm's law for finite resistivity and Hall current in cylindrical geometry. Some simplifications allow to obtain a Fokker-Planck type equation. Invariant solutions are obtained involving the effects of time-dependent flow and the Hall-current. Some interesting side results of this approach are new exact solutions that do not seem to have been reported in the literature.

Multiscale simulations for suspensions of rod-like molecules

We study the Doi model for suspensions of rod-like molecules. The Doi model couples a microscopic Fokker–Planck type equation (Smoluchowski equation) to the macroscopic Stokes equation. The Smoluchowski equation describes the evolution of the distribution of the rod orientations; it comes as a drift–diffusion equation on the sphere at every point in physical space.For sufficiently high macroscopic shear rates (high Deborah numbers), the solution of the coupled system develops internal layers in the macroscopic strain rate (the spurt phenomenon).In the high Deborah numbers regime, the drift term in the Smoluchowski equation is dominant. We thus introduce a finite-volume type discretization of the microscopic Smoluchowski equation which is motivated by transport-dominated PDEs.We carry out direct numerical simulations of the spurt phenomenon both in the dilute and concentrated regimes. Below the isotropic–nematic transition, the solution structure is identical to the one described by purely macroscopic models (JSO model). For higher concentrations, we observe the formation of microstructure coming from a position-dependent tumbling rate. We also investigate the 2-d stability of the spurted solution.

On the orientation of stiff fibres suspended in turbulent flow in a planar contraction

The influence of turbulence on the orientation state of a dilute density matched suspension of stiff fibres at high Reynolds number in a planar contraction is investigated. High-speed imaging and laser-Doppler velocimetry techniques are used to quantify fibre orientation distribution and turbulent characteristics. A nearly homogeneous isotropic grid-generated turbulent flow is introduced at the contraction inlet. Flow Reynolds number and inlet turbulent characteristics are varied in order to determine their effects on orientation distribution. The orientation anisotropy is shown to be accurately modelled by a Fokker-Planck type equation. Results show that rotational diffusion is highly influenced by inlet turbulent characteristics and decays exponentially with convergence ratio. Furthermore, the effect of turbulent energy production in the contraction is shown to be negligible. Also, the results show that the flow Reynolds number has negligible effect on the development of orientation anisotropy, and the influence of turbulence on fibre rotation is negligible for rotational Péclet number $\gt$10.