From continuous time random walks to the fractional fokker-planck equation

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.     

Generalized Einstein relation and the Metzler-Klafter conjecture in a composite-subdiffusive regime

In this paper, a generalized diffusion model driven by the composite-subdiffusive fractional Brownian motion (FBM) is employed. Based on this stochastic process, we derive a fractional Fokker-Planck equation (FFPE) and obtain its solution. It is proved that the Generalized Einstein Relation (GER) and the Metzler and Klafter conjecture on the asymptotic behavior of stretched Gaussian hold the FFPE in a composite-subdiffusive regime.     

Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces

We derive a fractional Fokker-Planck equation for subdiffusion in a general space- and time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.     

Exact solutions of the Fokker-Planck equations with moving boundaries

By means of time-dependent similarity transformations, we derive exact solutions of the Fokker-Planck equations with moving boundaries in the presence of: (1) a time-dependent linear force and (2) a time-dependent nonlinear force. The method of similarity transformation is simple and can be easily applied to more general Fokker-Planck equations. Furthermore, the knowledge of the exact solutions in closed form can be useful as a benchmark to test approximate numerical or analytical procedures.     

Exact propagator of the Fokker-Planck equation with a space-dependent diffusion coefficient and a time-dependent mean-reverting force

We have investigated the algebraic structure of the Fokker-Planck equation with a variable diffusion coefficient and a time-dependent mean-reverting force. Such a model could be useful to study the general problem of a Brownian walker with a space-dependent diffusion coefficient. We also show that this model is related to the Fokker-Planck equation with a constant diffusion coefficient and a time-dependent anharmonic potential of the form V(x, t) = ?a(t)x ² + b ln x, which has been widely applied to model different physical and biological phenomena, e.g. the study of neuron models and stochastic resonance in monostable nonlinear oscillators. Using the Lie algebraic approach we have derived the exact diffusion propagators for the Fokker-Planck equations associated with different boundary conditions, namely (i) the case of a single absorbing barrier, and (ii) the case of two absorbing barriers. These exact diffusion propagators enable us to study the time evolution of the corresponding stochastic systems. Received 23 October 2001 and Received in final form 24 December 2001     

Fractional Fokker-Planck equation for fractal media

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.     

The importance of the fluctuating force for dissipative tunneling,examplified in a time-dependent picture

We exploit a time-dependent picture to compare the penetrability obtained from a multidimensional Schrödinger equation with the one from a Fokker-Planck equation. The result for the latter one is manifestly different. The origin of this difference is studied and traced back to the fluctuating force which accompanies dissipation, and which is present for open systems.     

Separation of suspended particles by arrays of obstacles in microfluidic devices

The stochastic transport of suspended particles through a periodic pattern of obstacles in microfluidic devices is investigated by means of the Fokker-Planck equation and numerical simulations. Asymmetric arrays of obstacles have been shown to induce the continuous separation of DNA molecules, with particles of different size migrating in different directions within the microdevice (vector separation). We show that the separation of tracer particles only occurs in the presence of a permeating driving force with a nonzero normal component at the surface of the solid obstacles, and arises from differences in the local Peclet number of the particles. On the other hand, finite-size particles also exhibit nonzero, but small, migration angles in the case of nonpermeating fields. Monte Carlo simulations for different driving fields agree with the solutions to the Fokker-Planck equation.     

Weber electrodynamics,part I. general theory,steady current effects

The original Weber action at a distance theory, valid for slowly varying effects, is extended to time-retarded fields, valid for rapidly varying effects including radiation. A new law for the force on a charge moving in this field is derived (replacing the Lorentz force which violates Newton's third law). The limitations of the Maxwell theory are discussed. The Weber theory, in addition to predicting all of the usual electrodynamic results, predicts the following crucial results for slowly varying effects (where Maxwell theory fails): 1) the force on Ampere's bridge in agreement with the measurements of Moyssides and Pappas, 2) the tension required to rupture current carrying wires as observed by Graneau, 3) the force to drive the Graneau-Hering submarine, 4) the force to drive the mercury in Hering's pump, and 5) the force to drive the oscillations in a current carrying mercury wedge as observed by Phipps.     

Distribution function for classical and quantum systems far from thermal equilibrium

We consider a system composed of many subsystems which are coupled to individual reservoirs at different temperatures. We show how the solution of a many-dimensional Fokker-Planck equation may be reduced to a Fokker-Planck equation of dimensionn, wheren is the number of relevant constants of motion. We treat also a Fokker-Planck equation with continuously many variables and the time-dependent one. The usefulness of the present procedure to determine explicitly distribution functions is exhibited by several examples. If all temperatures are equal the Boltzman distribution function is obtained as a special case. Using the method of quantum-classical correspondence, the distribution function for quantum systems may be found.     

Diffusive energy growth in classical and quantum driven oscillators

We study the long-time stability of oscillators driven by time-dependent forces originating from dynamical systems with varying degrees of randomness. The asymptotic energy growth is related to ergodic properties of the dynamical system: when the autocorrelation of the force decays sufficiently fast one typically obtains linear diffusive growth of the energy. For a system with good mixing properties we obtain a stronger result in the form of a central limit theorem. If the autocorrelation decays slowly or does not decay, the behavior can depend on subtle properties of the particular model. We study this dependence in detail for a family of quasiperiodic forces. The solution involves the analysis of a small-denominator problem that can be treated by fairly elementary methods. In the special case of a periodic force the quantum stability problem can be expressed in terms of spectral properties of the Floquet operator. In the presence of resonances the spectrum is absolutely continuous. We find explicitly the eigenvalues and eigenfunctions for the nonresonant case.     

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.     

Subordinated diffusion and continuous time random walk asymptotics

Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Le?vy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit.     

Kinetic equation for a high intensity monomode laser

We generalize the Fokker-Planck equation for the laser radiation field distribution. Our kinetic equation is valid for a solid-state laser in the case of arbitrary coupling strength but low concentration in active material.     

长时相干效应与分形布朗运动

 我们研究了阻尼布朗粒子，在具有幂律长时相干C(t)~t^-β（0<β<1，1<β<2）的无规涨落力作用下的运动情况。我们发现它是作分形布朗运动，而不是作普通的布朗运动，而且，找出了分形布朗运动的有效Fokker-Planck方程，以及相应的精确解。于是第一次把长时相干效应和分形布朗运动建立了定量的联系。     

Dynamics of a three-level atom in a resonant light field

The paper is devoted to a consideration of the motion of a three-level atom in two resonant light waves. A kinetic equation of the Fokker-Planck type for the atomic distribution function is derived, which is valid when the recoil energy is small compared to the linewidths of the resonant transitions. The detailed behaviour of the radiation force and the diffusion tensor are studied numerically. The case of exact resonance and the nonresonant case are both considered. It is shown that a detuning from exact resonance results in a drastic decrease of the resonant light pressure force. For the detuning we determine the condition, under which an efficient action of the light pressure on a three-level atom takes place.     

Lie algebraic approach for Fokker-Planck dynamics with space-dependent diffusion and mean-reverting drift

Using the Lie algebraic approach we have derived the exact diffusion propagator of the Fokker-Planck equation with a time-dependent variable diffusion coefficient and a time-dependent mean-reverting force between two absorbing boundaries. The exact diffusion propagator not only enables us to study the time evolution of the corresponding stochastic system, but the knowledge of the propagator can also provide a benchmark for testing approximate numerical or analytical procedures. Furthermore, the Lie algebraic method is very simple and could be easily extended to the more general Fokker-Planck equations with well-defined algebraic structures. Received 18 December 2002 / Received in final form 3 March 2003 Published online 24 April 2003     

Fractional generalized Langevin equation approach to single-file diffusion

Fractional generalized Langevin equation with external force is used to model single-file diffusion. It is found that for external force that varies with power law the solution for such a fractional Langevin equation gives the correct short and long time behavior for the mean square displacement of single-file diffusion when appropriate choice of parameters associated with fractional generalized Langevin equation are used. By considering some special cases of the fractional generalized Langevin equation, a new class of closed analytic expressions for the mean square displacement of single-file diffusion can be obtained. The effective Fokker-Planck equation associated with single-file diffusion is briefly considered.