Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations

Consequences of the connection between nonlinear Fokker-Planck equations and entropic forms are investigated. A particular emphasis is given to the feature that different nonlinear Fokker-Planck equations can be arranged into classes associated with the same entropic form and its corresponding stationary state. Through numerical integration, the time evolution of the solution of nonlinear Fokker-Planck equations related to the Boltzmann-Gibbs and Tsallis entropies are analyzed. The time behavior in both stages, in a time much smaller than the one required for reaching the stationary state, as well as towards the relaxation to the stationary state, are of particular interest. In the former case, by using the concept of classes of nonlinear Fokker-Planck equations, a rich variety of physical behavior may be found, with some curious situations, like an anomalous diffusion within the class related to the Boltzmann-Gibbs entropy, as well as a normal diffusion within the class of equations related to Tsallis’ entropy. In addition to that, the relaxation towards the stationary state may present a behavior different from most of the systems studied in the literature.     

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.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.     

A Stochastic Differential Equation Model with Jumps for Fractional Advection and Dispersion

The path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation (SDE) driven by Brownian noise. This model may not be adequate for highly heterogeneous media. This paper extends the model to a general SDE driven by a Lévy noise. Particle paths follow a Markov process with long jumps. Their transition probability density solves a forward equation derived here via pseudo-differential operator theory and Fourier analysis. In particular, the SDE with stable driving noise has a space-fractional advection-dispersion equation (fADE) with variable coefficients as the forward equation. This result provides a stochastic solution to anomalous diffusion models, and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Matrix approach to discrete fractional calculus II: Partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny’s matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359–386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.     

Skellam Type Processes of Order k and Beyond

In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Skellam process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, Lévy measures, governing difference-differential equations of the introduced processes. Our results generalize the Skellam process and running average of Poisson process in several directions.     

A general nonlinear Fokker-Planck equation and its associated entropy

A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equation, comes out as a very general tool to describe phenomenologically systems presenting complex behavior, like anomalous diffusion, in the presence of external forces. Such an equation is characterized by a nonlinear diffusion term that may present, in general, two distinct powers of the probability distribution. Herein, we calculate the stationary-state distributions of this equation in some special cases, and introduce associated classes of generalized entropies in order to satisfy the H-theorem. Within this approach, the parameters associated with the transition rates of the original master-equation are related to such generalized entropies, and are shown to obey some restrictions. Some particular cases are discussed.     

Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation

We geometrically study the Legendre duality relation that plays an important role in statistical physics with the standard or generalized entropies. For this purpose, we introduce dualistic structure defined by information geometry, and discuss concepts arising in generalized thermostatistics, such as relative entropies, escort distributions and modified expectations. Further, a possible generalization of these concepts in a certain direction is also considered. Finally, as an application of such a geometric viewpoint, we briefly demonstrate several new results on the behavior of the solution to a nonlinear diffusion equation called the porous medium equation.     

On the way towards a generalized entropy maximization procedure

We propose a generalized entropy maximization procedure, which takes into account the generalized averaging procedures and information gain definitions underlying the generalized entropies. This novel generalized procedure is then applied to Rényi and Tsallis entropies. The generalized entropy maximization procedure for Rényi entropies results in the exponential stationary distribution asymptotically for q∈(0,1] in contrast to the stationary distribution of the inverse power law obtained through the ordinary entropy maximization procedure. Another result of the generalized entropy maximization procedure is that one can naturally obtain all the possible stationary distributions associated with the Tsallis entropies by employing either ordinary or q-generalized Fourier transforms in the averaging procedure.     

The Clock Paradox in a Static Homogeneous Gravitational Field1

The gedanken experiment of the clock paradox is solved exactly using the general relativistic equations for a static homogeneous gravitational field. We demonstrate that the general and special relativistic clock paradox solutions are identical and in particular that they are identical for finite acceleration. Practical expressions are obtained for proper time and coordinate time by using the destination distance as the key observable parameter. This solution provides a formal demonstration of the identity between the special and general relativistic clock paradox with finite acceleration and where proper time is assumed to be the same in both formalisms. By solving the equations of motion for a freely falling clock in a static homogeneous field elapsed times are calculated for realistic journeys to the stars. ¹ Both authors contributed equally to this paper.     

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.     

The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems. In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form. Received 26 February 1999     

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.     

Finite difference approximations for the fractional advection-diffusion equation

Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection-diffusion equation, the fractional Crank-Nicholson method (FCN) is given, which is based on shifted Grünwald-Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order ²(Δt). A numerical example is presented to confirm the conclusions.     

Nonlinear diffusion under a time dependent external force: q-maximum entropy solutions

Nonlinear diffusion equations provide useful models for a number of interesting phenomena, such as diffusion processes in porous media. We study here a family of nonlinear Fokker-Planck equations endowed both with a power-law nonlinear diffusion term and a drift term with a time dependent force linear in the spatial variable. We show that these partial differential equations exhibit exact time dependent particular solutions of the Tsallis maximum entropy (q-MaxEnt) form. These results constitute generalizations of previous ones recently discussed in the literature [C. Tsallis, D.J. Bukman, Phys. Rev. E 54, R2197 (1996)], concerning q-MaxEnt solutions to nonlinear Fokker-Planck equations with linear, time independent drift forces. We also show that the present formalism can be used to generate approximate q-MaxEnt solutions for nonlinear Fokker-Planck equations with time independent drift forces characterized by a general spatial dependence. Received 25 April 2001 and Received in final form 6 June 2001     

Path integral solution of linear second order partial differential equations I: the general construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.     

Analytical approach for the fractional differential equations by using the extended tanh method

In this study, we consider analytical solutions of space–time fractional derivative foam drainage equation, the nonlinear Korteweg–de Vries equation with time and space-fractional derivatives and time-fractional reaction–diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann–Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.     

Local order, entropy and predictability of financial time series

We consider time series of financial data as the Dow Jones Index with respect to the existence of local order. The basic idea is that in spite of the high stochasticity in average there might be special local situations where there local order exist and the predictability is considerably higher than in average. In order to check this assumption we discretise the time series and investigate the frequency of the continuation of definite words of length n first. We prove the existence of relatively long-range correlations under special conditions. The higher order Shannon entropies and the conditional entropies (dynamical entropies) are calculated, characteristic fluctuations are found. Instead of the dynamic entropies which yield mean values of the uncertainty/predictability we finally investigate the local values of the uncertainty/predictability and the distribution of these quantities. Received 19 January 2000     

Implementation of three reliable methods for finding the exact solutions of (2 + 1) dimensional generalized fractional evolution equations

In this study, we have implemented the three methods namely extended \((G^{\prime}/G)\)-expansion, extended \((1/G^{\prime})\)-expansion and \((G^{\prime}/G,\,\,1/G)\)-expansion methods to determine exact solutions for the (2 + 1) dimensional generalized time–space fractional differential equations. We use Conformable fractional derivative and its properties in this research to convert fractional differential equations to ordinary differential equations with integer order. By using above mentioned methods, three types of traveling wave solutions are successfully obtained which have been expressed by the hyperbolic, trigonometric, and rational function solutions. The considered methods and transformation techniques are efficient and consistent for solving nonlinear time and space-fractional differential equations.