Results for a fractional diffusion equation with a nonlocal term in spherical symmetry

We obtain time dependent solutions for a fractional diffusion equation containing a nonlocal term by considering the spherical symmetry and using the Green function approach. The nonlocal term incorporated in the diffusion equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution. In addition, a rich class of anomalous diffusion processes may be described from the results obtained here.     

A model for solid state diffusion

Classical self-diffusion is discussed in the context of a model single particle Hamiltonian containing a periodic potential and a stochastic time dependent potential. Solutions are given in the Gaussian white noise approximation in terms of a Fokker-Planck equation. The effects of spatial dependence in the time dependent potential are illustrated by calculations of the frequency dependent mobility in a simple one dimension example.     

Solutions for a non-Markovian diffusion equation

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.     

Chaotic diffusion for particles moving in a time dependent potential well

The chaotic diffusion for particles moving in a time dependent potential well is described by using two different procedures: (i) via direct evolution of the mapping describing the dynamics and; (ii) by the solution of the diffusion equation. The dynamic of the diffusing particles is made by the use of a two dimensional, nonlinear area preserving map for the variables energy and time. The phase space of the system is mixed containing both chaos, periodic regions and invariant spanning curves limiting the diffusion of the chaotic particles. The chaotic evolution for an ensemble of particles is treated as random particles motion and hence described by the diffusion equation. The boundary conditions impose that the particles can not cross the invariant spanning curves, serving as upper boundary for the diffusion, nor the lowest energy domain that is the energy the particles escape from the time moving potential well. The diffusion coefficient is determined via the equation of the mapping while the analytical solution of the diffusion equation gives the probability to find a given particle with a certain energy at a specific time. The momenta of the probability describe qualitatively the behavior of the average energy obtained by numerical simulation, which is investigated either as a function of the time as well as some of the control parameters of the problem.     

The time dependent Dirac-Frenkel-McLachlan variation of parameters: an NMR application

The time dependent variation of parameters solution to the time dependent Schr?dinger equation pioneered by Dirac, Frenkel, and McLachlan is described in terms useful for immediate application to complex time dependent problems in magnetic resonance. A benchmark comparison of the theory to one dimensional images and spin echo envelope signals in simple spatially varying magnetic fields with molecular diffusion is provided.     

Comparison between fixed and Gaussian steplength in Monte Carlo simulations for diffusion processes

We analyze the different degrees of accuracy of two Monte Carlo methods for the simulation of one-dimensional diffusion processes with homogeneous or spatial dependent diffusion coefficient that we assume correctly described by a differential equation. The methods analyzed correspond to fixed and Gaussian steplengths. For a homogeneous diffusion coefficient it is known that the Gaussian steplength generates exact results at fixed time steps Δt. For spatial dependent diffusion coefficients the symmetric character of the Gaussian distribution introduces an error that increases with time. As an example, we consider a diffusion coefficient with constant gradient and show that the error is not present for fixed steplength with appropriate asymmetric jump probabilities.     

Time dependent Ginzburg-Landau equations for strong coupling superconductors

We derive time dependent Ginzburg-Landau equations for strong coupling superconductors. It is shown that due to a certain separability of the order parameter the equation for it’s time dependent fluctuations is again of diffusion type. Strong coupling effects show up only in the numerical coefficients of the diffusion equation. We apply our findings to the problem of electrical resistivity in strong coupling superconducting materials above the transition temperatureT _c.     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.``     

The renewal equation for persistent diffusion

Persistent diffusion in one dimension, in which the velocity of the diffusing particle is a dichotomic Markov process, is considered. The flow is non-Markovian, but the position and the velocity together constitute a Markovian diffusion process. We solve the coupled forward Kolmogorov equations and the coupled backward Kolmogorov equations with appropriate initial conditions, to establish a generalized (matrix) form of the renewal equation connecting the probability densities and first passage time distributions for persistent diffusion.     

On the quantum mechanical description of scattering by a dissipative and diffusive scattering centre

A partial integro-differential equation is formulated for the Wigner transform of the quantum mechanical reduced density operator describing the time evolution of a “macroscopic” coordinate under the influence of coupling to a large number of “intrinsic” degrees of freedom. The equation contains integral operators which lead to energy dissipation and diffusion and reduces to a transport equation of the Fokker-Planck type if the form factors in the integrands are treated in appropriate (harmonic) approximations. The stationary solution of the partial integro-differential equation is obtained numerically for scattering by a conservative potential and by a dissipative and diffusive scattering centre in one spatial dimension.     

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     

Some remarks on the time-dependent Thomas-Fermi theory

We investigate the Thomas Fermi limit (?=0) of the density operator in phase space for a system of noninteracting fermions evolving from the ground state in a time dependent potential. The semiclassical calculations for model situations in one spatial dimension are compared with the solution of the time dependent Schrödinger equation. The role of time dependent invariants and the relation to the hydrodynamical formulation of quantum mechanics is pointed out.     

Hopping on a zig-zag course

We study the diffusion coefficient of Active Brownian particles in two dimensions. In addition to usual attributes of active motion we let the particles turn in preferred directions over random times. This angular motion is modeled by an effective Lorentz force with time dependent frequency switching between two values at exponentially distributed random times. The diffusion coefficient is calculated by the Taylor-Kubo formula where distributions found from a Fokker-Planck equation or from a continuous time random walk approach have been inserted for averaging. Eventually properties of the diffusion coefficient will be discussed.     

A generalized simple random walk in one dimension related to the Gaussian polynomials

A generalization of the relation between the simple random walk on a regular lattice and the diffusion equation in a continuous space is described. In one dimension we consider a random walk of a walker with exponentially decreasing mobility with respect to time. It has an exact solution of the conditional probability, that is expressed in terms of the Gaussian polynomials, a generalization of binomial coefficients. Taking a suitable continuum limit we obtain the corresponding transport equation from the recursion relation of the discrete random walk process. The kernel of this differential equation is also directly obtained from that conditional probability by the same continuum limit.     

First passage time statistics of Brownian motion with purely time dependent drift and diffusion

Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on time and are proportional to each other. First passage time statistics, such as the survival probabilities and first passage time densities are obtained analytically. The analysis includes the study of different functional forms of the time dependent drift and diffusion, including power-law time dependence and different periodic drivers. As a case study of these theoretical results, a stochastic model of water resources availability in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated.     

间歇湍流的分数阶动力学

间歇湍流意味着湍流涡旋并不充满空间,其维数介于2和3之间.湍流扩散为超扩散,且概率密度分布具有长尾特征.本文将流体力学的Navier-Stokes(NS)方程中的黏性项用分数阶的拉普拉斯算子表达.分析表明,分数阶拉普拉斯的阶数α和间歇湍流的维数D相联系.对于均匀各向同性的Kolmogorov湍流α=2,即用整数阶NS方程描述.而对于间歇性湍流,一定用分数阶的NS方程来描述.对于Kolmogorov湍流,扩散方差正比于t3,即Richardson扩散.而对于间歇性湍流,扩散方差要比Richardson扩散更强.     

Diffusion and reaction in random media and models of evolution processes

A diffusion equation including source terms, representing randomly distributed sources and sinks is considered. For quasilinear growth rates the eigenvalue problem is equivalent to that of the quantum mechanical motion of electrons in random fields. Correspondingly there exist localized and extended density distributions dependent on the statistics of the random field and on the dimension of the space. Besides applications in physics (nonequilibrium processes in pumped disordered solid materials) a new evolution model is discussed which considers evolution as hill climbing in a random landscape.We dedicate this work to the memory of Ilya M. Lifshitz.