generalized chapman kolmogorov equation a unifying approach to the description of anomalous transport in external fields the generalized chapman kolmogorov equation v m kenkre e w montroll and m f shlesinger j stat phys 9 45 1973 is discussed it is demonstrated that this equation unifies recently proposed kinetic equations of fractional order that describe anomalous transport in external fields as well as continuous time random walks the conditions under which the individual models can be established are discussed

fractional langevin model of memory in financial markets the separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after decades of statistical analysis of financial time series certain stylized facts have emerged including the statistics of stock price fluctuations having fat tails and their linear correlations in time being exceedingly short lived on the other hand the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law one explanation of this long term memory is that it is a consequence of the time scale separation between microscopic and macroscopic economic variables we propose a fractional langevin equation as a dynamical model of the observed memory in financial time series

fractional diffusion waiting time distributions and cattaneo type equations we discuss a generalized diffusion equation resulting from an additive two state process in combination with an asymptotically fractal asymptotic power law waiting time distribution the obtained equation is an extension to previously discussed fractional diffusion equations our description leads to a mean squared displacement which describes enhanced subballistic transport for long times the short time behavior however is of a ballistic nature this separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting time distribution this is also mirrored by the observation that for small times our generalized diffusion equation reduces to the standard cattaneo equation the asymptotic probability density is of compressed gaussian type and thus differs from the l evy tail generally found for these kinds of processes

explicit units and the equivariant tamagawa number conjecture let l k be a finite abelian extension of number fields of group g we study the tamagawa number t l k of h 0 spec l considered as a motive defined over k and with coefficients inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i g for a large class of extensions l k we interpret the conjectural vanishing of t l k in terms of the existence of s units in l satisfying a variety of explicit conditions these explicit conditions are in the same spirit as but are in general much finer than the conditions studied by rubin and popescu by using this approach we are able to complete the proof that t l k vanishes for all of the genus field extensions considered by frohlich in the course of proving this result we find that for certain extensions the vanishing of t l k is a refinement of the main result of solomon concerning wild euler systems

levy flights in quenched random force fields l evy flights characterized by the microscopic step index f are for fl2 the case of rare events considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f ensuremath 2 in the short range case and the critical fall off exponent 2f ensuremath 2 in the long range case by means of a dynamic renormalization group analysis based on the momentum shell integration method we determine flows fixed point and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient unlike the case of ordinary brownian motion in a quenched force field characterized by a single critical dimension or fall off exponent d 2 two critical dimensions appear in the l evy case a critical dimension or fall off exponent d f below which the diffusion coefficient exhibits anomalous scaling behavior i e algebraic spatial behavior and long time tails and a critical dimension or fall off exponent d 2f ensuremath 2 below which the force correlations characterized by a nontrivial fixed point become relevant as a general result we find in all cases that the dynamic exponent z characterizing the mean square displacement locks onto the l evy index f independent of dimension and independent of the presence of weak quenched disorder

solutions of a class of non markovian fokker planck equations we show that a formal solution of a rather general non markovian fokker planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the markovian equation with the same fokker planck operator this allows us to classify memory kernels into safe ones for which the solution is always a probability density and dangerous ones when this is not guaranteed the first situation describes random processes subordinated to a wiener process while the second one typically corresponds to random processes showing a strong ballistic component in this case the non markovian fokker planck equation is only valid in a restricted range of parameters initial and boundary conditions

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

linear response in complex systems ctrw and the fractional fokker planck equations abstract we consider the linear response of systems modelled by continuous time random walks ctrw and by fractional fokker planck equations under the influence of time dependent external fields we calculate the corresponding response functions explicitly the ctrw curve exhibits aging i e it is not translationally invariant in the time domain this is different from what happens under fractional fokker planck conditions

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

generalized theory of diffusive stresses associated with the time fractional diffusion equation and nonlocal constitutive equations for the stress tensor abstract the theory of diffusive stresses based on the time fractional diffusion equation with the liouville caputo derivative is discussed the diffusion process is characterized by the chemical potential tensor and the concentration tensor eliminating the chemical potential tensor and the concentration tensor from the constitutive equations for the stress tensor the space time nonlocal equations for the mean stress and the deviatoric stress are obtained with the kernel being the wright function

theories of thermal stresses based on space time fractional telegraph equations the generalized telegraph equations with time and space fractional derivatives are considered the corresponding theories of thermal stresses are formulated the proposed theories interpolate the classical thermoelasticity the theory of lord and shulman thermoelasticity without energy dissipation of green and naghdi and theories of fractional thermoelasticity proposed earlier the fundamental solution to the nonhomogeneous space time fractional telegraph equation as well as the corresponding thermal stresses are obtained in the axisymmetric case

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

systematic derivation of reaction diffusion equations with distributed delays and relations to fractional reaction diffusion equations and hyperbolic transport equations application to the theory of neolithic transition we introduce a general method for the systematic derivation of nonlinear reaction diffusion equations with distributed delays we study the interactions among different types of moving individuals atoms molecules quasiparticles biological organisms etc the motion of each species is described by the continuous time random walk theory analyzed in the literature for transport problems whereas the interactions among the species are described by a set of transformation rates which are nonlinear functions of the local concentrations of the different types of individuals we use the time interval between two jumps the transition time as an additional state variable and obtain a set of evolution equations which are local in time in order to make a connection with the transport models used in the literature we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so called generalized master equations the method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations hyperbolic reaction diffusion equations and delay differential reaction diffusion equations thus in the analysis of a given problem we can fit to the data the type of reaction diffusion equation and the corresponding physical and kinetic parameters the method is illustrated as a test case by the study of the neolithic transition we introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction diffusion equation for the population density we derive a delay evolution equation for the rate of advance of agriculture which illustrates an application of our analysis

tamagawa numbers for motives with noncommutative coefficients ii we provide corroborative evidence for the equivariant tamagawa number conjecture which was formulated in the first part of this article

stochastic processes with stable distributions in random environments the asymptotic behavior in random environments of random flights with stable distribution laws is analyzed by the field theoretic renormalization group random force fields with isotropic divergenceless curl free and unconstrained pair correlation functions with both finite and infinite correlation length are considered stability of the effective distribution laws in the scaling limit is determined and the scaling dimension of time is calculated in the ensuremath varepsilon mathit d mathit c d expansion where mathit d mathit c is the critical dimension of the model textcopyright 1996 the american physical society

competition between levy jumps and continuous drift motivated by certain birth death processes with strongly fluctuating birth rates we consider a level crossing problem for a random process being a superposition of a continuous drift to the left and jumps to the right the lengths of the corresponding jumps follow a one sided extreme levy law of index we concentrate on the case 0 1 and discuss the probability of crossing a left boundary extinction we show that this probability decays exponentially as a function of the initial distance to the boundary such behavior is universal for all 1 and is exemplified by an exact solution for the special case 12 the splitting probabilities are also discussed

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

dynamical surface structures in multiparticle correlated surface growths we investigate the scaling properties of the interface fluctuation width for the q mer and q particle correlated deposition evaporation models these models are constrained with a global conservation law that the particle number at each height is conserved modulo q in equilibrium the stationary roughness is anomalous but universal with the roughness exponent alpha 1 3 while the early time evolution shows nonuniversal behavior with the growth exponent beta varying with models and q nonequilibrium surfaces display diverse growing and stationary behaviors the q mer model shows a faceted structure while the q particle correlated model shows a macroscopically grooved structure

levy statistics in a hamiltonian system enhanced diffusion in a hamiltonian system is studied in terms of the continuous time random walk formulation for l evy walks the previous l evy walk scheme is extended i to include interruptions by periods of temporal localization and ii to describe motion in two dimensions we analyze a case of conservative motion in a two dimensional periodic potential numerical calculations of the mean squared displacements and the propagators for intermediate energies are consistent with the l evy walk description

levy flights in a steep potential well levy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial monomodal to a terminal bimodal distribution occurs chechkin et al phys rev e 67 010102 r 2003 in this paper we present a detailed study of levy flights in potentials of the type u x x c with c 2 apart from the bifurcation into bimodality we find the interesting result that for c 4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial condition and the building up of the two emerging side peaks which are characteristic for the stationary state thus for certain system parameters there exists a transient trimodal distribution of the levy flight these properties of levy flights in external potentials of the power law type can be represented by certain phase diagrams we also present details about the proof of multimodality and the numerical procedures to establish the probability distribution of the process

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

the fractional fokker planck equation on comb like model abstract the fractional fokker planck equation were used to describe the anomalous diffusion in external fields is derived using a comb like structure as a background model for the force free case the distribution function associated with space dependence diffusion coefficient along the backbone of the structure are obtained in a closed form of h function the operator method has been used to solve the fractional fokker planck equation taking the external field into account

inertial effects in anomalous dielectric relaxation of symmetrical top molecules the linear dielectric response of an assembly of noninteracting symmetrical top molecules each of which is free to rotate in space is evaluated in the context of fractional dynamics the infinite hierarchy of differentialrecurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional klein kramers equation on solving this hierarchy in terms of matrix continued fractions as in the normal rotational diffusion the complex dynamic susceptibility is obtained and is calculated for typical values of the model parameters for the limiting case of spherical top molecules the solution is obtained in terms of an ordinary continued fraction it is shown that the model can reproduce nonexponential anomalous dielectric relaxation behavior at low frequencies vt 1 where t is the debye relaxation time and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies in the far infrared region so that gordon s sum rule for integral dipolar absorption is satisfied

fractional diffusion exact representations of spectral functions for all the relevant transformed spaces i e fourier laplace and fourier laplace we present exact solutions of a fractional diffusion equation describing random transport on fractals the potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes in contrast to the well known asymptotic results the exact representations provide a much broader basis for comparison with data

on the remarkable spectrum of a non hermitian random matrix model a non hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum various attempts have been made to understand the spectrum but even its dimension is not known using the dyson schmidt equation we show that the spectrum consists of a non denumerable set of lines in the complex plane each line is the support of the spectrum of a periodic hamiltonian obtained by the infinite repetition of any finite sequence of the disorder variables our approach is based on the theory of words we make a complete study of all four letter words the spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe including this particular paper

a new class of long tailed pausing time densities for the ctrw we present some asymptotic results for the family of pausing time densities having the asymptotic t property t t ln1 t t 1 in particular we show that for this class of pausing time densities the mean squared displacement r2 t is asymptotically proportional to ln t t and the asymptotic distribution of the displacement has a negative exponential form

mean first passage time for anomalous diffusion when the random force acting on a particle diffusing in an interval 0 l and subjected to a constant external force is a gaussian white noise the brownian mean squared displacement is described by the seminal relation x 2 2 mathrm dt ensuremath gamma with ensuremath gamma 1 however for more complicated random forces the diffusion may be slower ensuremath gamma l1 subdiffusion or faster ensuremath gamma g1 superdiffusion than the normal diffusion for both these cases we calculated the mean free passage time mfpt the time needed to reach one of the traps at boundaries the simple formulas for the different diffusive regimes are compared quantitatively for the simplest case of the absence of an external field and for an initial position in the middle of the interval it turns out that the mfpt s for anomalous diffusion can be both larger or smaller than that for normal diffusion depending on the values of the length of the interval and the diffusion coefficient moreover the mfpt can show nonmonotonic changes with the degree of departure from normal diffusion

molecular switching with nonexponential relaxation patterns a random walk approach the transition from an initial locally stable configuration to a globally stable state in molecular switches is investigated in terms of a random walk model effectively taking the reaction pathway through a potentially rugged energy landscape into account exponential and nonexponential scenarios are discussed and the implications on measurable quantities are explored

anomalous diffusion in the nonasymptotic regime we analyze some properties of the one dimensional levy flights assuming that the one step transition rates depend on the flight length x as p alpha x equivalent to x alpha 2 for flights on a finite 2m 1 site lattice we can define an effective size dependent diffusion coefficient d alpha m equivalent to m 1 alpha 1 1 alpha if alpha are well defined provided that alpha r 3 these moments exhibit a power law singularity if alpha 1 and r 2 3 the short and intermediate time properties of the generalized mean square displacement are then studied numerically this work suggests the conditions under which the asymptotic analytical formulas obtained in the literature by the use of the generalized central limit theorem could be applied to finite time experiments these formulas should work much better if alpha is close to zero than in the alpha 1 neighborhood

even visiting random walks exact and asymptotic results in one dimension we reconsider the problem of even visiting random walks in one dimension this problem is mapped onto a non hermitian anderson model with binary disorder we develop very efficient numerical tools to enumerate and characterize even visiting walks the number of closed walks is obtained as an exact integer up to 1828 steps i e some 10535 walks on the analytical side the concepts and techniques of one dimensional disordered systems allow one to obtain explicit asymptotic estimates for the number of closed walks of 4k steps up to an absolute prefactor of the order of unity which is determined numerically all the cumulants of the maximum height reached by such walks are shown to grow as k1 3 with exactly known prefactors these results illustrate the tight relationship between even visiting walks trapping models and the lifshitz tails of disordered electron or phonon spectra

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

stable equilibrium based on levy statistics stochastic collision models approach we investigate equilibrium properties of two very different stochastic collision models i the rayleigh particle and ii the driven maxwell gas for both models the equilibrium velocity distribution is a levy distribution the maxwell distribution being a special case we show how these models are related to fractional kinetic equations our work demonstrates that a stable power law equilibrium which is independent of details of the underlying models is a natural generalization of maxwell s velocity distribution

universality of stretched gaussian asymptotic behaviour for the fractional fokker planck equation in external force fields we introduce a heterogeneous fractional fokker planck equation hffpe on heterogeneous fractal structure media describing systems involving external force fields the hffpe is shown to obey the generalized einstein relation and its stationary solution is the boltzmann distribution it is proven that the asymptotic shape of its solution is a stretched gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for constant and generic potentials with polar singularity at origin

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

the diffusion drift equation on comb like structure from the generalized scheme of random walks on the comb like structure one can obtain the fractional fokker planck equation in the domain of fractal time evolution with critical exponent 0 1 the operator method for the moments associated with the density p x t is used to solve the obtained equation it is shown that due to fingers diffusion has an anomalous character the first two moments increase sub linearly in time

continuous time random walks and traveling fronts we present a geometric approach to the problem of propagating fronts into an unstable state valid for an arbitrary continuous time random walk with a fisher kolmogorov petrovski piskunov growth reaction rate we derive an integral hamilton jacobi type equation for the action functional determining the position of reaction front and its speed our method does not rely on the explicit derivation of a differential equation for the density of particles in particular we obtain an explicit formula for the propagation speed for the case of anomalous transport involving non markovian random processes

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

aging properties of an anomalously diffusing particule we report new results about the two time dynamics of an anomalously diffusing classical particle as described by the generalized langevin equation with a frequency dependent noise and the associated friction the noise is defined by its spectral density proportional to 1 at low frequencies with 0 1 subdiffusion or 1 2 superdiffusion using laplace analysis we derive analytic expressions in terms of mittag leffler functions for the correlation functions of the velocity and of the displacement while the velocity thermalizes at large times slowly in contrast to the standard brownian motion case 1 the displacement never attains equilibrium it ages we thus show that this feature of normal diffusion is shared by a subdiffusive or superdiffusive motion we provide a closed form analytic expression for the fluctuation dissipation ratio characterizing aging

diffusion on random site percolation clusters theory and nmr microscopy experiments with model objects quasi two dimensional random site percolation model objects were fabricated based on computer generated templates samples consisting of two compartments a reservoir of mathrm h 2 mathrm o gel attached to a percolation model object which was initially filled with mathrm d 2 mathrm o were examined with nuclear magnetic resonance microscopy for rendering proton spin density maps the propagating proton deuteron interdiffusion profiles were recorded and evaluated with respect to anomalous diffusion parameters the deviation of the concentration profiles from those expected for unobstructed diffusion directly reflects the anomaly of the propagator for diffusion on a percolation cluster the fractal dimension of the random walk d w evaluated from the diffusion measurements on the one hand and the fractal dimension d f deduced from the spin density map of the percolation object on the other permits one to experimentally compare dynamical and static exponents approximate calculations of the propagator are given on the basis of the fractional diffusion equation furthermore the ordinary diffusion equation was solved numerically for the corresponding initial and boundary conditions for comparison the anomalous diffusion constant was evaluated and is compared to the brownian case some ad hoc correction of the propagator is shown to pay tribute to the finiteness of the system in this way anomalous solutions of the fractional diffusion equation could experimentally be verified

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

short time regime propagator in fractals the propagator p r t for fractals in the short time regime i e the probability of finding at distance r at time t a particle that diffuses in a fractal substrate when ensuremath xi r sqrt 2d t 1 d w ensuremath gg 1 is studied in order to elucidate its full functional form for finitely ramified fractals it is shown and for any other self similar media conjectured that the short time propagator is given by p r t ensuremath approx p 0 t ensuremath d s 2 ensuremath xi ensuremath alpha mathrm exp ensuremath c ensuremath xi ensuremath nu where ensuremath nu d w d w ensuremath 1 and ensuremath alpha ensuremath nu 2 ensuremath d f d f and d w being the fractal and random walk dimension of the medium respectively the value for ensuremath nu agrees with that generally accepted however our result for the as yet not well established value of ensuremath alpha differs from other recent proposals we have checked these various short time propagator proposals by comparing them to the short time propagator calculated numerically for the sierpinsky gasket our numerical results are precise enough to clearly support the validity of the short time propagator proposed here in particular the validity of our relation for ensuremath alpha and to rule out the others

bifurcation bimodality and finite variance in confined levy flights we investigate the statistical behavior of levy flights confined in a symmetric quartic potential well u x proportional variant x 4 at stationarity the probability density function features a distinct bimodal shape and decays with power law tails which are steep enough to give rise to a finite variance in contrast to free levy flights from a delta initial condition a bifurcation of the unimodal state is observed at t c 0 the nonlinear oscillator with potential u x ax 2 2 bx 4 4 a b 0 shows a crossover from unimodal to bimodal behavior at stationarity depending on the ratio a b

non hermitian tridiagonal random matrices and returns to the origin of a random walk we study a class of tridiagonal matrix models the q roots of unity models which includes the sign q 2 and the clock q models by feinberg and zee we find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides in the complex plane furthermore the averaged traces of mk are integers that count closed random walks on the line such that each site is visited a number of times multiple of q we obtain an explicit evaluation for them

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

fractional dynamics from the ordinary langevin equation we consider the usual langevin equation depending on an internal time this parameter is substituted by a first passage time of a self similar markov process then the gaussian process is parent and the hitting time process is directing the probability to find the resulting process at the real time is defined by the integral relationship between the probability densities of the parent and directing processes the corresponding master equation becomes the fractional fokker planck equation we show that the resulting process has non markovian properties all its moments are finite the fluctuation dissipation relation and the h theorem hold

on the first passage of one sided levy motions abstract we study the first passage problem for one sided levy motions how does such a motion started at the origin cross a barrier positioned at the point x x 0 since one sided levy motions are pure jump processes they always leap over barriers rather than crossing them continuously we hence explore the following issues i first passage times fpts how long would it take the motion to cross the barrier ii first passage leapovers fpls how far would the motion leap over the barrier and iii what is dependence between the fpts and the fpls formulae for the joint laplace transforms of the fpts and fpls are derived and the scaling limits of the fpts and fpls are computed the scaling limits turn out to display dramatically different behavior when i the underlying levy motion has finite cumulants and when ii the underlying levy motion is heavy tailed special attention is devoted to the investigation of the first passage problem for self similar one sided levy motions

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

enumeration of simple random walks and tridiagonal matrices we present some old and new results in the enumeration of random walks in one dimension mostly developed in work on enumerative combinatorics the relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial enumeration of paths it also seems promising for the theory of tridiagonal random matrices

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder dynamical generalizations of stretched exponential the asymptotic behavior of multichannel parallel relaxation processes for systems with dynamical disorder is investigated in the limit of a very large number of channels an individual channel is characterized by a state vector x which due to dynamical disorder is a random function of time a limit of the thermodynamic type in the x space is introduced for which both the volume available and the average number of channels tend to infinity but the average volume density of channels remains constant scaling arguments combined with a stochastic renormalization group approach lead to the identification of two different types of universal behavior of the relaxation function corresponding to nonintermittent and intermittent fluctuations respectively for nonintermittent fluctuations a dynamical generalization of the static huber s relaxation equation is derived which depends only on the average functional density of channels w t d w t the channels being classified according to their different relaxation rates w w t which are random functions of time for intermittent fluctuations a more complicated relaxation equation is derived which in addition to the average density of channels w t d w t depends also on a positive fractal exponent h which characterizes the fluctuations of the density of channels the general theory is applied for constructing dynamical analogs of the stretched exponential relaxation function for nonintermittent fluctuations the type of relaxation is determined by the regression dynamics of the fluctuations of the relaxation rate if the regression process is fast and described by an exponential attenuation function then after an initial stretched exponential behavior the relaxation process slows down and it is not fully completed even in the limit of very large times for self similar regression obeying a negative power law the relaxation process is less sensitive to the influence of dynamical disorder both for small and large times the relaxation process is described by stretched exponentials with the same fractal exponent as for systems with static disorder for large times the efficiency of the relaxation process is also slowed down by fluctuations similar patterns are found for intermittent fluctuations with the difference that for very large times and a slow regression process a crossover from a stretched exponential to a self similar algebraic relaxation function occurs some implications of the results for the study of relaxation processes in

anomalous diffusion stable processes and generalized functions the evolution equations in real space and time corresponding to a class of anomalous diffusion processes arc examined as special cases evolution equations corresponding to stable processes are derived using the theory of generalized functions recovering some known results differently interpreted and an evolution law for stable processes of order unity

fractional diffusion model for force distribution in static granular media we present the results of a numerical and an experimental investigation of the probability distribution of normal contact forces in static packs of particles with two different hardnesses force distributions are computed and compared with existing models and experimental data it is found that the probability distribution function of normal contact forces p f is well described by a semiempirical model derived from a fractional diffusion equation this model reproduces most of the features common to force distributions observed in experimental and numerical studies including the finite value for p f as the forces tend to zero the results indicate that the fractional model fits well both the numerical and experimental data over a wide range of particle deformations in contrast to the existing models these results provide an insight into the physics of granular media and complement previous findings

operators of fractional integration and their applications the main purpose of this paper is to present a systematic and historical account of the investigations carried out by various authors in the field of fractional calculus and its applications several interesting results relevant to the present investigation are also considered

levy random walks in finite systems l evy walks on finite intervals with absorbing boundaries are studied using analytic and monte carlo techniques the integral equations for l evy walks in infinite 1d systems are generalized to treat the evolution of the probability distribution on finite and semi infinite intervals in particular the near boundary behavior of the probability distribution and also its properties at asymptotically large times are studied the probability distribution is found to be discontinuous near the boundary for l evy walks in finite and semi infinite systems previous results for infinite systems and a previous scaling for semi infinite systems are reproduced the use of linear operator theory to solve the integral equations governing the evolution of the l evy walk implies that the probability distribution decays exponentially at large times for a jump distribution that satisfies ensuremath psi x ensuremath sim x ensuremath ensuremath alpha for large x the decay constant for the exponential decay is estimated and found to scale at large l as l 1 ensuremath ensuremath alpha for 2l ensuremath alpha l3 and l ensuremath 1 for 1l ensuremath alpha l2 in contrast to l ensuremath 2 for normal diffusion for 2l ensuremath alpha l3 the ratio of the decay constants of the first and second eigenfunctions is less than 4 for large l so that the second eigenfunction is relatively more important in describing the system s large time behavior than the corresponding eigenfunction for normal diffusion for 1l ensuremath alpha l2 the ratio of the decay constants may be greater or less than 4 the shapes of the eigenfunctions for the l evy processes are obtained numerically and the strong similarity between the first eigenfunction and its normal diffusion counterpart for 2 ensuremath lesssim ensuremath alpha l3 indicate that it would be difficult experimentally to distinguish such a l evy process on a finite interval from a normal diffusive system by considering only the asymptotic shape of the probability distribution for ensuremath alpha ensuremath lesssim 2 we observe significant differences between the first and second eigenfunctions and their normal diffusion counterparts for moderately large intervals the first eigenfunction is flatter with large boundary discontinuities while the second eigenfunction can differ from its normal diffusion counterpart in both its symmetry properties and number of nodes

on multiplicative galois structure invariants introduction let n k be a finite galois extension of number fields of group g let ko z g denote the grothendieck group of finitely generated projective left 7 g modules with cl z g the subgroup of elements which have rank 0 in c1 chinburg used the four term exact sequences of prescribed extension class introduced by tate in t2 pp 56 57 to define an invariant qm n k e cl z g of the multiplicative galois structure of n the class qm n k is also often denoted q n k 3 this invariant can be described in terms of local and global weil groups and has been conjectured by chinburg to equal the cassou nogues frohlich root number class c2 there is by now a considerable amount of evidence to support this conjecture cf d 4 the conjecture is in part motivated by tate s re formulation of the stark conjecture and specifically his construction of a natural generalization of the classical dirichlet unit regulator t2 more recently frohlich has defined an integral version of the stark tate regulator and by integrally reinterpreting much of the tate approach has studied elements in natural quotients of cl z g which explicitly relate the galois structure of unit lattices and the regular part of ideal class groups cf f3 f4 d 3 in this paper we shall show that the results of fr6hlich represent approximations to the conjecture of chinburg for example we show that the fine structure results of f4 are for the special classes of abelian extensions considered there the best approximations to a proof of chinburg s conjecture which can be obtained without analysis of extension class data moreover for certain classes of field extensions the relevant cohomological questions can be resolved and so we shall obtain a full proof of the chinburg conjecture for these extensions in this way the dependence of the conjecture on the role of the canonical cohomology classes of tate in defining qm n k is made very clear also taken in conjunction with recent work of snaith sn on the additive chinburg invariant q n k 2 a special case of our results provides an infinite family of wildly ramified abelian extensions for which all of the conjectures of chinburg are now completely verified cf remark 1 11 ii

coburn s lemma and the finite section method for random jacobi operators abstract we study the spectra and pseudospectra of semi infinite and bi infinite tridiagonal random matrices and their finite principal submatrices in the case where each of the three diagonals varies over a separate compact set say u v w c such matrices are sometimes termed stochastic toeplitz matrices a in the semi infinite case and stochastic laurent matrices a in the bi infinite case their spectra spec a and spec a are independent of a and a as long as a and a are pseudoergodic in the sense of davies 2001 20 which holds almost surely in the random case this was shown in davies 2001 20 for a that the same holds for a is one main result of this paper although the computation of and in terms of u v and w is intrinsically difficult we give upper and lower spectral bounds and we explicitly compute a set g that fills the gap between and in the sense that g we also show that the invertibility of one and hence all operators a implies the invertibility and uniform boundedness of the inverses of all finite tridiagonal square matrices with diagonals varying over u v and w this implies that the so called finite section method for the approximate solution of a system a x b is applicable as soon as a is invertible and that the finite section method for estimating the spectrum of a does not suffer from spectral pollution both results illustrate that tridiagonal stochastic toeplitz operators share important properties of classical toeplitz operators indeed one of our main tools is a new stochastic version of the coburn lemma for classical toeplitz operators saying that a stochastic tridiagonal toeplitz operator if fredholm is always injective or surjective in the final part of the paper we bound and compare the norms and the norms of inverses of bi infinite semi infinite and finite tridiagonal matrices over u v and w this in particular allows the study of the resolvent norms and hence the pseudospectra of these operators and matrices

levy scaling the diffusion entropy analysis applied to dna sequences we address the problem of the statistical analysis of a time series generated by complex dynamics with the diffusion entropy analysis dea n scafetta p hamilton and p grigolini fractals 9 193 2001 this method is based on the evaluation of the shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations rather than on the measurement of the variance of this diffusion process as done with the traditional methods we compare the dea to the traditional methods of scaling detection and prove that the dea is the only method that always yields the correct scaling value if the scaling condition applies furthermore dea detects the real scaling of a time series without requiring any form of detrending we show that the joint use of dea and variance method allows to assess whether a time series is characterized by levy or gauss statistics we apply the dea to the study of dna sequences and prove that their large time scales are characterized by levy statistics regardless of whether they are coding or noncoding sequences we show that the dea is a reliable technique and at the same time we use it to confirm the validity of the dynamic approach to the dna sequences proposed in earlier work

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

levy walk approach to anomalous diffusion abstract the transport properties of levy walks are discussed in the framework of continuous time random walks ctrw with coupled memories this type of walks may lead to anomalous diffusion where the mean squared displacement r2 t t with 1 we focus on the enhanced diffusion limit 1 in one dimension and present our results on r2 t the mean number of distinct sites visited s t and p r t the probability of being at position r at time t

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

growing surfaces with anomalous diffusion results for the fractal kardar parisi zhang equation in this paper i study a model for a growing surface in the presence of anomalous diffusion also known as the fractal kardar parisi zhang equation fkpz this equation includes a fractional laplacian that accounts for the possibility that surface transport is caused by a hopping mechanism of a levy flight it is shown that for a specific choice of parameters of the fkpz equation the equation can be solved exactly in one dimension so that all the critical exponents which describe the surface that grows under fkpz can be derived for that case afterwards the self consistent expansion sce is used to predict the critical exponents for the fkpz model for any choice of the parameters and any spatial dimension it is then verified that the results obtained using sce recover the exact result in one dimension at the end a simple picture for the behavior of the fractal kpz equation is suggested and the upper critical dimension of this model is discussed

self similar gaussian processes for modeling anomalous diffusion we study some gaussian models for anomalous diffusion which include the time rescaled brownian motion two types of fractional brownian motion and models associated with fractional brownian motion based on the generalized langevin equation gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean square displacement to vary with t ensuremath alpha 0l ensuremath alpha l2 however these processes have different properties thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism although the two versions of fractional brownian motion and time rescaled brownian motion all have the same probability distribution function the slepian theorem can be used to compare their first passage time distributions which are different finally in order to model anomalous diffusion with a variable exponent ensuremath alpha t it is necessary to consider the multifractional extensions of these gaussian processes

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function