polymer melt dynamics microscopic roots of fractional viscoelasticity the rheological properties of polymer melts and other complex macromolecular fluids are often successfully modeled by phenomenological constitutive equations containing fractional differential operators we suggest a molecular basis for such fractional equations in terms of the generalized langevin equation gle that underlies the renormalized rouse model developed by schweizer j chem phys 91 5802 1989 the gle describes the dynamics of the segments of a tagged chain under the action of random forces originating in the fast fluctuations of the surrounding polymer matrix by representing these random forces as fractional gaussian noise and transforming the gle into an equivalent diffusion equation for the density of the tagged chain segments we obtain an analytical expression for the dynamic shear relaxation modulus g t which we then show decays as a power law in time this power law relaxation is the root of fractional viscoelastic behavior

universality of high dimensional spanning forests and sandpiles we prove that the wired uniform spanning forest exhibits mean field behaviour on a very large class of graphs including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph several of our results are new even in the case of mathbb z d d ge 5 in particular we prove that every tree in the forest has spectral dimension 4 3 and walk dimension 3 almost surely and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1 2 respectively the past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest we obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth and is 1 on any bounded degree nonamenable graph we deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the abelian sandpile model are 2 and 1 2 respectively for any transitive graph with polynomial growth of dimension at least five and are 1 and 1 2 respectively for any bounded degree nonamenable graph in the case of mathbb z d d ge 5 some of our results regarding critical exponents recover earlier results of bhupatiraju et al electron j probab 22 85 51 2017 in this case we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant order multiplicative errors rather than the polylogarithmic order multiplicative errors present in that work

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

diffusion in heterogeneous media diffusion processes in heterogeneous media and biological systems in particular are riddled with the difficult theoretical issue of whether the true origin of anomalous behavior is renewal or memory or a special combination of the two accounting for the possible mixture of renewal and memory sources of subdiffusion is challenging from a computational point of view as well this problem is exacerbated by the limited number of techniques available for solving fractional diffusion equations with time dependent coefficients we propose an iterative scheme for solving fractional differential equations with time dependent coefficients that is based on a parametric expansion in the fractional index we demonstrate how this method can be used to predict the long time behavior of nonautonomous fractional differential equations by studying the anomalous diffusion process arising from a mixture of renewal and memory sources

on a generalization of mittag leffler function and its properties abstract let s and z be complex variables s the gamma function and s s s for any complex the generalized pochhammer symbol the principal aim of the paper is to investigate the function e q z n 0 q n n z n n where c re 0 re 0 re 0 and q 0 1 n this is a generalization of the exponential function exp z the confluent hypergeometric function z the mittag leffler function e z the wiman s function e z and the function e z defined by prabhakar for the function e q z its various properties including usual differentiation and integration laplace transforms euler beta transforms mellin transforms whittaker transforms generalised hypergeometric series form mellin barnes integral representation with their several special cases are obtained and its relationship with laguerre polynomials fox h function and wright hypergeometric function is also established

stochastic interpolation of sparsely sampled time series via multipoint fractional brownian bridges we propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and or temporal scales to this end we extend the notion of a fractional brownian bridge defined as fractional brownian motion with a given scaling hurst exponent h and with prescribed start and end points to a bridge process with an arbitrary number of intermediate and nonequidistant points determining the optimal value of the hurst exponent h opt appropriate to interpolate the sparse signal is a very important step of our method we demonstrate the validity of our method on a signal from fluid turbulence in a high reynolds number flow and discuss the implications of the non self similar character of the signal the method introduced here could be instrumental in several physical problems including astrophysics particle tracking and specific tailoring of surrogate data as well as in domains of natural and social sciences

symplectic integration of constrained hamiltonian systems by composition methods recent work reported in the literature suggests that for the long term integration of hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow in this paper the symplecticity of numerical integrators is investigated for constrained hamiltonian systems with holonomic constraints the following two results will be derived i it is shown that any first or second order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints based on this higher order methods can be derived by the same composition methods used for unconstrained problems ii leimkuhler and reich math comp 63 1994 pp 589 605 derived symplectic integrators based on dirac s reformulation of the constrained problem as an unconstrained hamiltonian system however although the unconstrained reformulation can be handled by direct application of any symplectic implicit runge kutta m

spectral dimension of trees with a unique infinite spine using generating functions techniques we develop a relation between the hausdorff and spectral dimension of trees with a unique infinite spine furthermore it is shown that if the outgrowths along the spine are independent and identically distributed then both the hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n we apply this new method to both critical non generic trees and the attachment and grafting model which is a special case of the vertex splitting model resulting in a simplified proof for the values of the hausdorff and spectral dimension for the former and novel results for the latter

differential geometry of viscoelastic models with fractional order derivatives viscoelastic materials with memory effect are studied based on the fractional rheonomic geometry the geometric objects are regarded as basic quantities of fractional viscoelastic models i e the metric tensor and torsion tensor are interpreted as the strain and the fractional strain rate respectively the generalized viscoelastic equations are expressed by the geometric objects especially the basic constitutive equations such as voigt and maxwell models can be derived geometrically from the generalized equation this leads to the fact that various viscoelastic models can be unified into one geometric expression

on the formulation of high frequency dissipative time stepping algorithms for nonlinear dynamics part i low order methods for two model problems and nonlinear elastodynamics abstract we present in this paper the development of a class of time stepping algorithms for nonlinear elastodynamics that exhibits the controllable numerical dissipation in the high frequency range required for the robust solution of the resulting numerically stiff systems to motivate and illustrate better the developments in this general case we present first the formulation and analysis of these methods for two simple model problems namely we consider a nonlinear elastic spring mass system and a simplified model of thin elastic beams as it is discussed in detail in this paper the conservation by the numerical algorithm of the momenta and corresponding relative equilibria of these characteristic hamiltonian systems with symmetry is of the main importance these conservation properties lead for a fixed and finite time step to a correct qualitative picture of the phase space where the discrete dynamics takes place even in the presence of the desired and controlled numerical dissipation of the energy this situation is contrasted with traditional dissipative numerical schemes which are shown through rigorous analyses to not only lose their dissipative character in the general nonlinear range but also the aforementioned conservation properties thus leading to a qualitatively distorted approximation of the phase dynamics the key for a successful algorithm in this context is the incorporation of the numerical dissipation in the internal modes of the motion while not affecting the group motions of the systems the algorithms presented in this work accomplish these goals the focus in this first part is given to first order methods representative numerical simulations ranging from applications in nonlinear structural dynamics to nonlinear continuum three dimensional elastodynamics are presented in the context of the finite element method to illustrate these ideas and results

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

fractional correlation functions in simple viscoelastic liquids abstract we develop a hydrodynamic formulation of fractional fluctuations in a viscoelastic liquid whose longitudinal modulus is a scalar and only depends on time the method is based on the introduction of fractional time derivatives in the hydrodynamic equations due to the viscoelastic memory coupled generalized langevin equations for the fluctuations of the state variables are formulated and solved analytically for a power law memory kernel with long correlation noise the associated fractional fokker planck equation ffpe is also derived and it is shown that it exhibits a fluctuation dissipation theorem fdt the explicit analytic diffusion coefficient is calculated for power law viscoelasticity and is shown to be subdiffusive the fractional correlations for longitudinal velocity and density fluctuations are analytically calculated and used to obtain the light scattering spectrum and the intermediate scattering function of the viscoelastic fluid our model calculation predicts that the fractional effects on these properties are not small effects and might be measurable

subordinated continuous time ar processes and their application to modeling behavior of mechanical system abstract many real data exhibit behavior adequate to subdiffusion processes very often it is manifested by so called trapping events the visible evidence of subdiffusion we observe not only in financial time series but also in technical data in this paper we propose a model which can be used for description of such kind of data the model is based on the continuous time autoregressive time series with stable noise delayed by the infinitely divisible inverse subordinator the proposed system can be applied to real datasets with short time dependence visible jumps and mentioned periods of stagnation in this paper we extend the theoretical considerations in analysis of subordinated processes and propose a new model that exhibits mentioned properties we concentrate on the main characteristics of the examined subordinated process expressed mainly in the language of the measures of dependence which are main tools used in statistical investigation of real data we present also the simulation procedure of the considered system and indicate how to estimate its parameters the theoretical results we illustrate by the analysis of real technical data

ergodicity breaking and localization we study the joint action of the non poisson renewal events npr yielding continuous time random walk ctrw with index ensuremath alpha l1 and two different generators of hurst coefficient h ensuremath ne 0 5 one generating fractional brownian motion fbm and another scaled brownian motion sbm we discuss the ergodicity breaking emerging from these joint actions and we find that in both cases the adoption of time averages leads to localization in the case of the joint action of npr and sbm localization occurs when sbm would produce subdiffusion the joint action of npr and fbm on the contrary may lead to localization when fbm is a source of superdiffusion the joint action of npr and fbm is equivalent to extending the ctrw to the case where the jumps of the runner are correlated and we argue that the the memory induced localization requires a refinement of the theoretical perspective about determinism and randomness

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

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

record length requirement of long range dependent teletraffic abstract this article contributes the highlights mainly in two folds on the one hand it presents a formula to compute the upper bound of the variance of the correlation periodogram measurement of teletraffic traffic for short with long range dependence lrd for a given record length t and a given value of the hurst parameter h theorems 1 and 2 on the other hand it proposes two formulas for the computation of the variance upper bound of the correlation periodogram measurement of traffic of fractional gaussian noise fgn type and the generalized cauchy gc type respectively corollaries 1 and 2 they may constitute a reference guideline of record length requirement of traffic with lrd in addition record length requirement for the correlation periodogram measurement of traffic with either the schuster type or the bartlett one is studied and the present results about it show that both types of periodograms may be used for the correlation measurement of traffic with a pre desired variance bound of correlation estimation moreover real traffic in the internet archive by the special interest group on data communication under the association for computing machinery of us acm sigcomm is analyzed in the case study in this topic

spectral dimension and random walks on the two dimensional uniform spanning tree we study the simple random walk on the uniform spanning tree on mathbb z 2 we obtain estimates for the transition probabilities of the random walk the distance of the walk from its starting point after n steps and exit times of both euclidean balls and balls in the intrinsic graph metric in particular we prove that the spectral dimension of the uniform spanning tree on mathbb z 2 is 16 13 almost surely

a fractional order model on new experiments of linear viscoelastic creep of hami melon this paper describes our experimental testing of linear viscoelastic creep behaviours in hami melon experimental data shows that hami melon has complex viscoelastic property which cannot be well described by the standard model consequently this study develops a fractional derivative model to describe such complex viscoelastic creep behaviours of hami melon the analytical creep function of the proposed fractional linear viscoelastic models is derived via the boltzmann superposition principle and discrete inverse laplace transform and then such analytical solutions are used to fit the experimental viscoelastic data of hami melon our study shows that the present fractional linear viscoelastic model with merely three parameters is more efficient and accurate than the generalised kelvin viscoelastic model of six parameters to describe the stress strain constitutive relations of hami melon it is noted that the present fractional model with adjustable parameters can also be used to describe creep damage

the loop erased random walk and the uniform spanning tree on the four dimensional discrete torus let x and y be points chosen uniformly at random from mathbb z n 4 the four dimensional discrete torus with side length n we show that the length of the loop erased random walk from x to y is of order n2 log n 1 6 resolving a conjecture of benjamini and kozma we also show that the scaling limit of the uniform spanning tree on mathbb z n 4 is the brownian continuum random tree of aldous our proofs use the techniques developed by peres and revelle who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d dimensional discrete torus for d 5 in combination with results of lawler concerning intersections of four dimensional random walks

a very efficient approach to compute the first passage probability density function in a time changed brownian model applications in finance abstract we propose a numerical method to compute the first passage probability density function in a time changed brownian model in particular we derive an integral representation of such a density function in which the integrand functions must be obtained solving a system of volterra equations of the first kind in addition we develop an ad hoc numerical procedure to regularize and solve this system of integral equations the proposed method is tested on three application problems of interest in mathematical finance namely the calculation of the survival probability of an indebted firm the pricing of a single knock out put option and the pricing of a double knock out put option the results obtained reveal that the novel approach is extremely accurate and fast and performs significantly better than the finite difference method

fractional generalized langevin equation approach to single file diffusion abstract fractional generalized langevin equation with external force is used to model single file diffusion it is found that for external force that varies with power law the solution for such a fractional langevin equation gives the correct short and long time behavior for the mean square displacement of single file diffusion when appropriate choice of parameters associated with fractional generalized langevin equation are used by considering some special cases of the fractional generalized langevin equation a new class of closed analytic expressions for the mean square displacement of single file diffusion can be obtained the effective fokker planck equation associated with single file diffusion is briefly considered

geometry of the uniform spanning forest transitions in dimensions 4 8 12 the uniform spanning forest usf in d is the weak limit of random uniformly chosen spanning trees in n n d pemantle 11 proved that the usf consists a s of a single tree if and only if d 4 we prove that any two components of the usf in d are adjacent a s if 5 d 8 but not if d 9 more generally let n x y be the minimum number of edges outside the usf in a path joining x and y in d then max n x y x y in mathbb z d left lfloor d 1 4 right rfloor textrm a s

smoothed dynamics of highly oscillatory hamiltonian systems abstract we consider the numerical treatment of hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential such systems arise e g in molecular dynamics simulations and the spatial discretization of hamiltonian partial differential equations since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size the numerical integration of such systems provides a challenging task it has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential this approach has e g been successfully applied to the bond stretching in molecular dynamics simulations in other cases such as the bond angle bending this methods fails due to the introduced rigidity here we give a careful analysis of the analytical problem by means of a smoothing operator this will lead us to the notion of the smoothed dynamics of a highly oscillatory hamiltonian system based on our analysis we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high frequency components in the solutions and thus allowing for larger time steps the new constrained formulation is hamiltonian and can be discretized by the well known shake method

smoothed langevin dynamics of highly oscillatory systems abstract we consider hamiltonian systems containing a strong constraining potential such systems were first investigated by rubin and ungar who derived the corresponding reduced equations of motion for the slow degree of freedom the purpose of this paper is two fold i we transform the highly oscillatory degree of freedom to action angle variables and use recent results on the truncation error of the corresponding normal form transformations to derive an estimate on the lifespan of the underlying adiabatic invariant ii we investigate thermally embedded hamiltonian systems i e langevin dynamics containing a strong constraining potential using again a transformation to action angle variables and applying the principle of averaging we derive reduced equations of motion for the slowly varying degrees of freedom this includes in particular the derivation of a correcting force term that stands for the coupling of the slow and fast degrees of motion in case of strong thermal coupling our reduced equations of motion suggest that the correcting force term can be approximated by the gradient of the fixman potential as used in statistical mechanics we also discuss smoothing in the context of constant temperature molecular dynamics and provide a numerical example

langevin formulation of a subdiffusive continuous time random walk in physical time systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power law scaling of the mean square displacement one of the most common models to describe such subdiffusive dynamics is the continuous time random walk ctrw stochastic trajectories of a ctrw can be described in terms of the subordination of a normal diffusive process by an inverse levy stable process here we propose an equivalent langevin formulation of a force free ctrw without subordination by introducing a different type of non gaussian noise we are able to express the ctrw dynamics in terms of a single langevin equation in physical time with additive noise we derive the full multipoint statistics of this noise and compare it with the scaled brownian motion sbm an alternative stochastic model describing subdiffusive dynamics interestingly these two noises are identical up to the second order correlation functions but different in the higher order statistics we extend our formalism to general waiting time distributions and force fields and compare our results with those of the sbm in the presence of external forces our proposed noise generates a different class of stochastic processes resembling a ctrw but with forces acting at all times

unbounded complex modulus of viscoelastic materials and the kramers kronig relations abstract the kramers kronig k k dispersion relations developed for the complex modulus of elasticity of solid viscoelastic materials connect the frequency dependences of the dynamic modulus and loss modulus whether the boundedness of the complex modulus at high infinite frequency is required or not for the applicability of the k k relations is investigated in this paper the derivation of the k k relations developed for the complex modulus is presented by examining the physical background of the relations it is shown that the k k relations can be applied even if the complex modulus is unbounded at high frequencies the fractional derivative kelvin model is used to demonstrate the application of the k k relations for a class of unbounded complex modulus

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

from the solutions of diffusion equation to the solutions of subdiffusive one starting with green s functions found for normal diffusion we construct exact time dependent green s functions for a subdiffusive equation with fractional time derivatives green s functions satisfy the boundary conditions involving a linear combination of fluxes and concentrations the method is particularly useful to calculate the concentration profiles in a multi part system where different kinds of transport occur in each part of it as an example we find the solution of the subdiffusive equation for the system composed of two parts with normal diffusion and subdiffusion

exponential tail bounds for loop erased random walk in two dimensions let m n be the number of steps of the loop erasure of a simple random walk on 2 from the origin to the circle of radius n we relate the moments of m n to es n the probability that a random walk and an independent loop erased random walk both started at the origin do not intersect up to leaving the ball of radius n this allows us to show that there exists c such that for all n and all k 1 2 e m k n and hence to establish exponential moment bounds for m n this implies that there exists c 0 such that for all n and all 0 p m n e m n 2e c using similar techniques we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any 0 such that for all n and 0 p m n 1 e m n ce c

ageing and confinement in non ergodic heterogeneous diffusion processes we study the effects of ageing the time delay between initiation of the physical process at t 0 and start of observation at some time t 0 a and spatial confinement on the properties of heterogeneous diffusion processes hdps with deterministic power law space dependent diffusivities d xd x 0 from analysis of the ensemble and time averaged mean squared displacements and the ergodicity breaking parameter quantifying the inherent degree of irreproducibility of individual realizations of the hdp we obtain striking similarities to ageing subdiffusive continuous time random walks with scale free waiting time distributions we also explore how both processes can be distinguished for confined hdps we study the long time saturation of the ensemble and time averaged particle displacements as well as the magnitude of the inherent scatter of time averaged displacements and contrast the outcomes to the results known for other anomalous diffusion processes under confinement

fractional derivative quantum fields at positive temperature abstract this paper considers fractional generalization of finite temperature klein gordon kg field and vector potential in covariant gauge and static temporal gauge fractional derivative quantum field at positive temperature can be regarded as a collection of infinite number of fractional thermal oscillators generalized riemann zeta function regularization and heat kernel techniques are used to obtain the high temperature expansion of free energy associated with the fractional kg field we also show that quantization of the fractional derivative fields can be carried out by using the parisi wu stochastic quantization

hierarchical analogues to fractional relaxation equations recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena such as stress strain relationships in polymeric materials in the rheological constitutive equations this implies the replacement of the first order derivatives by fractional derivatives here we show that such procedures have hierarchical mechanical analogues we focus on the generalized dashpot and the generalized maxwell model and display the corresponding arrangements our models allow a transparent interpretation of the parameters which enter the fractional equations and reveal that the internal dynamics are hierarchically constrained

ergodic properties of fractional brownian langevin motion we investigate the time average mean square displacement overline ensuremath delta 2 mathbf x t mathbf ensuremath int 0 t ensuremath ensuremath delta x t ensuremath ensuremath delta ensuremath x t ensuremath 2 d t ensuremath t ensuremath ensuremath delta for fractional brownian langevin motion where x t is the stochastic trajectory and ensuremath delta is the lag time unlike the previously investigated continuous time random walk model overline ensuremath delta 2 converges to the ensemble average x 2 ensuremath sim t 2h in the long measurement time limit the convergence to ergodic behavior is slow however and surprisingly the hurst exponent h frac 3 4 marks the critical point of the speed of convergence when hl frac 3 4 the ergodicity breaking parameter e b mathbf overline ensuremath delta 2 mathbf x t mathbf mathbf 2 ensuremath overline ensuremath delta 2 mathbf x t mathbf 2 mathbf overline ensuremath delta 2 mathbf x t mathbf 2 ensuremath sim k h ensuremath delta t ensuremath 1 when h frac 3 4 e b ensuremath sim frac 9 16 mathrm ln phantom rule 0 2em 0ex t ensuremath delta t ensuremath 1 and when frac 3 4 lhl1 e b ensuremath sim k h ensuremath delta 4 ensuremath 4h t 4h ensuremath 4 in the ballistic limit h ensuremath rightarrow 1 ergodicity is broken and e b ensuremath sim 2 the critical point h frac 3 4 is marked by the divergence of the coefficient k h fractional brownian motion as a model for recent experiments of subdiffusion of mrna in the cell is briefly discussed and a comparison with the continuous time random walk model is made

fokker planck type equations associated with fractional brownian motion controlled by infinitely divisible processes abstract in this paper we study the anomalous diffusion process driven by fractional brownian motion delayed by general infinitely divisible subordinator we show the analyzed process is the stochastic representation of the fokker planck type equation that describes the probability density function of an introduced model moreover we study main characteristics of the examined process the first two moments that allow us in special cases for classification of it as a system with accelerating subdiffusion property

characterization of sub gaussian heat kernel estimates on strongly recurrent graphs sub gaussian estimates for random walks are typical of fractal graphs we characterize them in the strongly recurrent case in terms of resistance estimates only without assuming elliptic harnack inequalities

homogenization of hamiltonian systems with a strong constraining potential abstract the paper studies hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale in particular we are interested in the limit situation where the size of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of the order o 1 the process of establishing an effective initial value problem for the limit positions will be called homogenization of the hamiltonian system this problem occurs in mechanics as the problem of realization of holonomic constraints as various singular limits in fluid flow problems in plasma physics as the problem of guiding center motion and in the simulation of biomolecules as the so called smoothing problem we suggest the systematic use of the notion of weak convergence in order to approach this problem this methodology helps to establish unified and short proofs of many known results which throw light on the inherent structure of the problem moreover we give a careful and critical review of the literature

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

fokker planck equation and path integral representation of the fractional ornstein uhlenbeck process with two indices this paper considers the fokker planck equation and path integral formulation of the fractional ornstein uhlenbeck process parametrized by two indices the effective fokker planck equation of this process is derived from the associated fractional langevin equation the path integral representation of the process is constructed and the basic quantities are evaluated

symplectic numerical integrators in constrained hamiltonian systems recent work reported in the literature suggests that for the long time integration of hamiltonian dynamical systems one should use methods that preserve the symplectic or canonical structure of the flow here we investigate the symplecticness of numerical integrators for constrained dynamics such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies this leads to a constrained hamiltonian system of smaller dimension previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space here it is shown that the very popular verlet method with shake type constraints is equivalent to the same method with rattle type constraints and that the latter is symplectic and time reversible this assumes that the iteration is carried to convergence we also demonstrate the global convergence of the verlet scheme in the presence of shake type and rattle type constraints we perform numerical experiments to compare these methods with the second order backward differentiation method commonly recommended for ordinary differential equations with constraints

path integral representation of fractional harmonic oscillator abstract fractional oscillator process can be obtained as the solution to the fractional langevin equation there exist two types of fractional oscillator processes based on the choice of fractional integro differential operators namely weyl and riemann liouville an operator identity for the fractional differential operators associated with the fractional oscillators is derived it allows the solution of fractional langevin equations to be obtained by simple inversion the relationship between these two fractional oscillator processes is studied the operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes relevant quantities such as two point and n point functions can be calculated from the generating functions

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

anomalous diffusion and long range correlations in the score evolution of the game of cricket we investigate the time evolution of the scores of the second most popular sport in the world the game of cricket by analyzing event by event the scores of more than 2000 matches we point out that the score dynamics is an anomalous diffusive process our analysis reveals that the variance of the process is described by a power law dependence with a superdiffusive exponent that the scores are statistically self similar following a universal gaussian distribution and that there are long range correlations in the score evolution we employ a generalized langevin equation with a power law correlated noise that describes all the empirical findings very well these observations suggest that competition among agents may be a mechanism leading to anomalous diffusion and long range correlation

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

invariant manifolds and the initialization problem for some atmospheric equations abstract an invariant manifold of quasi geostrophic states is shown to exist for a finite dimensional system of atmospheric equations there is another set of scalings for the parameters with respect to the rossby number which leads to the same limiting geostrophic equations but such that there need not be an invariant manifold also discussed are computational methods

subsequential scaling limits of simple random walk on the two dimensional uniform spanning tree the first main result of this paper is that the law of the rescaled two dimensional uniform spanning tree is tight in a space whose elements are measured rooted real trees continuously embedded into euclidean space various properties of the intrinsic metrics measures and embeddings of the subsequential limits in this space are obtained with it being proved in particular that the hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8 5 in addition the tightness result is applied to deduce that the annealed law of the simple random walk on the two dimensional uniform spanning tree is tight under a suitable rescaling for the limiting processes which are diffusions on random real trees embedded into euclidean space detailed transition density estimates are derived

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

nonstationary markovian replication process causing diverse diffusions we introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions a nonstationary markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker s displacement and the generalized telegrapher equation with time varying coefficients and we find that diffusivity can be determined by temporal changes of replication of an immediate step by controlling the replications we realize diverse diffusions such as alternating diffusion superdiffusion subdiffusion and marginal diffusion which originate from oscillating increasing decreasing and slowly increasing or decreasing replications with time respectively

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

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

first passage and first exit times of a bessel like stochastic process we study a stochastic process x t which is a particular case of the rayleigh process and whose square is the bessel process with various applications in physics chemistry biology economics finance and other fields the stochastic differential equation is dx t nd x t dt 2d dw t where w t is the wiener process the drift term can arise from a logarithmic potential or from taking x t as the norm of a multidimensional random walk due to the singularity of the drift term for x t 0 different natures of boundary at the origin arise depending on the real parameter n entrance exit and regular for each of them we calculate analytically and numerically the probability density functions of first passage times or first exit times nontrivial behavior is observed in the case of a regular boundary