trapping aspects in enhanced diffusion we study the superlinear diffusion x2 t t 1 in layered media containing random velocity fields the superlinear behavior holds in the case of random velocities along thex direction accompanied by diffusional motion in the space transverse to it the transverse space can be either euclidean fractal or ultrametric for a one dimensional transverse space we derive exact expressions for the higher moments of the displacement furthermore we investigate the propagatorp x t along thex direction and establish its scaling behavior our analysis highlights the resemblance between the stretched gaussian behavior of the propagator and the stretched exponential form of the survivial probability in the trapping problem both show late crossover behavior

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

relaxation dynamics of perturbed regular hyperbranched fractals we focus on perturbed regular hyperbranched fractals prhf which are rhf whose f coordinated centers fcc are traps we compute the mechanical properties storage and loss modulus and the average displacement in the framework of generalized gaussian structures by making use of the eigenvalue spectrum of the connectivity matrix we generalize the analysis to the case of a connectivity matrix perturbed by a diagonal and pure imaginary operator although the above cited observables in this new situation lose their original meaning they still give important information about the underlying structures and they could help to analyze other phenomena where complex operators are involved we obtain analytically the eigenvalue spectrum for prhf a drastic change was observed in the behavior of the studied quantities even for a very small perturbation strength however it is still possible to depict the scaling of the fractals in the intermediate time frequency domains

nonequilibrium fluctuations for a tagged particle in mean zero one dimensional zero range processes we prove a non equilibrium functional central limit theorem for the position of a tagged particle in mean zero one dimensional zero range process the asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation

a framework of verified eigenvalue bounds for self adjoint differential operators for eigenvalue problems of self adjoint differential operators a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues in the case of the laplacian operator by applying crouzeix raviart finite elements an efficient algorithm is developed to bound the eigenvalues for the laplacian defined in 1d 2d and 3d spaces moreover for nonconvex domains for which case there may exist singularities of eigenfunctions around re entrant corners the proposed algorithm can easily provide eigenvalue bounds by further adopting the interval arithmetic the explicit eigenvalue bounds from numerical computations can be mathematically correct

exact solution for mean first passage time on a pseudofractal scale free web the explicit determinations of the mean first passage time mfpt for trapping problem are limited to some simple structure e g regular lattices and regular geometrical fractals and determining mfpt for random walks on other media especially complex real networks is a theoretical challenge in this paper we investigate a simple random walk on the the pseudofractal scale free web psfw with a perfect trap located at a node with the highest degree which simultaneously exhibits the remarkable scale free and small world properties observed in real networks we obtain the exact solution for the mfpt that is calculated through the recurrence relations derived from the structure of psfw the rigorous solution exhibits that the mfpt approximately increases as a power law function of the number of nodes with the exponent less than 1 we confirm the closed form solution by direct numerical calculations we show that the structure of psfw can improve the efficiency of transport by diffusion compared with some other structure such as regular lattices sierpinski fractals and t graph the analytical method can be applied to other deterministic networks making the accurate computation of mfpt possible

dynamics of generalized gaussian polymeric structures in random layered flows we develop a formalism for the dynamics of a flexible branched polymer with arbitrary topology in the presence of random flows this is achieved by employing the generalized gaussian structure ggs approach and the matheron de marsily model for the random layered flow the expression for the average square displacement asd of the center of mass of the ggs is obtained in such flow the averaging is done over both the thermal noise and the external random flow although the formalism is valid for branched polymers with various complex topologies we mainly focus here on the dynamics of the flexible star and dendrimer we analyze the effect of the topology the number and length of branches for stars and the number of generations for dendrimers on the dynamics under the influence of external flow which is characterized by their root mean square velocity persistence flow length and flow exponent our analysis shows two anomalous power law regimes viz subdiffusive intermediate time polymer stretching and flow induced diffusion and superdiffusive long time flow induced diffusion the influence of the topology of the ggs is unraveled in the intermediate time regime while the long time regime is only weakly dependent on the topology of the polymer with the decrease in the value of the magnitude of the asd decreases while the temporal exponent of the asd increases in both the time regimes also there is an increase in both the magnitude of the asd and the crossover time from the subdiffusive to the superdiffusive regime with an increase in the total mass of the polymeric structure

mapping between hopping on hierarchical structures and diffusion on a family of fractals the authors present a simple analytical approach to studying diffusion on hierarchical and fractal structures they show that hopping on one dimensional hierarchical structures can be mapped onto diffusion on a family of loopless fractals

polymer transport in random flow the dynamics of polymers in a random smooth flow is investigated in the framework of the hookean dumbbell model the analytical expression of the time dependent probability density function of polymer elongation is derived explicitly for a gaussian rapidly changing flow when polymers are in the coiled state the pdf reaches a stationary state characterized by power law tails both for small and large arguments compared to the equilibrium length the characteristic relaxation time is computed as a function of the weissenberg number in the stretched state the pdf is unstationary and exhibits multiscaling umerical simulations for the two dimensional navier stokes flow confirm the relevance of theoretical results obtained for the correlated model

random walks in modular scale free networks with multiple traps extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale free and modular structure and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks in this paper we examine two cases of random walks performed on a class of modular scale free networks with multiple traps located at several given nodes we first derive a formula of the mean first passage time mfpt for a general network which is the mean of the expected time to absorption originating from a specific node averaged over all nontrap starting nodes although the computation is complex the expression of the formula is exact moreover the computational approach and procedure are independent of the number and position of the traps we then determine analytically the mfpt for the two random walks being considered the obtained analytical results are in complete agreement with the numerical ones our results show that the number and location of traps play an important role in the behavior of the mfpt since for both cases the mfpt grows as a power law function of the number of nodes but their exponents are quite different we demonstrate that the root of the difference in the behavior of mfpt is attributed to the modular and scale free topologies of the networks this work can deepen the understanding of diffusion on networks with modular and scale free architecture and motivate relevant studies for random walks running on complex random networks with multiple traps

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

a discrete random walk on the hypercube abstract in this paper we study the scaling for mean first passage time mfpt of random walks on the hypercube and obtain a closed form formula for the mfpt over all node pairs we also determine the exponent of scaling efficiency characterizing the random walks and compare it with those of the existing networks finally we study the random walks on the hypercube with a located trap and provide a solution of the kirchhoff index of the hypercube

standard random walks and trapping on the koch network with scale free behavior and small world effect a vast variety of real life networks display the ubiquitous presence of scale free phenomenon and small world effect both of which play a significant role in the dynamical processes running on networks although various dynamical processes have been investigated in scale free small world networks analytical research about random walks on such networks is much less in this paper we will study analytically the scaling of the mean first passage time mfpt for random walks on scale free small world networks to this end we first map the classical koch fractal to a network called koch network according to this proposed mapping we present an iterative algorithm for generating the koch network based on which we derive closed form expressions for the relevant topological features such as degree distribution clustering coefficient average path length and degree correlations the obtained solutions show that the koch network exhibits scale free behavior and small world effect then we investigate the standard random walks and trapping issue on the koch network through the recurrence relations derived from the structure of the koch network we obtain the exact scaling for the mfpt we show that in the infinite network order limit the mfpt grows linearly with the number of all nodes in the network the obtained analytical results are corroborated by direct extensive numerical calculations in addition we also determine the scaling efficiency exponents characterizing random walks on the koch network

efficiency of trapping processes in regular and disordered networks we use a markov method to study the efficiency of trapping processes involving both a random walker and a deep trap in regular and disordered networks the efficiency is gauged by the mean absorption time average of the mean number of steps performed by the random walker before being absorbed by the trap we compute this quantity in terms of different control parameters namely the length of the walker jumps the mobility of the trap and the degree of spatial disorder of the network for a proper choice of the system size we find in all cases a nonmonotonic behavior of the efficiency in terms of the corresponding control parameter we thus arrive at the conclusion that despite the decrease of the effective system size underlying the increase of the control parameter the efficiency is reduced as a result of an increase of the escape probability of the walker once it finds itself in the interaction zone of the trap this somewhat anti intuitive effect is very robust in the sense that it is observed regardless of the specific choice of the control parameter for the case of a ring lattice results for the mean absorption time in systems of arbitrary size are given in terms of a two parameter scaling function for the case of a mobile trap we deal with both trapping via a single channel walker trap overlap and via two channels walker trap overlap and walker trap crossing thereby generalizing previous work as for the disordered case our analysis concerns small world networks for which we see several crossovers of the absorption time as a function of the control parameter and the system size the methodology used may be well suited to exploring characteristic time scales of encounter controlled phenomena in networks with a few interacting elements and the effect of geometric constraints in nanoscale systems with a very small number of particles

random walk on a population of random walkers we consider a population of n labelled random walkers moving on a substrate and an excitation jumping among the walkers upon contact the label of the walker carrying the excitation at time t can be viewed as a stochastic process where the transition probabilities are a stochastic process themselves upon mapping onto two simpler processes the quantities characterizing can be calculated in the limit of long times and low walkers density the results are compared with numerical simulations several different topologies for the substrate underlying diffusion are considered

a geometric growth model interpolating between regular and small world networks we propose a geometric growth model which interpolates between one dimensional linear graphs and small world networks the model undergoes a transition from large to small worlds we study the topological characteristics by both theoretical predictions and numerical simulations which are in good accordance with each other our geometrically growing model is a complementarity for the static ws model

cyclization of a polymer first passage problem for a non markovian process we discuss a closure of a polymer chain released from a state with given end to end distance considering the situation as a first passage problem for a non markovian process we express the survival probability through the three time distribution of the end to end distance and evaluate it numerically by solving a volterra equation we also show how the wilemski fixmann approximation emerges within this scheme and discuss its geometrical nature

random walks on deterministic scale free networks exact results we study the random walk problem on a class of deterministic scale free networks displaying a degree sequence for hubs scaling as a power law with an exponent gamma log 3 log 2 we find exact results concerning different first passage phenomena and in particular we calculate the probability of first return to the main hub these results allow to derive the exact analytic expression for the mean time to first reach the main hub whose leading behavior is given by tau approximately v1 1 gamma where v denotes the size of the structure and the mean is over a set of starting points distributed uniformly over all the other sites of the graph interestingly the process turns out to be particularly efficient we also discuss the thermodynamic limit of the structure and some local topological properties

self diffusion for brownian motions with local interaction we derive explicitly the asymptotic law of the tagged particle process in a system of interacting brownian motions in the presence of a diffusive scaling in nonequilibrium the interaction is local and interpolates between the totally independent case noninteracting and the totally reflecting case and can be viewed as the limiting local version of an interaction through a pair potential as its support shrinks to zero we also prove the independence of two tagged particles in the limit

on the statistics of generalized gaussian structures collapse and random external fields we consider the statistics of generalized gaussian structures ggs exposed to a random external field a ggs comprises n monomers connected to each other by harmonic potentials when the spectral dimension d of a ggs exceeds the value of two its radius of gyration r becomes independent of its mass n the cross over into this collapse can be treated continuously by cross linking m precursor chains of length n in the stretched state to an object which we call a polymer bundle we demonstrate that an external field f applied to each monomer can unfold such a collapsed state in the case where every monomer has an individual randomly distributed charge the critical spectral dimension for the collapse is raised to four r scales like fnalpha with alpha 4 ds 2ds for ds 4

determining global mean first passage time of random walks on vicsek fractals using eigenvalues of laplacian matrices the family of vicsek fractals is one of the most important and frequently studied regular fractal classes and it is of considerable interest to understand the dynamical processes on this treelike fractal family in this paper we investigate discrete random walks on the vicsek fractals with the aim to obtain the exact solutions to the global mean first passage time gmfpt defined as the average of first passage time fpt between two nodes over the whole family of fractals based on the known connections between fpts effective resistance and the eigenvalues of graph laplacian we determine implicitly the gmfpt of the vicsek fractals which is corroborated by numerical results the obtained closed form solution shows that the gmfpt approximately grows as a power law function with system size number of all nodes with the exponent lies between 1 and 2 we then provide both the upper bound and lower bound for gmfpt of general trees and show that the leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size we also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs this study provides a comprehensive understanding of random walks on the vicsek fractals and general treelike networks

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

the exact solution of the mean geodesic distance for vicsek fractals vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important in this paper the exact formula for the mean geodesic distance of vicsek fractals is found the quantity is computed precisely through the recurrence relations derived from the self similar structure of the fractals considered the obtained exact solution exhibits that the mean geodesic distance approximately increases as a power law function of the number of nodes with the exponent equal to the reciprocal of the fractal dimension the closed form solution is confirmed by extensive numerical calculations

finding lie groups that reduce the order of discrete dynamical systems discrete dynamical systems of the form are considered where is an n component vector equations x f x define a mapping f from an n dimensional projective space into itself each component of f is a rational function i e a ratio of polynomials in n dynamical variables maeda showed that when f commutes with each transformation of a lie group a reduction in the order of the dynamical system results given a discrete dynamical system the difficulty is to find its continuous symmetries we present a way of using f invariant sets to find these symmetries the approach taken is to arrange groups in order of increasing complication and to characterize the set of dynamical systems admitting each group criteria are given for recognizing and reducing the order of systems admitting subgroups of the projective general linear group in n variables pgl n or certain lie subgroups of the cremona group of birational transformations in n variables quispel et al demonstrated the use of canonical group variables for achieving this reduction we develop canonical coordinates for several groups with elementary lie algebras and demonstrate reduction of order in each case results are used to reduce the order of several examples of recursion formulae taken from the literature on renormalizable lattice models

group theoretic reduction of laplacian dynamical problems on fractal lattices discrete forms of the schr umlt o dinger equation the diffusion equation the linearized landau ginzburg equation and discrete models for vibrations and spin dynamics belong to a class of laplacian based finite difference models real space renormalization of such models on finitely ramified regular fractals is known to give exact recursion relations it is shown that these recursions commute with lie groups representing continuous symmetries of the discrete models each such symmetry reduces the order of the renormalization recursions by one resulting in a system of recursions with one fewer variable group trajectories are obtained from inverse images of fixed and invariant sets of the recursions a subset of the laplacian finite difference models can be mapped by change of boundary conditions and time dependence to a diffusion problem with closed boundaries in such cases conservation of mass simplifies the group flow and obtaining the groups becomes easier to illustrate this the renormalization recursions for green functions on four standard examples are decoupled the examples are 1 the linear chain 2 an anisotropic version of dhar close quote s 3 simplex similar to a model dealt with by hood and southern 3 the fourfold coordinated sierpi acute n ski lattice of rammal and of domany ital et al and 4 more a form of the vicsek lattice prospects for applying the group theoretic method to more general dynamical systems are discussed copyright ital 1997 ital the american physical society less

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

spectral gap for zero range dynamics we give a lower bound on the spectral gap for symmetric zero range processes under some conditions on the rate function we show that the gap shrinks as n 2 independent of the density for the dynamics localized on a cube of size n d we follow the method outlined by lu and yau where a similar spectral gap is proved for kawasaki dynamics

on the diffusive nature of entropy flow in infinite systems remarks to a paper by guo papanicolau varadhan the hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy free energy arguments the methods of 13 are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise following 14 15 and 3 5 we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended the flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems 4

scaling of the first passage time and the survival probability on exact and quasi exact self similar structures the authors calculate the asymptotic behaviour of the moments of the first passage time and survival probability for random walks on an exactly self similar tree and on a quasi self similar comb by applying an exact decimation approach to the master equations for the hierarchical comb a transition from ordinary to anomalous diffusion occurs at r 2 where r is the ratio of teeth length in successive iterations of the structure in the anomalous regime r 2 the positive integer moments of the first passage time tq scale as ltau q with tau q 1 2q 1 ln r ln 2 where l is the linear distance from input to output the asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations they find that the characteristic time t in the asymptotic exponential decay of the survival probability exp t t scales as t approximately ltau with tau ln r2 ln 2 i e tau is distinct from tau 1 however substantial corrections to this asymptotic form for tau exist and these are needed to account for the recent simulation data of havlin and matan 1988

average receiving scaling of the weighted polygon koch networks with the weight dependent walk based on the weighted koch networks and the self similarity of fractals we present a family of weighted polygon koch networks with a weight factor r 0 r 1 we study the average receiving time art on weight dependent walk i e the walker moves to any of its neighbors with probability proportional to the weight of edge linking them whose key step is to calculate the sum of mean first passage times mfpts for all nodes absorpt at a hub node we use a recursive division method to divide the weighted polygon koch networks in order to calculate the art scaling more conveniently we show that the art scaling exhibits a sublinear or linear dependence on network order thus the weighted polygon koch networks are more efficient than expended koch networks in receiving information finally compared with other previous studies results i e koch networks weighted koch networks we find out that our models are more general

determining mean first passage time on a class of treelike regular fractals relatively general techniques for computing mean first passage time mfpt of random walks on networks with a specific property are very useful since a universal method for calculating mfpt on general graphs is not available because of their complexity and diversity in this paper we present techniques for explicitly determining the partial mean first passage time pmfpt i e the average of mfpts to a given target averaged over all possible starting positions and the entire mean first passage time emfpt which is the average of mfpts over all pairs of nodes on regular treelike fractals we describe the processes with a family of regular fractals with treelike structure the proposed fractals include the t fractal and the peano basin fractal as their special cases we provide a formula for mfpt between two directly connected nodes in general trees on the basis of which we derive an exact expression for pmfpt to the central node in the fractals moreover we give a technique for calculating emfpt which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials making use of the proposed methods we obtain analytically the closed form solutions to pmfpt and emfpt on the fractals and show how they scale with the number of nodes in addition to exhibit the generality of our methods we also apply them to the vicsek fractals and the iterative scale free fractal tree and recover the results previously obtained

polymer dynamics in time dependent matheron de marsily flows an exactly solvable model we introduce a new model of random layered media extending the matheronde marsily model here we allow for the flows to change in time for such layered structures we solve exactly the equations of motion for single particles and also for polymers modelled as rouse chains the results show a rich variety of dynamical patterns

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

kemeny s constant and the effective graph resistance kemeny s constant and its relation to the effective graph resistance has been established for regular graphs by palacios et al 1 based on the moore penrose pseudo inverse of the laplacian matrix we derive a new closed form formula and deduce upper and lower bounds for the kemeny constant furthermore we generalize the relation between the kemeny constant and the effective graph resistance for a general connected undirected graph

large deviation principle for interacting brownian motions we prove the large deviation principle for the empirical process in a system of locally interacting brownian motions in the nonequilibrium such a phenomenon has been proven only for two lattice systems the symmetric simple exclusion process and the zero range process therefore we have achieved the third result in this context and moreover the first result for the diffusion type interacting particle system 2016 wiley periodicals inc

explicit bound for quadratic lagrange interpolation constant on triangular finite elements abstract for the quadratic lagrange interpolation function an algorithm is proposed to provide explicit and verified bound for the interpolation error constant that appears in the interpolation error estimation the upper bound for the interpolation constant is obtained by solving an eigenvalue problem along with explicit lower bound for its eigenvalues the lower bound for interpolation constant can be easily obtained by applying the rayleigh ritz method numerical computation is performed to demonstrate the sharpness of lower and upper bounds of the interpolation constants over triangles of different shapes an online computing demo is available at http www xfliu org onlinelab

scaling limits of a tagged particle in the exclusion process with variable diffusion coefficient we prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in z with variable diffusion coefficient the scaling limits are obtained from a similar result for the current through 1 2 for a zero range process with bond disorder for the clt we prove convergence to a fractional brownian motion of hurst exponent 1 4

zero constant formula for first passage observables in bounded domains in this letter we develop an analytical approach which provides an explicit determination of mean first passage times mfpts for random walks in bounded domains for a wide class of transport processes in particular we derive for the first time explicit expressions of mfpts for emblematic models of transport in complex media such as diffusion on deterministic and random fractals this approach relies on a scale invariance hypothesis and a large volume expansion of the mfpt which actually proves to be very accurate even for small system sizes as shown by numerical simulations this explicit determination of mfpts can be straightforwardly generalized to further useful first passage observables such as occupation times and splitting probabilities

elastic scattering by deterministic and random fractals self affinity of the diffraction spectrum the diffraction spectrum of coherent waves scattered from fractal supports is calculated exactly the fractals considered are of the class generated iteratively by successive dilations and translations and include generalizations of the cantor set and sierpinski carpet as special cases also randomized versions of these fractals are treated the general result is that the diffraction intensities obey a strict recursion relation and become self affine in the limit of large iteration number with a self affinity exponent related directly to the fractal dimension of the scattering object applications include neutron scattering x rays optical diffraction magnetic resonance imaging electron diffraction and he scattering which all display the same universal scaling textcopyright 1996 the american physical society

unfold dynamics of generalized gaussian structures we consider the unfold dynamics of generalized gaussian structures ggss exposed to different kinds of external forces a ggs consists of n monomers connected by harmonic springs into a network when its spectral dimension d s exceeds the critical value of 2 the ggs is in a collapsed state sommer and blumen j phys a 28 6669 1995 showed that collapsed structures can be unfolded under external forces they demonstrate this for the case where each monomer is exposed to a force with a random direction then networks with a spectral dimension up to 4 become unfolded in the present paper we focus on the dynamics of such unfold processes we investigate ggss exposed to different kinds of external forces pulling one monomer uncorrelated forces long range correlated forces and diblocklike forces we show that external perturbations that act only on a few monomers are not able to unfold a collapsed structure on the other hand more general kinds of forces lead to a stretching of ggss even for d s g2 as long as d s l d c where d c depends on the kind of the force field in general during the unfold process the size r of the ggs grows via a power law r ensuremath propto t ensuremath alpha 0l ensuremath alpha l 1 where ensuremath alpha depends on d s as well as on the kind of force field that is applied

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

eigentime identities of flower networks with multiple branches abstract for the fractal networks the eigentime identity is the expected time for a walker going from one node to another in this paper we study a family of flower networks that have k parallel paths with lengths m 1 m 2 m k let c t be the eigentime identity of the flower networks in generation t then the obtained result shows that the eigentime identity is c t i 1 k m i i 1 k m i 1 t

analysis of diffusion and trapping efficiency for random walks on non fractal scale free trees abstract in this paper the discrete random walks on the recursive non fractal scale free trees nfsft are studied and a kind of method to calculate the analytic solutions of the mean first passage time mfpt for any pair of nodes the mean trapping time mtt for any target node and mean diffusing time mdt for any starting node are proposed furthermore we compare the trapping efficiency and diffusion efficiency between any two nodes of nfsft by using the mtt and the mdt as the measures of trapping efficiency and diffusion efficiency respectively and find the best or worst trapping sites and the best or worst diffusion sites the results show that the two hubs of nfsft are not only the best trapping site but also the worst diffusion site and that the nodes which are the farthest nodes from the two hubs are not only the worst trapping sites but also the best diffusion sites furthermore we find that the ratio between the maximum and minimum of mtt grows logarithmically with network order but the ratio between the maximum and minimum of mdt is almost equal to 1 the results imply that the trap s position has great effect on the trapping efficiency but the position of starting node has little effect on diffusion efficiency finally the simulation for random walks on nfsft is done and it is consistent with the derived results

nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps we prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium the scaling limit is a time inhomogeneous process of independent increments related to the solution of a fractional heat equation

first passage times for random walks in bounded domains we present a novel computational method of first passage times between a starting site and a target site of regular bounded lattices we derive accurate expressions for all the moments of this first passage time validated by numerical simulations their range of validity is discussed we also consider the case of a starting site and two targets in addition we present the extension to continuous brownian motion these results are of great relevance to any system involving diffusion in confined media

quantum walks and trapping on regular hyperbranched fractals the present work studies the behaviour of continuous time quantum walks on regular hyperbranched fractals whose centre is a trap we focus on the variations of the eigenvalue spectrum of the transfer operator by tuning the trap strength from zero to infinity we show that the degenerate eigenvalues are independent from the trap strength and can be obtained analytically due to this the mean survival probability is just in the intermediate range affected by the trap strength moreover because of the presence of real eigenvalues the asymptotical probability of being outside the trap is not zero

efficiency of encounter controlled reaction between diffusing reactants in a finite lattice topology and boundary effects abstract the role of dimensionality euclidean vs fractal spatial extent boundary effects and system topology on the efficiency of diffusion reaction processes involving two simultaneously diffusing reactants is analyzed we present numerically exact values for the mean time to reaction as gauged by the mean walklength before reactive encounter obtained via application of the theory of finite markov processes and via monte carlo simulation as a general rule we conclude that for sufficiently large systems the efficiency of diffusion reaction processes involving two synchronously diffusing reactants two walker case relative to processes in which one reactant of a pair is anchored at some point in the reaction space one walker plus trap case is higher and is enhanced the lower the dimensionality of the system this differential efficiency becomes larger with increasing system size and for periodic systems its asymptotic value may depend on the parity of the lattice imposing confining boundaries on the system enhances the differential efficiency relative to the periodic case while decreasing the absolute efficiencies of both two walker and one walker plus trap processes analytic arguments are presented to provide a rationale for the results obtained the insights afforded by the analysis to the design of heterogeneous catalyst systems is also discussed

global first passage times of fractal lattices the global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time t we find simple expressions for the mean global first passage time for five fractals the d dimensional sierpinski gasket t fractal hierarchical percolation model mandelbrot given curve and a deterministic tree we also find an exact expression for the second moment and show that the variance of the first passage time var t scales with the number of nodes within the fractal n such that var t similar to n 4 d where d is the spectral dimension

uniqueness of very weak solutions for a fractional filtration equation abstract we prove existence and uniqueness of distributional bounded solutions to a fractional filtration equation in r d with regards to uniqueness it was shown even for more general equations in 22 that if two bounded solutions u w of 1 1 satisfy u w l 1 r d 0 t then u w we obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions for nonnegative initial data we first show that a minimal solution exists and then that any other solution must coincide with it a similar procedure is carried out for sign changing solutions as a consequence distributional solutions have locally finite energy

eigentime identity of the weighted scale free triangulation networks for weight dependent walk abstract the eigenvalues of the normalized laplacian matrix of a network provide information on its structural properties and some relevant dynamical aspects in particular for weight dependent walk in order to get the eigentime identity for weight dependent walk we need to obtain the eigenvalues and their multiplicities of the laplacian matrix firstly the model of the weighted scale free triangulation networks is constructed then the eigenvalues and their multiplicities of transition weight matrix are presented after the recursive relationship of those eigenvalues at two successive generations are given consequently the laplacian spectrum is obtained finally the analytical expression of the eigentime identity indicating that the eigentime identity grows sublinearly with the network order is deduced

fractal models for diffusion controlled aggregation a two dimensional fractal model is constructed for diffusion controlled deposition on a surface the fractal geometry of the deposit and the power law behaviour of the quantities characterising the non equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process a qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly

anomalous diffusion in a random velocity field we construct diffusions in random velocity fields which present anomalous superdiffusive behavior the mean square displacement can be made to have any power lawt for 1 2 higher moments and characteristic functions are also investigated

exact mean first passage time on the t graph we consider a simple random walk on the t fractal and we calculate the exact mean time taug to first reach the central node i0 the mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph except i0 by means of analytic techniques based on decimation procedures we find the explicit expression for taug as a function of the generation g and of the volume v of the underlying fractal our results agree with the asymptotic ones already known for diffusion on the t fractal and more generally they are consistent with the standard laws describing diffusion on low dimensional structures

extended vicsek fractals laplacian spectra and their applications extended vicsek fractals evf are the structures constructed by introducing linear spacers into traditional vicsek fractals here we study the laplacian spectra of the evf in particularly the recurrence relations for the laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing laplacian eigenvalues this quantity characterizes the large scale properties such as the gyration radius of the polymeric structures or the global mean first passage time for the random walk processes introduction of the linear spacers leads to local heterogeneities which reveal themselves for example in the dynamics of evf under external forces

explicit ldp for a slowed rw driven by a symmetric exclusion process we consider a random walk rw driven by a simple symmetric exclusion process sse rescaling the rw and the sse in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle the corresponding large deviation rate function can be split into two components the rate function of the sse and the one of the rw given the path of the sse these components have different structures gaussian and poissonian respectively and to overcome this difficulty we make use of the theory of orlicz spaces in particular the component of the rate function corresponding to the rw is explicit

what is the alternative to the alexander orbach relation the alexander orbach ao relation d w 2d f d s connecting the fractal dimension of a random walk s rw trajectory d w or the exponent of anomalous diffusion 2 d w on a fractal structure with the fractal and spectral dimension of the structure itself plays a key role in discussion of dynamical properties of complex systems including living cells and single biomolecules this relation however does not hold universally and breaks down for some structures like diffusion limited aggregates and eden trees we show that the alternative to the ao relation is the explicit dependence of the coefficient of the anomalous diffusion on the system s size i e the absence of its thermodynamical limit the prerequisite for its breakdown is the dependence of the local structure of possible steps of the rw on the system s size the discussion is illustrated by the examples of diffusion on a koch curve ao conform and on a cantor dust violating ao relation

hydrodynamic limit for a zero range process in the sierpinski gasket we consider a system of random walks on graph approximations of the sierpinski gasket coupled with a zero range interaction we prove that the hydrodynamic limit of this system is given by a nonlinear heat equation on the sierpinski gasket

influences of degree inhomogeneity on average path length and random walks in disassortative scale free networks various real life networks exhibit degree correlations and heterogeneous structure with the latter being characterized by power law degree distribution p k k where the degree exponent describes the extent of heterogeneity in this paper we study analytically the average path length apl of and random walks rws on a family of deterministic networks recursive scale free trees rsfts with negative degree correlations and various 2 1 ln 3 ln 2 with an aim to explore the impacts of structure heterogeneity on the apl and rws we show that the degree exponent has no effect on the apl d of rsfts in the full range of d behaves as a logarithmic scaling with the number of network nodes n i e d ln n which is in sharp contrast to the well known double logarithmic scaling d ln ln n previously obtained for uncorrelated scale free networks with 2 3 in addition we present that some scaling efficiency exponents of random walks are reliant on the degree exponent

infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains we study the positivity and regularity of solutions to the fractional porous me dium equations ut sum 0 in 0 for m 1 and s 0 1 with dirichlet boundary data u 0 in 0 n and nonnegative initial condition u 0 u0 0 our first result is a quantitative lower bound for solutions that holds for all positive times t 0 as a consequence we find a global harnack principle stating that for any t 0 solutions are comparable to ds m where d is the distance to this is in sharp contrast with the local case s 1 where the equation has finite speed of propagation after this we study the regularity of solutions we prove that solutions are classical in the interior c in x and c1 in t and establish a sharp cxs m regularity estimate up to the boundary our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form ut f u 0 in both in bounded and unbounded domains 2016 wiley periodicals inc

critical behaviour of the rouse model for gelling polymers it is shown that the traditionally accepted rouse values for the critical exponents at the gelation transition do not arise from the rouse model for gelling polymers the true critical behaviour of the rouse model for gelling polymers is obtained from spectral properties of the connectivity matrix of the fractal clusters that are formed by the molecules the required spectral properties are related to the return probability of a blind ant random walk on the critical percolating cluster the resulting scaling relations express the critical exponents of the shear stress relaxation function and hence those of the shear viscosity and of the first normal stress coefficient in terms of the spectral dimension ds of the critical percolating cluster and the exponents and of the cluster size distribution

eigenvalue spectrum of transition matrix of dual sierpinski gaskets and its applications the eigenvalue spectrum of the transition matrix of a network contains much information about its structural properties and is related to the behavior of various dynamical processes performed on it in this paper we study the eigenvalues of the transition matrix of the dual sierpinski gaskets embedded in d dimensional euclidean spaces we obtain all the eigenvalues as well as their corresponding degeneracies by making use of the spectral decimation technique we then apply the obtained eigenvalues to determine the exact number of spanning trees in the studied fractals and derive an explicit formula of the eigentime identity for random walks taking place on them

close or connected distance and connectivity effects on transport in networks laboratoire de physique th eorique de la mati ere condens ee umr cnrs upmc universit e pierre et marie curie 4 place jussieu 75255 paris cedex dated february 25 2011 we develop an analytical approach which provides the dependence of the mean rst passage time mfpt for random walks on complex networks both on the target connectivity and on the source target distance our approach puts forward two strongly di erent behaviors depending on the type compact or non compact of the random walk in the case of non compact exploration we showthat the mfpt scales linearly with the inverse connectivity of the target and is largely independentof the starting point on the contrary in the compact case the mfpt is controlled by the source target distance and we nd that unexpectedly the target connectivity becomes irrelevant for remotetargets

random walks on complex trees we study the properties of random walks on complex trees we observe that the absence of loops is reflected in physical observables showing large differences with respect to their looped counterparts first both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case second the mean first passage time mfpt displays a logarithmic degree dependence in contrast to the inverse degree shape exhibited in looped networks this deviation can be ascribed to the dominance of source target topological distance in trees to show this we study the distance dependence of a symmetrized mfpt and derive its logarithmic profile obtaining good agreement with simulation results these unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees

explicit eigenvalue bounds of differential operators defined by symmetric positive semi definite bilinear forms abstract recently the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues in this paper the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi definite bilinear forms as an application the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections

first passage properties of the erdos renyi random graph we study the mean time for a random walk to traverse between two arbitrary sites of the erd s renyi random graph we develop an effective medium approximation that predicts that the mean first passage time between pairs of nodes as well as all moments of this first passage time are insensitive to the fraction p of occupied links this prediction qualitatively agrees with numerical simulations away from the percolation threshold near the percolation threshold the statistically meaningful quantity is the mean transit rate namely the inverse of the first passage time this rate varies non monotonically with p near the percolation transition much of this behaviour can be understood by simple heuristic arguments

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

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

distinct scalings for mean first passage time of random walks on scale free networks with the same degree sequence in general the power law degree distribution has profound influence on various dynamical processes defined on scale free networks in this paper we will show that power law degree distribution alone does not suffice to characterize the behavior of trapping problems on scale free networks which is an integral major theme of interest for random walks in the presence of an immobile perfect absorber in order to achieve this goal we study random walks on a family of one parameter denoted by q scale free networks with identical degree sequence for the full range of parameter q in which a trap is located at a fixed site we obtain analytically or numerically the mean first passage time mfpt for the trapping issue in the limit of large network order number of nodes for the whole class of networks the mfpt increases asymptotically as a power law function of network order with the exponent obviously different for different parameter q which suggests that power law degree distribution itself is not sufficient to characterize the scaling behavior of mfpt for random walks at least trapping problem performed on scale free networks

deterministic fractals extracting additional information from small angle scattering data the small angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space in the fractal region the curve i q q d is found to be log periodic with good accuracy and the period is equal to the scaling factor of the fractal here d and i q are the fractal dimension and the scattering intensity respectively the number of periods of this curve coincides with the number of fractal iterations we show that the log periodicity of i q q d in the momentum space is related to the log periodicity of the quantity g r r 3 ensuremath d in the real space where g r is the pair distribution function the minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space it is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets however they remain quite pronounced even at sufficiently large values of polydispersity a generalized self similar vicsek fractal with controllable fractal dimension is introduced and its scattering properties are studied to illustrate the above findings in contrast with the usual methods the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region but also the fractal iteration number the scaling factor and the number of structural units from which the fractal is composed

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed

estimation of interpolation error constants for the p0 and p1 triangular finite elements abstract we give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles for the piecewise linear one we mainly analyze the conforming case but some results are also given for the non conforming case we obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles in particular we explicitly determine the babuska aziz constant which plays an essential role in the interpolation error estimation of the linear triangular finite element the equation for determination is the transcendental equation tan 0 so that the solution can be numerically obtained with desired accuracy and verification such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions

analytic expression for the mean time to absorption for a random walker on the sierpinski gasket the exact analytic expression for the mean time to absorption or mean walk length for a particle performing a random walk on a finite sierpinski gasket with a trap at one vertex is found to be t n 3 n 5 n 1 4 5 n ensuremath 3 n 3 n 1 1 where n denotes the generation index of the gasket and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket in terms of the number n n of sites on the gasket and the spectral dimension stackrel ifmmode tilde else fi d of the gasket the precise asymptotic behavior for large n n is t n ensuremath rightarrow 1 3 2n n 2 stackrel ifmmode tilde else fi d ensuremath sim n 1 464 this serves as a partial check on our result as it is a intermediate between the known results t ensuremath sim n 2 d 1 and t ensuremath sim n mathrm ln n d 2 for random walks on d dimensional euclidean lattices and b consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice