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

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

transport upscaling in heterogeneous aquifers what physical parameters control memory functions 1 power law tailing is often observed in the breakthrough curves btcs of tracer tests tailing is attributed to heterogeneity of aquifer properties and cannot be properly modeled by means of the homogeneous advection dispersion equation mass transfer models e g continuous time random walk method multirate mass transfer or fractional order advection dispersion equations using memory have been widely applied for reproducing observed tails the relationship between memory parameters obtained from btc fitting and the parameters characterizing the heterogeneity of hydraulic properties is still unclear here we investigate the conditions under which heterogeneity produces the type of tailing observed in the field and how memory functions are influenced by measurable heterogeneity parameters e g variance variogram or integral scale of the underlying transmissivity field we find that the slope of a btc in a log log plot is mainly influenced by the connectivity of the underlying permeability field but is insensitive to its variance the slope btc reaches asymptotically 2 as connectivity increases we conclude that an appropriate choice of the memory function allows reproducing the spreading caused by hydraulic heterogeneity but not necessarily the rate of mixing

heavy tailed log hydraulic conductivity distributions imply heavy tailed log velocity distributions 1 equations of contaminant transport describing non gaussian dispersion of solute in heterogeneous porous media have been developed by several authors e g berkowitz and scher 1995 1998 benson 1998 2001 berkowitz et al 2000 and baeumer et al 2005 the equations assume that solute particle movement in time and or space can be described by heavy tailed probability distributions e g levy stable distributions this assumption requires a mechanism for solute particles to move through heterogeneous porous media in a manner that can be approximated with heavy tailed probability distributions this work investigates whether a heavy tailed levy stable distribution of increments in log hydraulic conductivity results in a heavy tailed distribution of increments in log velocity a heavy tailed distribution of increments in log velocity would be a mechanism for solute particle movement in time or space that can be described by a heavy tailed probability distribution a computer algorithm was written to generate realistic hydraulic conductivity fields the algorithm modifies the spectral synthesis method for generating fractional brownian motion i e saupe 1988 and brewer and wheatcraft 1994 and produces self similar hydraulic conductivity fields with a heavy tailed levy stable distribution of increments in log hydraulic conductivity a monte carlo approach was adopted in which 100 hydraulic conductivity fields were generated for each of the following values of 0 8 1 1 1 4 and 1 7 the usgs finite difference groundwater code modflow was used to calculate the velocity fields and the increments in longitudinal log hydraulic conductivity and log velocity from the fields were analyzed to determine whether they were consistent with a heavy tailed levy stable probability density function pdf after determining that the distributions of increments in log hydraulic conductivity and log velocity were consistent with a levy stable pdf the tail parameters describing each pair of hydraulic conductivity and velocity fields were estimated with the nolan 1997 maximum likelihood estimator the relationship between the two tail parameters was then determined there are three conclusions of this research first mandelbrot and pp plots confirm that levy stable distributions of increments in log hydraulic conductivity give rise to levy stable distributions of increments in log velocity second the tail parameter for the increments in log velocity is typically smaller indicating a heavier probability tail than the corresponding tail parameter for the increments in log hydraulic conductivity third there is a positive statistical relation between the two tail parameters

from a large deviations principle to the wasserstein gradient flow a new micro macro passage we study the connection between a system of many independent brownian particles on one hand and the deterministic diffusion equation on the other for a fixed time step h 0 a large deviations rate functional j h characterizes the behaviour of the particle system at t h in terms of the initial distribution at t 0 for the diffusion equation a single step in the time discretized entropy wasserstein gradient flow is characterized by the minimization of a functional k h we establish a new connection between these systems by proving that j h and k h are equal up to second order in h as h 0 this result gives a microscopic explanation of the origin of the entropy wasserstein gradient flow formulation of the diffusion equation simultaneously the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow

chaos or noise difficulties of a distinction in experiments the dynamical behavior of systems is reflected in time series due to the finiteness of the observational data set it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension these restrictions limit our ability to distinguish between signals generated by different systems such as regular chaotic or stochastic ones when analyzed from a time series point of view we propose to classify the signal behavior without referring to any specific model as stochastic or deterministic on a certain scale of the resolution ensuremath epsilon according to the dependence of the ensuremath epsilon ensuremath tau entropy h ensuremath epsilon ensuremath tau and the finite size lyapunov exponent ensuremath lambda ensuremath epsilon on ensuremath epsilon

a mass balance based numerical method for the fractional advection dispersion equation theory and application 1 the inherent heterogeneity of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement and one approach to model solute transport through such media is the continuous time random walk ctrw one special asymptotic case of the ctrw is the fractional advection dispersion equation fade which has proven to be a promising alternative to model anomalous dispersion and has been increasingly used in hydrology to model chemical transport in both surface and subsurface water most practical problems in hydrology have complicated initial and boundary conditions and need to be solved numerically but the numerical solution of the fade is not trivial in this paper we present a finite volume approach to solve the fade where the spatial derivative of the dispersion term is fractional we also give methods to solve different boundary conditions often encountered in practical applications the linear system resulting from the temporal spatial discretization is solved using a semi implicit scheme the numerical method is derived on the basis of mass balance and its accuracy is tested against analytical solutions the method is then applied to simulate tracer movement in a stream and a near saturated hillslope in a naturally structured upland podzol field in northeast scotland

asymptotic distributions of continuous time random walks a probabilistic approach we provide a systematic analysis of the possible asymptotic distributions o one dimensional continuous time random walks ctrws by applying the limit theorems of probability theory biased and unbiased walks of coupled and decoupled memory are considered in contrast to previous work concerning decoupled memory and levy walks we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean obtaining asymmetric levy stable limits suprisingly it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution we discuss interesting properties of walks with an infinite mean waiting time between successive jumps

random walk particle tracking simulations of non fickian transport in heterogeneous media derivations of continuum nonlocal models of non fickian anomalous transport require assumptions that might limit their applicability we present a particle based algorithm which obviates the need for many of these assumptions by allowing stochastic processes that represent spatial and temporal random increments to be correlated in space and time be stationary or non stationary and to have arbitrary distributions the approach treats a particle trajectory as a subordinated stochastic process that is described by a set of langevin equations which represent a continuous time random walk ctrw convolution based particle tracking cbpt is used to increase the computational efficiency and accuracy of these particle based simulations the combined ctrw cbpt approach enables one to convert any particle tracking legacy code into a simulator capable of handling non fickian transport

a discrete velocity stationary wigner equation this paper is concerned with the one dimensional stationary linear wigner equation a kinetic formulation of quantum mechanics specifically we analyze the well posedness of the boundary value problem on a slab of the phase space with given inflow data for a discrete velocity model we find that the problem is uniquely solvable if zero is not a discrete velocity otherwise one obtains a differential algebraic equation of index 2 and hence the inflow data make the system overdetermined

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

evolution of the system with multiplicative noise the governed equations for the order parameter one and two time correlators are obtained for systems with white multiplicative noise we consider the noise whose amplitude depends on stochastic variable as xa where 0 a 1 it turns out that the equation for autocorrelator includes an anomalous average of the power law function with the fractional exponent 2a determination of this average for the stochastic system with a self similar phase space is performed it is shown that at when the system is disordered the correlator behaves in the course of time non monotonically whereas the autocorrelator increases monotonically at the phase portrait of the system divides into two domains at small initial values of the order parameter the system evolves to a disordered state as above within the ordered domain it is attracted to the point with finite values of the autocorrelator and order parameter the long time asymptotes are defined to show that within the disordered domain the autocorrelator decays hyperbolically and the order parameter behaves as a power law function with fractional exponent 2 1 a correspondingly within the ordered domain the behaviour of both dependencies is exponential with an index proportional to

some insights in superdiffusive transport abstract in a wide range of systems the relaxation in response to an initial pulse has been experimentally found to follow a nonlinear relationship for the mean squared displacement of the kind x 2 t t where may be greater or smaller than 1 such phenomena have been described under the generic term of anomalous diffusion levy flights stochastic processes lead to superdiffusive behaviour 1 2 and have been recently proposed to model among the others the subsurface contaminant spread in highly heterogeneous media under the effects of water flow in this paper within the continuous time random walk ctrw approach to anomalous diffusion we compare the analytical solution of the approximated fractional diffusion equation fde with the monte carlo one obtained by simulating the superdiffusive behaviour of an ensemble of particle in a medium we show that the two are neatly different as the process approaches the standard diffusive behaviour we argue that this is due to a truncation in the fourier space expansion introduced by the fde approach we propose a second order correction to this expansion and numerically solve the ctrw model under this hypothesis the accuracy of the results thus obtained is validated through monte carlo simulation over all the superdiffusive range the same kind of discrepancy is shown to occur also in the derivation of the fractional moments of the distribution analogous corrections are proposed and validated through the monte carlo approach

spatial behavior of anomalous transport we present a general derivation of one dimensional spatial concentration distributions for anomalous transport regimes such transport can be captured in the framework of a continuous time random walk with a broad transition time distribution this general theory includes a fokker planck equation as a particular limiting case all of the concentration profiles as well as the associated temporal first passage time distributions can be written in terms of a single special function that belongs to the class of fox functions in addition we consider the first two moments of the spatial concentration distributions and determine not only their scaling behavior with time but also the coefficients and correction terms

theory of powders starting from the observations that powders have a large number of particles and reproducible properties we show how statistical mechanics applies to powders the volume of the powder plays the role of the energy in conventional statistical mechanics with the hypothesis that all states of a specified volume are equally probable we introduce a variable x the compactivity which is analogous to the temperature in thermodynamics some simple models are considered which demonstrate how the problems involved can be tackled using the concept of compactivity there is an increasing interest in applying the methods of statistical mechanics and of transport theory to systems which are neither atomistic nor in equilibrium but which still fulfil a remaining tenet of statistical physics which is that systems can be completely defined by a very small number of parameters and can be constructed in a reproducible way powders fall into this category if a powder consists for example of uniform cubes of salt and is poured into a container falling at low density uniformly from a great height one expects a salt powder of a certain density repeating the preparation reproduces the same density a treatment such as shaking the powder by a definite routine produces a new density and the identical routine applied to another sample of the initial powder will result in the same final density clearly a maxwell demon could arrange the little cubes of nacl to make a material of different properties to that of our experiment but if such demonics are ignored and we restrict ourselves to extensive operations such as stirring shaking compressing all actions which do not act on grains individually then well defined states of the powder result in this paper we will set up a framework for describing the state of the powder basing our development on anologies with statistical mechanics some attempts have been made to apply information theory ideas directly to powders

existence and uniqueness for nonlinear boundary value problems in kinetic theory a boundary value problem for the stationary nonlinear boltzmann equation in a slab has been examined in a weightedl space it has been proved that the problem possesses a unique solution for boundary data small enough the proof is based on the implicit function theorem it has also been shown that for the linearized problem the fredholm alternative applies

theory of anomalous chemical transport in random fracture networks we show that dominant aspects of chemical particle transport in fracture networks char22 non gaussian propagation char22 result from subtle features of the steady flow field distribution through the network this is an outcome of a theory based on a continuous time random walk formalism structured to retain the key space time correlations of particles as they are advected across each fracture segment the approach is designed to treat the complex geometries of a large variety of fracture networks and multiscale interactions monte carlo simulations of steady flow in these networks are used to determine the distribution of velocities in individual fractures as a function of their orientation the geometry and velocity distributions are used in conjunction with particle mixing rules to map the particle movement between fracture intersections onto a joint probability density ensuremath psi mathbf r t the chemical concentration plume and breakthrough curves can then be calculated analytically particle tracking simulations on these networks exhibit the same non gaussian profiles demonstrating quantitative agreement with the theory the analytic plume shapes display the same basic behavior as extensive field observations at the columbus air force base mississippi the quantitative correlation between the time dependence of the mean and standard deviation of the field plumes and their shape is predicted by the theory

flow and travel time statistics in highly heterogeneous porous media 1 in this paper we present flow and travel time ensemble statistics based on a new simulation methodology the adaptive fup monte carlo method afmcm as a benchmark case we considered two dimensional steady flow in a rectangular domain characterized by multi gaussian heterogeneity structure with an isotropic exponential correlation and lnk variance y2 up to 8 advective transport is investigated using the travel time framework where lagrangian variables e g velocity transverse displacement or travel time depend on space rather than on time we find that eulerian and lagrangian velocity distributions diverge for increasing lnk variance due to enhanced channeling transverse displacement is a nonnormal for all y2 and control planes close to the injection area but after xiy 20 was found to be nearly normal even for high y2 travel time distribution deviates from the fickian model for large lnk variance and exhibits increasing skewness and a power law tail for large lnk variance the slope of which decreases for increasing distance from the source no anomalous features are found second moment of advective transport is analyzed with respect to the covariance of two lagrangian velocity variables slowness and slope which are directly related to the travel time and transverse displacement variance which are subsequently related to the longitudinal and transverse dispersion we provide simple estimators for the eulerian velocity variance travel time variance slowness and longitudinal dispersivity as a practical contribution of this analysis both two parameter models considered the advection dispersion equation and the lognormal model provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media we identify the need for further theoretical and experimental scrutiny of early arrival times and the need for computing higher order moments for a more accurate characterization of the travel time probability density function a brief discussion is presented on the challenges and extensions for which afmcm is suggested as a suitable approach

evolution of a stochastic system within the framework of tsallis statistics abstract the general approach of a nonlinear fokker planck equation is applied to investigate the behavior of main statistical moments of a stochastic system it was shown that the system described by tsallis statistics can undergo transitions inherent to multiplicative noise induced transitions the scaling laws of motion are defined to identify an anomalous time behavior of the system with multiplicative noise

measure solutions of the steady boltzmann equation in a slab it is shown that the steady boltzmann equation in a slab 0 a has solutionsx x such that the ingoing boundary measures 0 0 and 0 can be prescribed a priori the collision kernel is truncated such that particles with smallx component of the velocity have a reduced collision rate

passage times of asymmetric anomalous walks with multiple paths we investigate the transient and the long time behaviour of asymmetric anomalous walks in heterogeneous media two types of disorder are worked out explicitly weak and strong disorder in addition the occurrence of disordered multiple paths is considered we calculate the first passage time distribution of the associated stochastic transport process we discuss the occurrence of the crossover from a power law to an exponential decay for the long time behaviour of the distribution of the first passage times of disordered biased walks

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

generalized diffusion equation with fractional derivatives within renyi statistics by using the zubarev nonequilibrium statistical operator method and the liouville equation with fractional derivatives a generalized diffusion equation with fractional derivatives is obtained within the renyi statistics averaging in generalized diffusion coefficient is performed with a power distribution with the renyi parameter q

a boundary value problem for the two dimensional broadwell model it is shown that a certain boundary value problem for the steady two dimensional broadwell model on a rectangle has a solution the boundary conditions specify the ingoing particle densities on each side of the rectangle

origins of anomalous transport in heterogeneous media structural and dynamic controls anomalous or non fickian transport is ubiquitous in the context of tracer migration in geological formations we quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous porous medium under uniform in the mean flow conditions we focus on anomalous transport which arises in the complex flow patterns of lognormally distributed hydraulic conductivity k fields with several decades of k values transport in the domains is determined by a particle tracking technique and characterized by breakthrough curves btcs the btc averaged over multiple realizations demonstrates anomalous transport in all cases which is accounted for entirely by a power law distribution t 1 of local transition times the latter is contained in the probability density function t of transition times embedded in the framework of a continuous time random walk ctrw a unique feature of our analysis is the derivation of t as a function of parameters quantifying the heterogeneity of the domain in this context we first establish the dominance of preferential pathways across each domain and characterize the statistics of these pathways by forming a particle visitation weighted histogram hw k of the hydraulic conductivity by converting the ln k dependence of hw k into time we demonstrate the equivalence of hw k and t and delineate the region of hw k that forms the power law of t this thus defines the origin of anomalous transport analysis of the preferential pathways clearly demonstrates the limitations of critical path analysis and percolation theory as a basis for determining the origin of anomalous transport furthermore we derive an expression defining the power law exponent in terms of the hw k parameters the equivalence between hw k and t is a remarkable result particularly given the nature of the k heterogeneity the complexity of the flow field within each realization and the statistics of the particle transitions

continuous time random walks for analyzing the transport of a passive tracer in a single fissure 1 the selection of the correct model for describing contaminant transport in fractured media remains a challenging problem in subsurface hydrology in this work the experimental data corresponding to nonsorbing tracer runs in natural fissures of moreno et al 1985 are analyzed the advection dispersion equation ade model yields a poor fit in the late time regions of the experimental breakthrough curves denoting the existence of anomalous or non fickian transport continuous time random walks ctrw are proposed for describing this kind of transport using two of the most studied forms for the transition rate probability the asymptotic and truncated power law models the ctrw approach produces better fits to the experimental data than the ade model in all the simulations a moderately dispersive transport was found which is interpreted as being as a result of the effects of the small and intermediate scale heterogeneities

loeb solutions of the boltzmann equation existence problems for the boltzmann equation constitute a main area of research within the kinetic theory of gases and transport theory the present paper considers the spatially periodic case with l1 initial data the main result is that the loeb subsolutions obtained in a preceding paper are shown to be true solutions the proof relies on the observation that monotone entropy and finite energy imply loeb integrability of non standard approximate solutions and uses estimates from the proof of the h theorem two aspects of the continuity of the solutions are also considered

on nonlinear stationary half space problems in discrete kinetic theory we show that every steady discrete velocity model of the boltzmann equation on the real line i d dx fi ci f which satisfies anh theorem and for which all i 0 has solutions on the half line 0 which take prescribed non negativefi o if i 0 and approach a certain manifold of maxwellians asx such solutions give the density distribution in a knudsen boundary layer in the discrete velocity case

boundary value problems for the steady boltzmann equation we discuss steady boundary value problems for the boltzmann equation with inflow and diffusive boundary conditions in one two and three dimensions with suitable truncations of the collision kernel general existence and uniqueness results are obtained if the domain is sufficiently small in one dimension the existence of solutions on general intervals is obtained by abstract fixed point theory

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

upscaling transport with mass transfer models mean behavior and propagation of uncertainty 1 the choice of an adequate large scale conceptual transport model constitutes a major challenge associated with the upscaling of solute transport among the different alternatives to the classical advection dispersion model the multirate mass transfer model has been proposed as a valuable and convenient alternative to model the large scale behavior of solute transport this paper evaluates the use of mass transfer models as a constitutive equation for upscaling solute transport to achieve this we compare monte carlo simulations of solute transport at two different support scales transport simulations performed at the smallest scale represent a set of reference transport solutions described at a high resolution which are contrasted against transport simulations obtained using an upscaled model low resolution several formulations of the multirate mass transfer model which differ in the type of memory function single rate double rate and truncated power law are used as a constitutive transport equation the large scale scenario represents a simplified model obtained by partially homogenizing the reference solution results show that the double rate and the truncated power law mass transfer models are capable of properly describing the ensemble average behavior of the main features associated with the integrated breakthrough curves however the uncertainty associated with the upscaled mass transfer models was substantially smaller than that attributed to the reference solution importantly the cumulative distribution function of concentrations associated with the upscaled model follows a distribution similar to the reference solution but with smaller statistical dispersion the reason is that while appropriate memory functions can be used to preserve the residence time distribution of mass particles during upscaling the lack of memory in space prevents the model from reproducing mass fluxes in all directions specifically the reproduction of mass fluxes taking place at the interface between two homogenized blocks of the upscaled model is not satisfied thus providing a poor description of the spatial distribution of mass particles in a given realization

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

continuous time random walks revisited first passage time and spatial distributions abstract we investigate continuous time random walk ctrw theory which often assumes an algebraic decay for the single transition time probability density function pdf t t 1 for large times t in this form is a constant 0 moreover simulated fptd distributions are observed to approximate closely different levy stable distributions with growing as travel distance increases the modifications of ctrw distributions also point to the limitations of fractional derivative equation fde approaches appearing in the literature we propose an alternative form of a fde corresponding to our ctrw distributions in the biased 1d case for all 0 1

moment analysis for spatiotemporal fractional dispersion 1 the evolution of the first five nonnegative integer order spatial moments corresponding to the mass mean variance skewness and kurtosis are investigated systematically for spatiotemporal nonlocal fractional dispersion three commonly used fractional order transport equations including the time fractional advection dispersion equation time fade the fractal mobile immobile mim equation and the fully fractional advection dispersion equation ffade are considered analytical solutions verify our numerical results and reveal the anomalous evolution of the moments following adams and gelhar s 1992 work on the classical ade we find that a simultaneous analysis of all moments is critical in discriminating between different nonlocal models the evolution of dispersion among the subdiffusive to superdiffusive rates is then further explored numerically by a non markovian random walk particle tracking method that can be used for any heterogeneous boundary or initial value problem in three dimensions both the analytical and the numerical results also show the similarity at the early time and the difference at the late time of moment growth for solutes in different phases mobile versus total described by the mim models further simulations of the 1 d bromide snapshots measured at the made experiments using all three models with parameters fitted by the observed zeroth to fourth moments indicate that 1 both the time and space nonlocality strongly affect the solute transport at the made site 2 all five spatial moments should be considered in transport model selection and calibration because those up to the variance cannot effectively discriminate between nonlocal models and 3 the log concentration should be used when evaluating the plume leading edge and the effects of space nonlocality

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

self organized criticality within fractional lorenz scheme abstract the theory of a flux steady state related to avalanche formation is presented for the simplest model of a sand pile within the framework of the lorenz approach the stationary values of sand velocity and sand pile slope are derived as functions of a control parameter driven sand pile slope the additive noise of above values are introduced for building a phase diagram where the noise intensities determine both avalanche and non avalanche domains as well as mixed one corresponding to the soc regime the last domain is crucial to affect of the noise intensity of the vertical component of sand velocity and especially sand pile slope to address to a self similar behavior a fractional feedback is used as an efficient ingredient of the modified lorenz system in the spirit of edwards paradigm an effective thermodynamics is introduced to determine a distribution over an avalanche ensemble with negative temperature steady state behavior of the moving grains number as well as non extensive values of entropy and energy is studied in detail the power law distribution over the avalanche size is described within a fractional lorenz scheme where the energy noise plays a crucial role this distribution is shown to be a solution of both fractional and nonlinear fokker planck equation as a result we obtain new relations between the exponent of the size distribution fractal dimension of phase space characteristic exponent of multiplicative noise number of governing equations dynamical exponents and non extensivity parameter

transport equation evaluation of coupled continuous time random walks the transport behavior of a migrating particle in a disordered medium is exhibited in the solution of a transport equation derived from a coupled continuous time random walk ctrw a core aspect of ctrw is the spectrum of transitions in displacement s and time t s t that characterizes the disordered system which determine the transport in many applications the ctrw approach has successfully accounted for the anomalous or non fickian nature of the particle plume propagation based on a power law dependence t in a decoupled p s t approximation to s t for example this power law dependence in t derives from the complex darcy flow fields in geological formations recently the fully coupled ctrw was analyzed using a particle tracking approach demonstrating that the decoupled approximation is valid only for a compact distribution of s in this paper we solve the nonlocal in time transport equation with a s t containing a power law dependence in both s a levy like distribution and t which necessitates the strong s t coupling we show enhanced transport behavior relative to the plume propagation behavior reported in the literature that derives from the rare large displacements in s limited by the transition t the interplay between the two coupled power laws is clearly shown in the changes in the breakthrough curves in the arrival times dispersion and dependence on the velocity v s t distribution similar enhancements are exhibited in the particle tracking results

a boundary value problem for discrete velocity models on considere deux problemes aux valeurs limites non lineaires concernant un ecoulement stationnaire a une dimension entre des plaques paralleles

how self organized criticality works a unified mean field picture we present a unified dynamical mean field theory based on the single site approximation to the master equation for stochastic self organized critical models in particular we analyze in detail the properties of sandpile and forest fire ff models in analogy with other nonequilibrium critical phenomena we identify an order parameter with the density of active sites and control parameters with the driving rates depending on the values of the control parameters the system is shown to reach a subcritical absorbing or supercritical active stationary state criticality is analyzed in terms of the singularities of the zero field susceptibility in the limit of vanishing control parameters the stationary state displays scaling characteristics of self organized criticality soc we show that this limit corresponds to the breakdown of space time locality in the dynamical rules of the models we define a complete set of critical exponents describing the scaling of order parameter response functions susceptibility and correlation length in the subcritical and supercritical states in the subcritical state the response of the system to small perturbations takes place in avalanches we analyze their scaling behavior in relation with branching processes in sandpile models because of conservation laws a critical exponents subset displays mean field values ensuremath nu frac 1 2 and ensuremath gamma 1 in any dimensions we treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects we present numerical simulations that confirm our results in the case of the forest fire model our approach can distinguish between different regimes soc ff and deterministic ff studied in the literature and determine the full spectrum of critical exponents