a note on the markov property of stochastic processes described by nonlinear fokker planck equations abstract we study the markov property of processes described by generalized fokker planck equations that are nonlinear with respect to probability densities such as mean field fokker planck equations and fokker planck equations related to generalized thermostatistics we show that their transient solutions describe non markov processes in contrast stationary solutions can describe markov processes as a result nonlinear fokker planck equations can be used to model transient non markov processes that converge to stationary markov processes

a hybrid method for hydrodynamic kinetic flow part ii coupling of hydrodynamic and kinetic models in this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic kinetic problems in which the knudsen number may span from equilibrium to highly rarefied regimes our approach is characterized by using the full boltzmann equation for the kinetic regime the compressible euler equations for equilibrium with a buffer zone in which the bgk es equation is used to represent the transition between fully kinetic to equilibrium flows in this fashion the boltzmann solver is used only when the collision integral is non stiff and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities thus in principle the same mesh size and time steps can be used in the whole computation moreover the time step is limited only by convective terms since the boltzmann solver is applied only in wholly kinetic regimes we use the reduced noise dsmc scheme we have proposed in part i of the present work this ensures a smooth exchange of information across the different domains with a natural way to construct interface numerical fluxes several tests comparing our hybrid scheme with full boltzmann dsmc computations show the good agreement between the two solutions on a wide range of knudsen numbers

fractional laplacian time space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency frequency dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2 the standard time domain partial differential equation models can describe merely two extreme cases of frequency independent and frequency squared dependent attenuations the otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation in this study a linear integro differential equation wave model was developed for the anomalous attenuation by using the space fractional laplacian operation and the strategy is then extended to the nonlinear burgers equation a new definition of the fractional laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional laplacian under the szabo s smallness approximation where attenuation is assumed to be much smaller than the wave number the linear model is found consistent with arbitrary frequency power law dependency

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

associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise we study associative memory based on temporal coding in which successful retrieval is realized as an entrainment in a network of simple phase oscillators with distributed natural frequencies under the influence of white noise the memory patterns are assumed to be given by uniformly distributed random numbers on 0 2 ensuremath pi so that the patterns encode the phase differences of the oscillators to derive the macroscopic order parameter equations for the network with an extensive number of stored patterns we introduce an effective transfer function by assuming a fixed point equation of the form of the thouless anderson palmer equation which describes the time averaged output as a function of the effective time averaged local field properties of the networks associated with synchronization phenomena for a discrete symmetric natural frequency distribution with three frequency components are studied based on the order parameter equations and are shown to be in good agreement with the results of numerical simulations two types of retrieval states are found to occur with respect to the degree of synchronization when the size of the width of the natural frequency distribution is changed

ellipsoidal bgk model for polyatomic molecules near maxwellians a dichotomy in the dissipation estimate abstract we consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal bgk model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic maxwellian we observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter reaches zero accordingly we employ two different sets of micro macro system to derive the full coercivity and close the energy estimate

locally solving fractional laplacian viscoacoustic wave equation using hermite distributed approximating functional method abstractaccurate seismic modeling in realistic media serves as the basis of seismic full waveform inversion and imaging recently viscoacoustic seismic modeling incorporating attenuation effects has been performed by solving a fractional laplacian viscoacoustic wave equation in this equation attenuation being spatially heterogeneous is represented partially by the spatially varying power of the fractional laplacian operator previously approximated by the global fourier method we have developed a local spectral approach based on the hermite distributed approximating functional hdaf method to implement the fractional laplacian in the viscoacoustic wave equation our approach combines local methods simplicity and global methods accuracy several numerical examples are developed to evaluate the feasibility and accuracy of using the hdaf method for the frequency independent q fractional laplacian wave equation

option pricing formulas based on a non gaussian stock price model options are financial instruments that depend on the underlying stock we explain their non gaussian fluctuations using the nonextensive thermodynamics parameter q a generalized form of the black scholes bs partial differential equation and some closed form solutions are obtained the standard bs equation q 1 which is used by economists to calculate option prices requires multiple values of the stock volatility known as the volatility smile using q 1 5 which well models the empirical distribution of returns we get a good description of option prices using a single volatility

markov models of non gaussian exponentially correlated processes and their applications we consider three different methods of generating non gaussian markov processes with given probability density functions and exponential correlation functions all models are based on stochastic differential equations a number of analytically treatable examples are considered the results obtained can be used in different areas such as telecommunications and neurobiology

discrete singular convolution finite subdomain method for the solution of incompressible viscous flows this paper proposes a discrete singular convolution finite subdomain method dsc fsm for the analysis of incompressible viscous flows in multiply connected complex geometries the dsc algorithm has its foundation in the theory of distributions a block structured grid of fictitious overlapping interfaces is designed to decompose a complex computational geometry into a finite number of subdomains in each subdomain the governing navier stokes equations are discretized by using the dsc algorithm in space and a third order runge kutta scheme in time information exchange between fictitious overlapping zones is realized by using the dsc interpolating algorithm the taylor problem with decaying vortices could be solved to machine precision with an excellent comparison against the exact solution the reliability of the proposed method is tested by simulating the flow in a lid driven cavity the utility of the dsc fsm approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results compare well with the numerical and experimental data available in the literature

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

numerical comparison between the boltzmann and es bgk models for rarefied gases abstract rarefied gas flows obey the boltzmann equation but numerical simulations of this equation are not always possible so that simpler models have been introduced the es bgk equation is one of these models it gives the correct transport coefficients for the navier stokes approximation so that boltzmann or es bgk simulations are expected to give the same results for dense gases but in the case of a rarefied flow complete numerical comparisons are needed in this paper we present numerical comparisons between the two models in transitional regimes where the es bgk model is expected to be useful for reentry flows around a compression ramp and a plate we also emphasize that the es bgk model gives flow predictions closer to the boltzmann result than the simpler bgk model

generation of compound non gaussian random processes with a given correlation function a compound representation of random processes is considered each independent component of such a process is considered as the solution of the proper stochastic differential equation sde this guarantees that the process obtained is stationary and ergodic the analytical expressions are developed for nonlinear coefficients of the generating sde theoretical results are compared with numerical simulation

a law of large numbers for moderately interacting diffusion processes we consider two special models of interacting diffusion processes and derive in the limit as the number of different processes tends to infinity and the interaction is rescaled in a suitable moderate way a law of large numbers for the empirical processes as limit dynamics we obtain certain nonlinear diffusion equations

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

microscopically implicit macroscopically explicit schemes for the bgk equation in this work a new class of numerical methods for the bgk model of kinetic equations is introduced the schemes proposed are implicit with respect to the distribution function while the macroscopic moments are evolved explicitly in this fashion the stability condition on the time step coincides with a macroscopic cfl evaluated using estimated values for the macroscopic velocity and sound speed thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either with the technique proposed here the updating of the distribution function requires the solution of a linear system of equations even though the bgk model is highly non linear thus the proposed schemes are particularly effective for high or moderate mach numbers where the macroscopic cfl condition is comparable to accuracy requirements we show results for schemes of order 1 and 2 and the generalization to higher order is sketched

numerical method for the nonlinear fokker planck equation a practical method based on distributed approximating functionals dafs is proposed for numerically solving a general class of nonlinear time dependent fokker planck equations the method relies on a numerical scheme that couples the usual path integral concept to the daf idea the high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear fokker planck equation and the method is compared with the accurate k point stirling interpolation formula finite difference method the approach is also used successfully to solve a nonlinear self consistent dynamic mean field problem for which both the cumulant expansion and scaling theory have been found by drozdov and morillo phys rev e 54 931 1996 to be inadequate to describe the occurrence of a long lived transient bimodality the standard interpretation of the transient bimodality in terms of the flat region in the kinetic potential fails for the

cauchy problem for the ellipsoidal bgk model of the boltzmann equation ellipsoidal bgk model es bgk is a generalized version of the original bgk model designed to reproduce the physically correct prandtl number in the navier stokes limit in this paper we study the cauchy problem for the es bgk model under the condition of finite initial mass energy and entropy equivalence type estimates for the temperature tensor are crucially used

multivariate ornstein uhlenbeck processes with mean field dependent coefficients application to postural sway we study the transient and stationary behavior of many particle systems in terms of multivariate ornstein uhlenbeck processes with friction and diffusion coefficients that depend nonlinearly on process mean fields mean field approximations of this kind of system are derived in terms of fokker planck equations in such systems multiple stationary solutions as well as bifurcations of stationary solutions may occur in addition strictly monotonically decreasing steady state autocorrelation functions that decay faster than exponential functions are found which are used to describe the erratic motion of the center of pressure during quiet standing

numerical treatment of stochastic models used in statistical systems and financial markets in this paper by means of the variational iteration method numerical solutions are computed for some stochastic models without any linearization or weak assumptions two stochastic models the fokker planck equation for non equilibrium statistical systems and the black scholes model for pricing stock options are solved numerically in this approach the solution is found in the form of a convergent series with easily computed components the behavior of the approximate solutions is shown graphically

nonlinear fokker planck equations and generalized entropies abstract we consider maximum entropy solutions to nonlinear fokker planck equations within general thermostatistical formalisms we show that the family of generalized nonextensive entropies introduced by tsallis is the only one that has an associated family of nonlinear fokker planck equations endowed with time dependent maxent solutions of a generalized gaussian type consequently the natural association that arises between tsallis thermostatistical formalism and special families of nonlinear fokker planck equations is not a universal feature of general thermostatistical formalisms but a particular feature of the tsallis one

on a general link between anomalous diffusion and nonextensivity stochastic processes of systems with composable entropy measures are studied the processes are described by fokker planck equations which are nonlinear with respect to their probability densities and whose diffusion coefficients are derived from the entropy measures in questions in nonextensive extensive systems anomalous normal diffusion is found

stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations abstract we discuss two fundamental aspects of fokker planck equations that are nonlinear with respect to probability densities first we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions second we demonstrate that nonlinear fokker planck equations can be interpreted as linear fokker planck equations describing nonlinear families of markov diffusion processes we exploit this finding in order to derive complete hierarchies of probability densities from nonlinear fokker planck equations

modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional laplacians we evaluated a time domain wave equation for modeling acoustic wave propagation in attenuating media the wave equation was derived from kjartansson s constant q constitutive stress strain relation in combination with the mass and momentum conservation equations our wave equation expressed by a second order temporal derivative and two fractional laplacian operators described very nearly constant q attenuation and dispersion effects the advantage of using our formulation of two fractional laplacians over the traditional fractional time derivative approach was the avoidance of time history memory variables and thus it offered more economic computations in numerical simulations we formulated the first order constitutive equations with the perfectly matched layer absorbing boundaries the temporal derivative was calculated with a staggered grid finite difference approach the fractional laplacians are calculated in the spatial frequency domain using a fourier pseudospectral implementation we validated our numerical results through comparisons with theoretical constant q attenuation and dispersion solutions field measurements from the pierre shale and results from 2d viscoacoustic analytical modeling for the homogeneous pierre shale we also evaluated different formulations to show separated amplitude loss and dispersion effects on wavefields furthermore we generalized our rigorous formulation for homogeneous media to an approximate equation for viscoacoustic waves in heterogeneous media we then investigated the accuracy of numerical modeling in attenuating media with different q values and its stability in largecontrast heterogeneous media finally we tested the applicability of our time domain formulation in a heterogeneous medium with high attenuation

classical solutions for the ellipsoidal bgk model with fixed collision frequency abstract we establish the existence of global in time smooth solutions for the ellipsoidal bgk model which is a variant of the bgk model for the boltzmann equation designed to yield the correct prandtl number in the hydrodynamic approximation at the navier stokes level for this we carefully design a function space which captures the growth of the solution in a weighted sobolev norm and show that the ellipsoidal relaxation operator is lipschitz continuous in the induced metric this approach is restricted to the case when the collision frequency does not depend on the macroscopic field but no smallness on the initial data is required

nonlinear fokker planck equations whose stationary solutions make entropy like functionals stationary abstract the present study extends the correspondence principle of martinez et al that establishes a link between nonlinear fokker planck equations nlfpes and the variational principle approach of the theory of canonical ensembles by virtue of the extended correspondence principle we reobtain results of kaniadakis and quarati for bose and fermi systems and find relations similar to those derived by plastino and plastino and recently by borland concerning the entropy of nonextensive thermostatistics introduced in physics by tsallis moreover we propose nlfpes related to the renyi entropy and to the entropy proposed by sharma and mittal the latter comprises tsallis entropy and the renyi entropy

two stage coarsening mechanism in a kinetically constrained model of an attractive colloid we study an attractive version of the east model using the real space renormalization group rg introduced by stella et al the former is a kinetically constrained model with an ising like interaction between excitations and shows striking agreement with the phenomonology of attractive colloidal systems we find that the rg predicts two nonuniversal dynamic exponents which suggests that in the out of equilibrium regime the model coarsens via a two stage mechanism we explain this mechanism physically and verify this prediction numerically in addition we predict that the characteristic relaxation time of the model is a nonmonotonic function of attraction strength again in agreement with numerical results

n dimensional nonlinear fokker planck equation with time dependent coefficients an n dimensional nonlinear fokker planck equation is investigated here by considering the time dependence of the coefficients where drift controlled and source terms are present we exhibit the exact solution based on the generalized gaussian function related to the tsallis statistics furthermore we show that a rich class of diffusive processes including normal and anomalous ones can be obtained by changing the time dependence of the coefficients

numerical solutions of fokker planck equations with drift admitting jumps we develop a finite difference scheme based on a grid staggered by flux points and solution points to solve fokker planck equations with drift admitting jumps to satisfy the matching conditions at the jumps i e the continuities of the propagator and the probability current the jumps are set to be solution points and used to divide the solution domain into subdomains while the values of the probability current at flux points are obtained within each subdomain the values of its first derivative at solution points are evaluated by using stencils across the subdomains several benchmark problems are solved numerically to show the validity of the proposed scheme

how to quantify deterministic and random influences on the statistics of the foreign exchange market it is shown that price changes of the u s dollar german mark exchange rates upon different delay times can be regarded as a stochastic marcovian process furthermore we show how kramers moyal coefficients can be estimated from the empirical data finally we present an explicit fokker planck equation which models very precisely the empirical probability distributions in particular their non gaussian heavy tails

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

stochastic evolution of systems of particles obeying an exclusion principle abstract we present numerical solutions of a nonlinear fokker planck equation to describe a fermi gas the model equation g kaniadakis p quarati phys rev e 49 1994 5103 includes a nonlinear term due to the pauli exclusion principle which has been disregarded up to now a new numerical method to study the time evolution of the distribution function is presented the method can be applied to any potential and allows us to describe accurately and efficiently a fermi gas in this work we focus on the differences between the results of the linear case in which the effects of the exclusion principle are not considered and the nonlinear case in which these effects are taken into account

a langevin approach for the microscopic dynamics of nonlinear fokker planck equations abstract for stochastic processes defined by nonlinear fokker planck equations a microscopic dynamics is proposed in terms of generalized langevin equations these langevin equations can for example be used to model stochastic processes with mean field interactions and random walks related to the generalized thermostatistics suggested by tsallis

integral kinetic method for one dimension the spherical case the integral method to numerically calculate the time evolution of kinetic systems is discussed and improved for one dimensional problems the new approach is applied to the solution of the spherically symmetric one component plasma kinetic evolution the results are compared with those obtained by means of the finite difference solution to the equivalent fokker planck kinetic equation

a mini tutorial on measure valued markov processes and nonlinear martingale problems as a reply to mccauley s comment abstract one goal of this mini tutorial is to provide an introduction into the theory of measure valued markov processes and nonlinear martingales defined by strongly nonlinear fokker planck equations and to discuss the physical relevance of the associated processes another goal is to reply to mccauley s comment on t d frank physica a 331 391 2004 the tutorial addresses in detail two approaches found in physics and mathematics the first approach exploits a mapping between linear and nonlinear fokker planck equations the second approach exploits martingale theory several examples of markov processes and martingales in quantum mechanical nonextensive and self organizing systems defined by nonlinear fokker planck equations are discussed

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

convergence of a semi lagrangian scheme for the bgk model of the boltzmann equation recently a new class of semi lagrangian methods for the bgk model of the boltzmann equation has been introduced f filbet and g russo kinet relat models 2 2009 pp 231 250 g russo and p santagati a new class of large time step methods for the bgk models of the boltzmann equation arxiv 1103 5247 p santagati high order semi lagrangian methods for the bgk model of the boltzmann equation ph d thesis university of catania italy 2007 these methods work in a satisfactory way either in a rarefied or a fluid regime moreover because of the semi lagrangian feature the stability property is not restricted by the cfl condition these aspects make them very attractive for practical applications in this paper we prove that the discrete solution of the scheme converges in a weighted l 1 norm to the unique smooth solution by deriving an explicit error estimate

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

two efficient modeling schemes for fractional laplacian viscoacoustic wave equation abstractrecently a decoupled fractional laplacian viscoacoustic wave equation has been developed based on the constant q model to describe wave propagation in heterogeneous media we have developed two efficient modeling schemes to solve the decoupled fractional laplacian viscoacoustic wave equation both schemes can cope with spatial variable order fractional laplacians conveniently and thus are applicable for modeling viscoacoustic wave propagation in heterogeneous media both schemes are based on fast fourier transform and have a spectral accuracy in space the first scheme solves a modified wave equation with constant order fractional laplacians instead of spatial variable order fractional laplacians due to separate discretization of space and time the first scheme has only first order accuracy in time differently the second scheme is based on an analytical wave propagator and has a higher accuracy in time to increase computational efficiency of the second modeling scheme we have adopted the

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

numerical solution of the stationary fpk equation using shannon wavelets abstract the fokker planck kolmogorov fpk equation governs the probability density function p d f of the dynamic response of a particular class of linear or non linear system to random excitation this paper proposes a numerical method for calculating the stationary solution of the fpk equation which is based upon the weighted residual approach using shannon wavelets as shape functions the method is developed here for an n dimensional system and its relationship with the distributed approximating functional daf approach is investigated for the purposes of validation numerical results obtained using the proposed method are compared with available exact solutions and numerical solutions for some non linear oscillators for the systems considered excellent results over the main body and tails of the marginal distributions are obtained the accuracy and efficiency of the method are investigated in comparison to the finite element method fem

finite genome length corrections for the mean fitness and gene probabilities in evolution models using the hamilton jacobi equation approach to study genomes of length l we obtain 1 l corrections for the steady state population distributions and mean fitness functions for horizontal gene transfer model as well as for the diploid evolution model with general fitness landscapes our numerical solutions confirm the obtained analytic equations our method could be applied to the general case of nonlinear markov models

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

quantum scaling approach to nonequilibrium models stochastic nonequilibrium exclusion models are treated using a real space scaling approach the method exploits the mapping between nonequilibrium and quantum systems and it is developed to accommodate conservation laws and duality symmetries yielding exact fixed points for a variety of exclusion models in addition it is shown how the asymmetric simple exclusion process in one dimension can be written in terms of a classical hamiltonian in two dimensions using a suzuki trotter decomposition

accurate viscoelastic modeling by frequency domain finite differences using rotated operators the viscoelastic wave equation is an integro differential equation that requires special methods when using time domain numerical finite difference methods in the frequency domain the integral terms are easily represented by complex valued elastic media properties there are further significant advantages to using the frequency domain if the forward or the inverse problem requires modeling or inverting a large number of prestack source gathers numerical modeling is expensive for seismic data because of the large number of wavelenghths typically separating sources from receivers which results in a need for a large number of grid points a major obstacle to using frequency domain methods is the consequent storage requirements to reduce these we maximize the accuracy and simultaneously minimize the spatial extent of the numerical operators we achieve this by extending earlier published methods introduced for the viscoacoustic case to the viscoelastic case this requires the formulation of two new nume

existence uniqueness and positivity of solutions for bgk models for mixtures abstract we consider kinetic models for a multi component gas mixture without chemical reactions in the literature one can find two types of bgk models in order to describe gas mixtures one type has a sum of bgk type interaction terms in the relaxation operator for example the model described by klingenberg pirner and puppo 20 which contains well known models of physicists and engineers for example hamel 16 and gross and krook 15 as special cases the other type contains only one collision term on the right hand side for example the well known model of andries aoki and perthame 1 for each of these two models 20 and 1 we prove existence uniqueness and positivity of solutions in the first part of the paper in the second part we use the first model 20 in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by dellacherie 11

towards a comprehensive theory of brain activity coupled oscillator systems under external forces abstract recently jirsa et al and haken discussed a theory comprising the brain wave equation proposed by nunez the wilson cowan ermentrout cowan model and hopfield networks this theory was applied to model findings obtained in an experiment that relates brain activity and behavior the so called julliard experiment in previous works of jirsa et al and frank et al the focus was on the brain wave aspect recently we conducted similar experiments the results obtained are modeled in this paper in terms of coupled oscillator systems coupled oscillator systems are considered to represent the wilson cowan ermentrout cowan model aspect of the unifying theory building on a model proposed by haken kelso and bunz and the theory of weakly coupled oscillators established by winfree and by kuramoto we derive a nonlinear fokker planck equation whose stationary solutions mimic the neocortical brain activity observed in our experiments

on the linear discrepancy model and risky shifts in group behavior a nonlinear fokker planck perspective using a nonlinear fokker planck perspective we re formulate the linear discrepancy model proposed by boster and colleagues that describes the emergence of risky shifts during group decision making analytical expressions for the stationary case are derived and risky shifts are obtained by monte carlo simulations striking similarities with the kuramoto model for group synchronization are pointed out

a constant fractional order viscoelastic wave equation and its numerical simulation scheme abstractefficient modeling schemes currently exist to handle the spatially variable order fractional laplacians in the fractional laplacian viscoacoustic wave equation the simplest approach is to change the spatially variable order fractional laplacians into a linear combination of several constant fractional order laplacians we generalize the constant fractional order scheme to a spatially variable fractional order viscoelastic wave equation and develop an almost equivalent constant fractional order viscoelastic wave equation our constant fractional order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order thus our scheme simulates seismic wave propagation in viscoelastic media with sharp q contrasts well the fast fourier transform is used in the approximation of the fractional laplacians which improves the spectral accuracy in space several simulation examples are performed to verify that the numerical solution of a homogeneous q model obtai

nonlinear equation for anomalous diffusion unified power law and stretched exponential exact solution the nonlinear diffusion equation ensuremath partial ensuremath rho ensuremath partial t d stackrel ifmmode tilde else fi ensuremath delta ensuremath rho ensuremath nu is analyzed here where stackrel ifmmode tilde else fi ensuremath delta ensuremath equiv 1 r d ensuremath 1 ensuremath partial ensuremath partial r r d ensuremath 1 ensuremath ensuremath theta ensuremath partial ensuremath partial r and d ensuremath theta and ensuremath nu are real parameters this equation unifies the anomalous diffusion equation on fractals ensuremath nu 1 and the spherical anomalous diffusion for porous media ensuremath theta 0 an exact point source solution is obtained enabling us to describe a large class of subdiffusion ensuremath theta g 1 ensuremath ensuremath nu d normal diffusion ensuremath theta 1 ensuremath ensuremath nu d and superdiffusion ensuremath theta l 1 ensuremath ensuremath nu d furthermore a thermostatistical basis for this solution is given from the maximum entropic principle applied to the tsallis entropy

h theorem for a mean field model describing coupled oscillator systems under external forces this article studies the asymptotic behavior of solutions of fokker planck equations describing mean field approximations of weakly coupled oscillator systems subjected to external forces using an h theorem we show that transient probability densities converge to stationary ones furthermore stability criteria are derived for the stationary solutions of these fokker planck equations the obtained results are applied to a model that combines the haken kelso bunz model and the models of weakly coupled oscillators proposed by winfree and kuramoto the stability criteria based on the h theorem agree with those derived in our earlier analyses

green functions and langevin equations for nonlinear diffusion equations a comment on markov processes hurst exponents and nonlinear diffusion equations by bassler et al abstract we discuss two central claims made in the study by bassler et al k e bassler g h gunaratne j l mccauley physica a 369 2006 343 bassler et al claimed that green functions and langevin equations cannot be defined for nonlinear diffusion equations in addition they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones we review bottom up and top down approaches that have been used in the literature to derive green functions for nonlinear diffusion equations and in doing so show that the first claim needs to be revised we show that the second claim as well needs to be revised to this end we point out similarities and differences between non autonomous linear fokker planck equations and autonomous nonlinear fokker planck equations in this context we raise the question whether bassler et al s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes such external entities can easily be eliminated when taking self organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear fokker planck equations

complete description of a generalized ornstein uhlenbeck process related to the nonextensive gaussian entropy abstract we consider a generalized ornstein uhlenbeck process described by a nonlinear fokker planck equation related to the one parametric nonextensive gaussian entropy which is a special case of the two parametric sharma mittal entropy we derive the entire hierarchy of distribution functions for that process and in doing so derive a complete description of a stochastic process related to a nonextensive entropy measure

memory index of first passage time a simple measure of non markovian character on developpe la theorie pour le second moment de la variable de premier temps de passage des flots statistiques a memoire et on l applique a un mouvement brownien libre non markovien

cauchy problem for the ellipsoidal bgk model for polyatomic particles abstract we establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal bgk model which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level

one dimensional contact process duality and renormalization we study the one dimensional contact process in its quantum version using a recently proposed real space renormalization technique for stochastic many particle systems exploiting the duality and other properties of the model we can apply the method for cells with up to 37 sites after suitable extrapolation we obtain exponent estimates that are comparable in accuracy with the best known in the literature

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

a coupled ordinates method for solution acceleration of rarefied gas dynamics simulations non equilibrium rarefied flows are frequently encountered in a wide range of applications including atmospheric re entry vehicles vacuum technology and microscale devices rarefied flows at the microscale can be effectively modeled using the ellipsoidal statistical bhatnagar gross krook esbgk form of the boltzmann kinetic equation numerical solutions of these equations are often based on the finite volume method fvm in physical space and the discrete ordinates method in velocity space however existing solvers use a sequential solution procedure wherein the velocity distribution functions are implicitly coupled in physical space but are solved sequentially in velocity space this leads to explicit coupling of the distribution function values in velocity space and slows down convergence in systems with low knudsen numbers furthermore this also makes it difficult to solve multiscale problems or problems in which there is a large range of knudsen numbers in this paper we extend the coupled ordinates method comet previously developed to study participating radiative heat transfer to solve the esbgk equations in this method at each cell in the physical domain distribution function values for all velocity ordinates are solved simultaneously this coupled solution is used as a relaxation sweep in a geometric multigrid method in the spatial domain enhancements to comet to account for the non linearity of the esbgk equations as well as the coupled implementation of boundary conditions are presented the methodology works well with arbitrary convex polyhedral meshes and is shown to give significantly faster solutions than the conventional sequential solution procedure acceleration factors of 5 9 are obtained for low to moderate knudsen numbers on single processor platforms

derivation of nonlinear fokker planck equations by means of approximations to the master equation nonlinear fokker planck equations fpes are derived as approximations to the master equation in cases of transitions among both discrete and continuous sets of states the nonlinear effects introduced through the transition probabilities are argued to be relevant for many real phenomena within the class of anomalous diffusion problems the nonlinear fpes obtained appear to be more general than some previously proposed on a purely phenomenological basis ones in spite of this the same kind of solution applies i e it is shown that the time dependent tsallis s probability distribution is a solution of both equations obtained either from discrete or continuous sets of states and that the corresponding stationary solution is in the infinite time limit a stable solution

theory and modeling of constant q p and s waves using fractional time derivatives i have developed and solved the constant q model for the attenuation of p and s waves in the time domain using a new modeling algorithm based on fractional derivatives the model requires time derivatives of order m 2 applied to the strain components where m 0 1 and 1 tan 1 1 q with q the p wave or s wave quality factor the derivatives are computed with the grunwald letnikov and central difference fractional approximations which are extensions of the standard finite difference operators for derivatives of integer order the modeling uses the fourier method to compute the spatial derivatives and therefore can handle complex geometries and general material property variability i verified the results by comparison with the 2d analytical solution obtained for wave propagation in homogeneous pierre shale moreover the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experiment

ergodicity of nonlinear markov operators on the finite dimensional space abstract a nonlinear markov chain is a discrete time stochastic process whose transition matrices may depend not only on the current state of the process but also on the current distribution of the process in this paper we study strong and uniform ergodicity of nonlinear markov operators defined by stochastic hypermatrices higher order matrix we introduce dobrushin s ergodicity coefficient for a stochastic hypermatrix which enables to study ergodicity of nonlinear markov operators by introducing a notion of scrambling stochastic hypermatrix we study the strong ergodicity of scrambling nonlinear markov operators

numerical solutions of nonlinear wave equations accurate stable numerical solutions of the nonlinear sine gordon equation are obtained with particular consideration of initial conditions that are exponentially close to the phase space homoclinic manifolds earlier local grid based numerical studies have encountered difficulties including numerically induced chaos for such initial conditions the present results are obtained using the recently reported distributed approximating functional method for calculating spatial derivatives to high accuracy and a simple explicit method for the time evolution the numerical solutions are chaos free for the same conditions employed in previous work that encountered chaos moreover stable results that are free of homoclinic orbit crossing are obtained even when initial conditions are within 10 27 of the phase space separatrix value p it also is found that the present

low frequency reflection data augmentation by an inpainting method 1d acoustic media abstractthe augmentation of low frequencies is an important problem for inverting seismic reflection data through an acoustic scattering approach we have characterized how the data down to zero frequency grows in importance with increasing depth and the size of the velocity contrast we have developed a hermite distributed approximation functional hdaf inpainting method that provides a least squares fitting of the acoustic reflection data down to zero frequency the hdaf data fitting is applied to the first born approximation for compact velocity contrasts the top and bottom depth velocities are equal several fitting constraints are derived using a taylor expansion of the reflection data near zero frequency they provide a coupling with the first born approximation of the velocity contrast the method can be extended to the general geophysical case in which the final velocity i e is different from the starting velocity by use of the distorted wave born approximation numerical results for 1d t

exact time dependent solutions of the renyi fokker planck equation and the fokker planck equations related to the entropies proposed by sharma and mittal abstract exact time dependent solutions representing generalized one dimensional ornstein uhlenbeck processes are derived from nonlinear fokker planck equations and the corresponding diffusion processes are studied the corresponding systems are related to the renyi entropy and entropies proposed by sharma and mittal in the former case normal diffusion is found with a diffusion rate depending on the order of the renyi entropy in the latter case we find anomalous diffusion processes

a generalization of the fourier pseudospectral method the fourier pseudospectral ps method is generalized to the case of derivatives of nonnatural order fractional derivatives and irrational powers of the differential operators the generalization is straightforward because the calculation of the spatial derivatives with the fast fourier transform is performed in the wavenumber domain where the operator is an irrational power of the wavenumber modeling constant q propagation with this approach is highly efficient because it does not require memory variables or additional spatial derivatives the classical acoustic wave equation is modified by including those with a space fractional laplacian which describes wave propagation with attenuation and velocity dispersion in particular the example considers three versions of the uniform density wave equation based on fractional powers of the laplacian and fractional spatial derivatives

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

nonlinear fokker planck equation exhibiting bifurcation phenomena and generalized thermostatistics a nonlinear fokker planck equation exhibiting bifurcation phenomena is proposed within the framework of generalized thermostatistics the nonlinearity responsible for the occurrence of bifurcation of solutions is assumed to be of the form appearing in the standard mean field model a liapunov function is defined that takes the form of free energy involving generalized entropies of tsallis and an h theorem is proved to show that the free energy which is bounded below continues to decrease until the system approaches one of the equilibrium distributions the h theorem ensures instead of uniqueness of the equilibrium distribution global stability of the system in that either one of multisolutions must be approached for large times local stability analysis is conducted and the second order variation of the liapunov function is computed to find its relevant part whose sign governs stability of the equilibrium distribution of the system the case with a bistable potential is investigated as an example of confirming the theory to give the bifurcation diagram displaying the order parameter as a function of the coefficient of the nonlinear diffusion term

viscoacoustic modeling and imaging using low rank approximation abstracta constant q wave equation involving fractional laplacians was recently introduced for viscoacoustic modeling and imaging this fractional wave equation has a convenient mixed domain space wavenumber formulation which involves the fractional laplacian operators with a spatially varying power we have applied the low rank approximation to the mixed domain symbol which enables a space variable attenuation specified by the variable fractional power of the laplacians using the proposed approximation scheme we formulated the framework of the q compensated reverse time migration q rtm for attenuation compensation numerical examples using synthetic data demonstrated the improved accuracy of using low rank wave extrapolation with a constant q fractional laplacian wave equation for seismic modeling and migration in attenuating media low rank q rtm applied to viscoacoustic data is capable of producing images comparable in quality with those produced by conventional rtm from acoustic data

first order phase transition in a nonequilibrium growth process we introduce a simple continuous model for nonequilibrium surface growth the dynamics of the system is defined by the kardar parisi zhang equation with a morse like potential representing a short range interaction between the surface and the substrate the mean field solution displays a nontrivial phase diagram with a first order transition between a growing and a bound surface associated with a region of coexisting phases and a tricritical point where the transition becomes second order numerical simulations in three dimensions show quantitative agreement with mean field results and the features of the phase space are preserved even in two dimensions