analysis of asymptotic preserving dg imex schemes for linear kinetic transport equations in a diffusive scaling in this paper some theoretical aspects will be addressed for the asymptotic preserving discontinuous galerkin implicit explicit dg imex schemes recently proposed in j jang f li j m qiu and t xiong high order asymptotic preserving dg imex schemes for discrete velocity kinetic equations in a diffusive scaling http arxiv org abs 1306 0227 2013 submitted for kinetic transport equations under a diffusive scaling we will focus on the methods that are based on discontinuous galerkin dg spatial discretizations with the p k polynomial space and a first order implicit explicit imex temporal discretization and apply them to two linear models the telegraph equation and the one group transport equation in slab geometry in particular we will establish uniform numerical stability with respect to knudsen number varepsilon using energy methods as well as error estimates for any given varepsilon when varepsilon rightarrow 0 a rigorous asymptotic analysis of the schemes is also obta

a smooth transition model between kinetic and diffusion equations this paper presents a model which provides a smooth transition between a kinetic and a diffusion domain the idea is to use a buffer zone in which both diffusion and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point this model avoids the delicate issue of finding the interface condition or iteration in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a new asymptotic preserving method for this model is introduced and numerical examples are used to validate this new model and the new numerical method

the milne and kramers problems for the boltzmann equation of a hard sphere gas existence uniqueness and properties of asymptotic behavior are proved for solutions of the milne and kramers problems for the linearized boltzmann equation for a gas of hard spheres the proof uses energy like estimates and follows the analysis of bardos santos and sentis for the equations of neutron transport

the radiative transfer equations diffusion approximation under accretiveness and compactness assumptions abstract when the mean free path of photons is small the radiative transfer equations may be approximated by a nonlinear angular diffusion equation in order to find the diffusion equation one uses accretiveness assumptions but theses assumptions impose some conditions on the opacity to overcome this inconvenient we propose to study the solution of the radiative transfer equations by compactness techniques

on fluid limit for the semiconductors boltzmann equation abstract this paper is devoted to the derivation of non linear drift diffusion equations from the semiconductor boltzmann equation collisions are taken into account through the non linear pauli operator but we do not assume relation on the cross section such as the so called detailed balance principle in turn equilibrium states are implicitly defined this article follows and completes the contribution of mellet monatsh math 134 4 2002 305 329 where the electric field is given and does not depend on time here we treat the self consistent problem the electric potential satisfying the poisson equation by means of a hilbert expansion we shall formally derive the asymptotic model in the general case we shall then rigorously prove the convergence in the one dimensional case by using a modified hilbert expansion

interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains abstract we investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions through formal derivations we show that nonlocal diffusion interface problems converge to their classical local counterparts in the limit of vanishing nonlocality a central focus of this paper is local nonlocal interface problems or interface problems with sharp transitions between local and nonlocal domains such problems can be cast as instances of a nonlocal interface problem with a finite horizon in certain regions and a vanishing horizon in other regions we derive a local nonlocal interface problem and utilize conservation principles to obtain local nonlocal interface conditions comparisons between nonlocal local and local nonlocal interface problems are presented analytically and numerically with a focus on multiscale aspects of nonlocal models induced by their inherent length scales

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

diffusion approximation for linear transport with multiplying boundary conditions we consider the diffusion limit of a suitably rescaled model transport equation in a slab with multiplying boundary conditions as the scaling parameter varepsilon tends to zero we show that for sufficiently smooth data the solution converges in the l2 norm for each t 0 to the solution of a diffusion equation with robin boundary conditions corresponding to an incoming flux the derivation of the diffusive limit is based on an asymptotic expansion which is rigorously justified

two dimensional approximations of three dimensional eigenvalue problems in plate theory the eigenvalues and eigenfunctions corresponding to the three dimensional equations for the linear elastic equilibrium of a clamped plate of thickness 2 are shown to converge in a specific sense to the eigenvalues and eigenfunctions of the well known two dimensional biharmonic operator of plate theory as approaches zero in the process it is found in particular that the displacements and stresses are indeed of the specific forms usually assumed a priori in the literature it is also shown that the limit eigenvalues and eigenfunctions can be equivalently characterized as the leading terms in an asymptotic expansion of the three dimensional solutions in terms of powers of the method presented here applies equally well to the stationary problem of linear plate theory as shown elsewhere by p destuynder

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

on a system of nonlinear boltzmann equations of semiconductor physics a system of kinetic equations of semiconductors is studied these are vlasov poisson equations with additional nonlinear boltzmann operators first h theorems for these integral operators are given and their null space is determined then the existence and uniqueness of smooth solutions is proven the proof relies on the construction of a sequence of solutions for a suitable linearized system which converges strongly

large scale dynamics of the persistent turning walker model of fish behavior this paper considers a new model of individual displacement based on fish motion the so called persistent turning walker ptw model which involves an ornstein uhlenbeck process on the curvature of the particle trajectory the goal is to show that its large time and space scale dynamics is of diffusive type and to provide an analytic expression of the diffusion coefficient two methods are investigated in the first one we compute the large time asymptotics of the variance of the individual stochastic trajectories the second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories the kinetic model is a fokker planck type equation posed in an extended phase space involving the curvature among the kinetic variables we show that both methods lead to the same value of the diffusion constant we present some numerical simulations to illustrate the theoretical results

the fractional laplacian operator on bounded domains as a special case of the nonlocal diffusion operator we analyze a nonlocal diffusion operator having as special cases the fractional laplacian and fractional differential operators that arise in several applications in our analysis a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem we demonstrate that when sufficient conditions on certain kernel functions hold the solution of the nonlocal equation converges to the solution of the fractional laplacian equation on bounded domains as the nonlocal interactions become infinite we also introduce a continuous galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite volume constraints

initial and boundary conditions for diffusive linear transport problems the initial and boundary layer analyses of an asymptotic expansion that yields a diffusive description of linear particle transport are carried out in some generality this yields initial and boundary conditions that apply to the diffusion equation previously reported in the literature effects treated include boundary curvature the variation of the transport boundary condition along the bounding surface and spatial and temporal variations of the interaction coefficients cross sections in the initial and boundary layers

optimal distributed control of nonlocal steady diffusion problems a control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied the control is the right hand side forcing function and the objective of control is a standard matching functional a recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system when sufficient conditions on certain kernel functions and the volume constraints hold the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived we demonstrate the convergence as the nonlocal interactions vanish of the optimal nonlocal state to the optimal state of a local pde constrained control problem we also define continuous and discontinuous galerkin finite element discretizations of the optimality system for which we derive a priori error estimates numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained f

a new asymptotic preserving scheme based on micro macro formulation for linear kinetic equations in the diffusion limit we propose a new numerical scheme for linear transport equations it is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts we also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part by using a suitable time semi implicit discretization our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes it is asymptotic preserving in the following sense when the mean free path of the particles is small our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model a uniform stability property is proved for the simple telegraph model various boundary conditions are studied our method is validated in one dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes

an asymptotic induced scheme for nonstationary transport equations in the diffusive limit an asymptotic induced scheme for nonstationary transport equations with the diffusion scaling is developed the scheme works uniformly for all ranges of mean free paths it is based on the asymptotic analysis of the diffusion limit of the transport equation a theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven moreover numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

control of modeling errors in the coupling of linear transport and diffusion models we consider in this paper the initial boundary value problem for the 1d neutron transport equation with isotropic scattering set in some bounded interval with inflow boundary conditions the usual parabolic scaling yields the diffusive limit a surrogate model coupling transport and diffusion equations is then introduced in order to accurately assess the value of specific quantities of interest the control of the quality of the computation with respect to such a quantity of interest is performed by means of an approach for modeling error estimation combined with an adaptive strategy to control the surrogate model if necessary

analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit we present a mathematical analysis of the asymptotic preserving scheme proposed in m lemou and l mieussens siam j sci comput 31 2008 pp 334 368 for linear transport equations in kinetic and diffusive regimes we prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles this property is satisfied under an explicitly given cfl condition this condition tends to a parabolic cfl condition for small mean free paths and is close to a convection cfl condition for large mean free paths our analysis is based on very simple energy estimates

stochastic one dimensional lorentz gas on a lattice we stud ya one dimensional stochastic lorentz gas wher ea light particle moves i na fixed array of nonidentical random scatterers arranged i na lattice each scatterer is characterized b ya random transmissionreflection coefficient we consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables a symbolic program is presented which generates the exact velocity autocorrelation functio n vacf in terms of the moments of the transmission coefficients the vacf is found for different types of disorder for times up to 20 collision times we then conside ra specific type of disorder a two state lorentz gas in which two types of scatterers are arranged randomly i na lattice the na lattice point is occupied b ya scatterer whose transmission coefficient is with probability p or with probability 1 p a perturbation expansion with respect to is derived the 2 term in this expansion shows that the vacf oscillates with time the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor the coarse grained vacf decays for long times like t 32 which is similar to the decay of the vacf of the random lorentz gas wit ha single type of scatterer the perturbation results and the exact one s found up to 20 collision times show good agreement

diffusion approximation and computation of the critical size this paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation to obtain this result we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size the paper also contains some simple numerical examples and some new results for the milne problem introduction the computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts first the computation of the critical size is much easier for the diffusion approximation than for the original transport equation second for the critical size one can consider a host media x nxq with x0 given and 17 a positive number the problem of the critical size is then reduced to the computation of the parameter zj it turns out that when the transport operator is almost conservative the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate on the other hand the physical boundary condition for the diffusion approxi mation is of the form

the brownian motion as the limit of a deterministic system of hard spheres we provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard spheres as the number of particles n goes to infinity and their diameter varepsilon simultaneously goes to 0 in the fast relaxation limit alpha n varepsilon d 1 rightarrow infty with a suitable diffusive scaling of the observation time as suggested by hilbert in his sixth problem we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions we use indeed the linear boltzmann equation to describe one tagged particle in a gas close to global equilibrium our proof is based on the fundamental ideas of lanford the main novelty here is the detailed study of the branching process leading to explicit estimates on pathological collision trees

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

on the time evolution of macroscopic systems autodiffusion a basse densite et theoreme de lanford correlations dans l etat initial auto diffusion a densite finie le gaz de lorentz et le theoreme de bunimovich et sinai

the milne problem for the radiative transfer equations with frequency dependence we study the following stationary frequency dependent transport equation 28xf a t f bv t o x o m o u e 1 1 tr t bv t fjdm 8 0 r x 1 1 2 f 2 w 2 0 u e j0 1 where bv is the well known planck function appearing in astrophysics we are able to describe the asymptotic behavior of f and t for x large when a t is of the special form a t tr m k t our method relies mainly on the monotonicity of the nonlinearity the proof does not use any linearization of the equation in particular no smallness assumption on the data yp in any sense is required resume nous etudions l equation de transport stationnaire avec dependance en frequence 2dxf a t f bv t o x o m o u e 1 1 tr t bv t fjdm 8 0 r x 1 1 2 8 2 w 2 o u e 0 1 lorsque a t est de la forme particuliere a t a m k t nous savons decrire le comportement asymptotique de f et t pour x grand notre methode repose principalement sur la monotonie de la non linearite la preuve n utilise aucune linearisation de l equation en particulier nous n avons besoin d aucune hypothese de petitesse d aucune sorte sur la donnee y

homogenization of transport equations this article is devoted to the mathematical analysis of various formulas giving the equivalent absorption and scattering cross section for mixed materials in linear transport theory we begin with a general result on the treatment of high frequency oscillations in linear transport equations which is partly based upon the velocity averaging results and is the analogue for transport equations of the compensated compactness class of results the case of periodic inhomogeneities is then studied in detail in particular we show the essential difference with periodic homogenization of diffusion equations due to small divisor problems these results were announced in f golse c r acad sci paris ser i math 305 1987 pp 801 804 f golse mathematical aspects of fluid and plasma dynamics g toscani v boffi and s rionero eds lecture notes in math 1460 springer verlag berlin new york 1991 pp 152 169 finally we treat a case of stochastic inhomogeneities in linear transport theory insp

weak turbulence plasma induced by two scale homogenization abstract this paper is devoted to the homogenization of the quasilinear theory of the plasma turbulence described by the vlasov poisson system it is shown that the homogenization limit in the sense of two scale limit of the distribution function satisfies the linear vlasov poisson equations moreover the limit distribution function can be decomposed into the mean and the fluctuation parts and the mean part the equilibrium distribution function is shown to be the solution of the nonlocal quasilinear velocity space diffusion equation we also investigate the landau damping from the point of view of homogenization through the two scale limit

asymptotic analysis and coupling conditions for kinetic and hydrodynamic equations abstract this article is concerned with domain decomposition for kinetic and hydrodynamic equations coupling conditions at the interface between the equations are developed and investigated in particular for nonequilibrium situations at the interface new coupling conditions are developed by considering interface layers this leads to kinetic half space problems a fast procedure to solve these problems is given

a ribbon hopf algebra approach to the irreducible representations of centralizer algebras the brauer birman wenzl and type a iwahori hecke algebras abstract we show how the ribbon hopf algebra structure on the drinfel d jimbo quantum groups of types a b c and d can be used to derive formulas giving explicit realizations of the irreducible representations of the iwahori hecke algebras of type a and the birman wenzl algebras we use this derivation to give explicit realizations of the irreducible representations of the brauer algebras as well the derivation is accomplished by way of a combination of techniques from operator algebras quantum groups and the theory of 3 manifold invariants although our applications are in the cases of the quantum groups of types a b c and d most of the aspects of our approach apply in the general setting of ribbon hopf algebras

moment methods for the semiconductor boltzmann equation on bounded position domains galerkin methods for the semiconductor boltzmann equation based on moment expansions for a discretization in the velocity direction are studied for the moment equations boundary conditions are proposed which are analogues to inflow and respectively reflecting boundary conditions for the boltzmann equation stability and an error estimate are proved for an expansion in terms of hermite polynomials finally an adaptive numerical implementation is introduced and results of numerical experiments are presented

physical nature of higher order mutual information intrinsic correlations and frustration this paper studies some properties and implications of higher order mutual information functions which should serve for the analysis of general complex systems we note that the higher order mutual information can either be positive or negative depending on the correlation among ensembles two opposite types of correlations are discussed in connection with the concept of frustration simple examples are presented to demonstrate that our concepts are especially helpful in understanding the nature of correlations in frustrated systems the higher order mutual information provides an appropriate measure of the frustration effect

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

the nonaccretive radiative transfer equations existence of solutions and rosseland approximation we present an existence theory and an asymptotic analysis for the radiative transfer equations ue t xuee ue e2 ue ue 0 inx 1 ue x sn k ue t 0 u0 where u u t x t r x x rn 1 sn and ue t x 1 sn ue t x d we prove that even if has a singularity 0 1 has a solution ue l r x sn as e 0 we show that ue converges pointwise to a function u l r x solution of the degenerate parabolic equation u t f u 0 in x u x k u t 0 u0 this is achieved without any monotonicity assumption on and therefore one cannot use the theory of nonlinear contraction semigroups

on infinite volume mixing in the context of the long standing issue of mixing in infinite ergodic theory we introduce the idea of mixing for observables possessing an infinite volume average the idea is borrowed from statistical mechanics and appears to be relevant at least for extended systems with a direct physical interpretation we discuss the pros and cons of a few mathematical definitions that can be devised testing them on a prototypical class of infinite measure preserving dynamical systems namely the random walks

new models for the solution of intermediate regimes in transport theory and radiative transfer existence theory positivity asymptotic analysis and approximations in this paper we propose derive and establish the mathematical foundations of new models for the solution of intermediate regimes in transport theory and radiative transfer these new models consist of coupling the transport equations with their diffusion approximations our mathematical theory includes the existence theory the positivity of the solutions and the asymptotic analysis we also give the rate of the asymptotic decay in order to solve the new coupled problem we propose to use the transmission time marching algorithm introduced and studied in refs 10 13 15 we then study the convergence of the resulting algorithm these studies are based in an essential way on the methods we introduced in refs 14 15

lyapunov statistics and mixing rates for intermittent systems we consider here a recent conjecture stating that correlation functions and tail probabilities of finite time lyapunov exponents would have the same power law decay in weakly chaotic systems we demonstrate that this conjecture fails for a generic class of maps of the pomeau manneville type we show further that typically the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes our approaches are firmly based on rigorous results particularly the aaronson darling kac theorem and are also confirmed by exhaustive numerical simulations

derived equivalences and sl2 categorifications for uq gln abstract we give a construction of sl 2 categorifications for u q gl n for generic q and for q a root of unity

fractional diffusion limit for collisional kinetic equations this paper is devoted to diffusion limits of linear boltzmann equations when the equilibrium distribution function is a maxwellian distribution it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion equation in this paper we consider situations in which the equilibrium distribution function is a heavy tailed distribution with infinite variance we then show that for an appropriate time scale the small mean free path limit gives rise to a fractional diffusion equation

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

computation of interface reflection and regular or diffuse transmission of the planar symmetric radiative transfer equation with isotropic scattering and its diffusion limit in this paper we develop a numerical scheme for the interface problem in the planar symmetric radiative transfer equation with isotropic scattering such problems arise in the modeling of the propagation of energy density for waves in heterogeneous media with weak random fluctuation in the high frequency regime the idea following the earlier work of jin and wen for regular transmission and reflection is to build the interface condition which characterizes the reflection and transmission into the numerical flux the new contribution of this article is to deal with the diffuse transmission at the interface the positivity of the scheme is proven moreover we show via numerical examples that this new scheme is able to capture the correct regular or diffuse transmission and reflection through the interface and as the mean free path goes to zero capture the diffusion limit with the correct interface conditions

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

fluctuation of the entropy production for the lorentz gas under small external forces in this paper we study the physical and statistical properties of the periodic lorentz gas with finite horizon driven to a non equilibrium steady state by the combination of non conservative external forces and deterministic thermostats a version of this model was introduced by chernov eyink lebowitz and sinai and subsequently generalized by chernov and the third author non equilibrium steady states for these models are srb measures and they are characterized by the positivity of the steady state entropy production rate our main result is to establish that the entropy production in this context equal to the phase space contraction satisfies the gallavotti cohen fluctuation relation the main tool needed in the proof is the family of anisotropic banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques

anomalous diffusion limit induced on a kinetic equation using a compactness argument based on the velocity averaging lemma of golse et al it is shown that the limiting behavior of a kinetic linearized bgk gas model confined between two plates with maxwell boundary conditions when the distance between the plates goes to zero under a suitable anomalous scaling is diffusive we do not require the use of central limit theorems as in the method of borgers et al

information theoretic characterization of frustrated systems abstract a recently proposed measure of the frustration effect is discussed in detail in the ising model on the kagome lattice with nearest neighbor and next nearest neighbor interactions the measure quantifies the degree of correlation dominated by the frustration effect it manifests the difference between frustrated and unfrustrated systems its negative sign indicating the presence of frustration we study the behavior of the measure for different values of interactions and show that it is responsible for the understanding of phase transitions in frustrated systems the reentrance phenomenon is discussed in terms of the measure

the convergence of numerical transfer schemes in diffusive regimes i discrete ordinate method in highly diffusive regimes the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path epsilon tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis a numerical scheme for solving this problem has an epsilon 1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small epsilon thus degrading its performance in diffusive regimes in this paper we show that whenever the discrete ordinate method has the correct diffusion limit both in the interior and at the boundaries its solutions converge to the solution of the transport equation uniformly in epsilon our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete ordinate method

asymptotic preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings in this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs the schemes are asymptotic preserving ap in the sense that they preserve the diffusive limits of the equations in discrete setting without requiring excessive refinement of the discretization our stochastic ap schemes are extensions of the well developed deterministic ap schemes to handle the random inputs we employ generalized polynomial chaos gpc expansion and combine it with stochastic galerkin procedure we apply the gpc galerkin scheme to a set of representative hyperbolic and transport equations and establish the ap property in the stochastic setting we then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic ap schemes

existence and uniqueness of weak solutions for nonlocal parabolic problems via the galerkin method abstract fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non classical phenomena with spatial heterogeneities throughout the applied science and engineering a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional laplacians on bounded domains in r n we develop the galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems moreover we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

nonunitary transformations and hydrodynamic limit in the one dimensional hard point gas abstract the ergodic theoretical formalism of nonunitary transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit the one dimensional hard point system is used as an illustration both methods rest on the k mixing property of the physical systems considered we show that the specific form of the operator is important only on the microscopic time scale on the macroscopic scale it acts like a projector with broken time reversal symmetry we also discuss the connection between the kolmogorov sinai k s entropy of the model dynamical system the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate

the statistics of the trajectory of a certain billiard in a flat two torus we consider a billiard in the punctured torus obtained by removing a small disk of radius 0 from the flat torus 2 with trajectory starting from the center of the puncture in this case the phase space is given by the range of the velocity only let and respectively r denote the first exit time length of the trajectory and respectively the number of collisions with the side cushions when 2 is being identified with 0 1 2 we prove that the probability measures on 0 associated with the random variables and r are weakly convergent as varepsilon rightarrow 0 and explicitly compute the densities of the limits

diffusion approximations and domain decomposition method of linear transport equations in this paper we construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations we treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist the diffusion equations and their data are derived from asymptotic and layer analysis which allows general scattering kernels and general data we apply the half space solver in 20 to resolve the boundary layer equation and obtain the boundary data for the diffusion equation the algorithms are validated by numerical experiments and also by error analysis for the pure diffusive scaling case

an asymptotic preserving scheme for linear kinetic equation with fractional diffusion limit we present a new asymptotic preserving scheme for the linear boltzmann equation which under appropriate scaling leads to a fractional diffusion limit our scheme rests on novel micro macro decomposition to the distribution function which splits the original kinetic equation following a reshuffled hilbert expansion as opposed to classical diffusion limit a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter our idea is while solving the macro micro part in a truncated velocity domain truncation only depends on numerical accuracy to incorporate an integrated tail over the velocity space that is beyond the truncation and its major component can be precomputed once with any accuracy such an addition is essential to drive the solution to the correct asymptotic limit numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes

recurrence properties of a special type of heavy tailed random walk in the proof of the invariance principle for locally perturbed periodic lorentz process with finite horizon a lot of delicate results were needed concerning the recurrence properties of its unperturbed version these were analogous to the similar properties of simple symmetric random walk however in the case of lorentz process with infinite horizon the analogous results for the corresponding random walk are not known either in this paper these properties are ascertained for the appropriate random walk this happens to be in the non normal domain of attraction of the normal law as a tool an estimation of the remainder term in the local limit theorem for the corresponding random walk is computed

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

reentrant transition of the ising model on the centred square lattice the ising model on the centred square lattice union jack lattice is studied by vdovichenko s method for a more general case than vaks et al 1966 the reentrant transition of the system is interpreted in terms of effective interactions a method of associating a factor 1 to some corners of a loop in vdovichenko s method 1965 is adopted by which one can obtain the expression of the free energy by calculating a finite number of determinants which consist only of elements of integers

a two surface problem of the electron flow in a semiconductor on the basis of kinetic theory a steady flow of electrons in a semiconductor between two parallel plane ohmic contacts is studied on the basis of the semiconductor boltzmann equation assuming a relaxation time collision term and the poisson equation for the electrostatic potential a systematic asymptotic analysis of the boltzmann poisson system for small knudsen numbers scaled mean free paths is carried out in the case where the debye length is of the same order as the distance between the contacts and where the applied potential is of the same order as the thermal potential a system of drift diffusion type equations and their boundary conditions is obtained up to second order in the knudsen number a numerical comparison is made between the obtained system and the original boltzmann poisson system

micro macro schemes for kinetic equations including boundary layers we introduce a new micro macro decomposition of collisional kinetic equations in the specific case of the diffusion limit which naturally incorporates the incoming boundary conditions the idea is to write the distribution function f in all its domain as the sum of an equilibrium adapted to the boundary which is not the usual equilibrium associated with f and a remaining kinetic part this equilibrium is defined such that its incoming velocity moments coincide with the incoming velocity moments of the distribution function a consequence of this strategy is that no artificial boundary condition is needed in the micro macro models and the exact boundary condition on f is naturally transposed to the macro part of the model this method provides an asymptotic preserving numerical scheme that generates a very good approximation of the space boundary values at the diffusive limit without any mesh refinement in the boundary layers our numerical results are in very good agreement with the exact so c

the milne problem for high field kinetic equations half space problems of the linear boltzmann equation with a constant driving force are considered such problems model boundary layers between kinetic zones and fluid zones described by a high field limit of the boltzmann equation existence uniqueness and asymptotic behavior of solutions are studied for positive and negative driving forces in the positive case the force field accelerates the particles and we show that the solution of the half space problem is determined only by the inflow data in contrast for negative forces the behavior at infinity has to be prescribed in order to insure uniqueness due to the nonvanishing forces the problem does not possess any entropy the existence and uniqueness issues are dealt with by supersolution techniques while the asymptotic behavior is analyzed by semiexplicit integration of the equations along the characteristics in the case of relaxation time approximation a fast numerical method for computing the asymptotic state method is presented and tested