about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

global boundedness of moments of solutions of the boltzmann equation for forces of infinite range this paper proves the global boundedness in time of higher moments to a weak solution of the non linear space homogeneous boltzmann equation for inverse kth power forces with k 5 in the course of the proof a new collisional estimate of v s v s v s v s is obtained where v v are the velocities before and v v are the velocities after a binary collision

exact stationary and non stationary solutions to inelastic maxwell model with infinite energy the one dimensional inelastic boltzmann equation with a constant collision rate the maxwell model is considered it is shown that for special values of restitution parameter there exists a stationary solution with the characteristic function in the form e p log z z where p is a periodic function the corresponding distribution function belongs to a one special class of stochastic processes termed as a generalized stable in the probability theory the fourier transform of the non stationary equation has the solution bigl 1 p log z z bigr e q log z z it is proved that this solution is a characteristic function if periodic functions p q satisfy some not very restrictive conditions the stationary and non stationary solutions correspond to a gas with infinite temperature

local properties of self similar solutions to smoluchowski s coagulation equation with sum kernels the regularity of the scaling profiles to smoluchowski s coagulation equation is studied when the coagulation kernel k is given by k x y x y with 0 1 more precisely is c smooth on 0 and decays exponentially fast for large x furthermore the singular behaviour of x as x 0 is identified thus giving a rigorous proof of physical conjectures

self similar solutions of the boltzmann equation and their applications we consider a class of solutions of the boltzmann equation with infinite energy using the fourier transformed boltzmann equation we prove the existence of a wide class of solutions of this kind they fall into subclasses labelled by a parameter a and are shown to be asymptotic in a very precise sense to the self similar one with the same value of a and the same mass specializing to the case of a maxwell isotropic cross section we give evidence to the effect that the only self similar closed form solutions are the bkw mode and the two solutions recently found by the authors all the self similar solutions discussed in this paper are eternal i e they exist for t which shows that a recent conjecture cannot be extended to solutions with infinite energy eternal solutions with finite moments of all orders and different from a maxwellian are also studied it is shown that these solutions cannot be positive moreover all such solutions partly negative must be asymptotically for large negative times close to the exact eternal solution of bkw type

eternal solutions of the boltzmann equation near travelling maxwellians abstract it is shown in this paper that the cauchy problem of the boltzmann equation with a cut off soft potential and an initial datum close to a travelling maxwellian has a unique positive eternal solution this eternal solution is exponentially decreasing at infinity for all t consequently the moments of any order are finite this result gives a negative answer to the conjecture of villani in the spatially inhomogeneous case

well posedness and large time behaviour for the non cutoff kac equation with a gaussian thermostat we consider here a kac equation with a gaussian thermostat in the case of a non cutoff cross section under the sole assumptions of finite mass and finite energy for the initial data we prove the existence of a global in time solution for which mass and energy are preserved then via fourier transform techniques we show that this solution is smooth unique and converges to the corresponding stationary state

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation we first consider the boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates this is the simplest boltzmann equation having the compressible euler equations as a scaling limit for it we prove a stability result for theh theorem which says that when the entropy production is small the solution of the spatially homogeneous boltzmann equation is necessarily close to equilibrium in the entropie sense and therefore strongl 1 sense we use this to prove that solutions to the spatially homogeneous boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes every initial condition with finite entropy andp th velocity moment for some p 2 belongs to such a class we then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below which covers a wide class of slightly modified physical collision kernels these results are the basis of a study of the relation between scaling limits of solutions of the boltzmann equation and hydrodynamics which will be developed in subsequent papers the program is described here

infinite energy solutions to the homogeneous boltzmann equation the goal of this work is to present an approach to the homogeneous boltzmann equation for maxwellian molecules with a physical collision kernel that allows us to construct unique solutions to the initial value problem in a space of probability measures defined via the fourier transform in that space the second moment of a measure is not assumed to be finite so infinite energy solutions are not a priori excluded from our considerations moreover we study the large time asymptotics of solutions and in a particular case we give an elementary proof of the asymptotic stability of self similar solutions obtained by a v bobylev and c cercignani j stat phys 106 2002 1039 1071 2009 wiley periodicals inc

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

solutions to complex smoothing equations we consider smoothing equations of the form begin aligned x mathop limits text law sum j ge 1 t j x j c end aligned where c t 1 t 2 ldots is a given sequence of random variables and x 1 x 2 ldots are independent copies of x and independent of the sequence c t 1 t 2 ldots the focus is on complex smoothing equations i e the case where the random variables x c t 1 t 2 ldots are complex valued but also more general multivariate smoothing equations are considered in which the t j are similarity matrices under mild assumptions on c t 1 t 2 ldots we describe the laws of all random variables x solving the above smoothing equation these are the distributions of randomly shifted and stopped levy processes satisfying a certain invariance property called u alpha stability which is related to operator semi stability the results are applied to various examples from applied probability and statistical physics

nonlinear complementarity problem and galerkin method etude du probleme de complementarite non lineaire dans des espaces de dimensions infinies en utilisant l approximation de galerkin et une generalisation de l hypothese de karamardian on obtient plusieurs resultats d existence pour une forme speciale du probleme ou f x x l x t x l etant un operateur lineaire et t un operateur non lineaire

scaling solutions of smoluchowski s coagulation equation we investigate the structure of scaling solutions of smoluchowski s coagulation equation of the formc k t s t k s t wherec k t is the concentration of clusters of sizek at timet s t is the average cluster size and x is a scaling function for the rate constantk i j in smoluchowski s equation we make the very general assumption thatk i j is a homogeneous function of the cluster sizesi andj k i j a k ai aj for alla 0 but we restrict ourselves to kernels satisfyingk i j j 0 asj we show that gelation occurs if 1 and does not occur if 1 for all gelling and nongelling models we calculate the time dependence ofs t and we derive an equation for x we present a detailed analysis of the behavior of x at large and small values ofx for all models we find exponential large x behavior x ax e x asx and for different kernelsk i j algebraic or exponential small x behavior x bx or x exp cx asx 0

contractive metrics for a boltzmann equation for granular gases diffusive equilibria we quantify the long time behavior of a system of partially inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures existence uniqueness boundedness of moments and regularity of a steady state are derived from this basic property the solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures then we prove a uniform bound in time on sobolev norms of the solution provided the initial datum has a finite norm in the corresponding sobolev space these results are then combined using interpolation inequalities to obtain exponential convergence to the diffusive equilibrium in the strong l1 norm as well as various sobolev norms

hausdorff operators on the power weighted hardy spaces abstract in this paper we study the high dimensional hausdorff operators on the power weighted hardy spaces we establish their sharp boundedness on power weighted hardy spaces and local hardy spaces particularly our results solve an open question posted by liflyand in 16

exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in banach spaces abstract in this paper we generalize the concept of exceptional family of elements for a completely continuous field from hilbert spaces to banach spaces and we study the solvability of the variational inequalities with respect to a mapping f that is from a closed convex cone of a banach space b to the dual space b by applying the generalized projection operator k and by using the leray schauder type alternative

networks based on collisions among mobile agents we investigate in detail a recent model of colliding mobile agents phys rev lett 96 088702 used as an alternative approach to construct evolving networks of interactions formed by the collisions governed by suitable dynamical rules the system of mobile agents evolves towards a quasi stationary state which is apart small fluctuations well characterized by the de nsity of the system and the residence time of the agents the residence time defines a collision rate and by varying the collision ra te the system percolates at a critical value with the emergence of a giant cluster whose critical exponents are the ones of t wodimensional percolation further the degree and clusteri ng coefficient distributions and the average path length sho w that the network associated with such a system presents non trivial features which depending on the collision rule enables one not only to recover the main properties of standard networks such as exponential random and scale free networks but also to obtain other topological structures namely we show a specific example where the obtained structure has topological fe atures which characterize accurately the structure and evolution of social networks in different contexts ranging from networks of acquaintances to networks of sexual contacts

global existence of classical solutions to the inelastic vlasov poisson boltzmann system we study the global existence and uniqueness of classical solutions to the inelastic vlasov poisson boltzmann system for a soft potential in the near vacuum regime for the global existence of classical solutions we assume reasonable conditions on the restitution coefficient which represents the character of inelastic collisions we use the smallness of initial data and an algebraically decaying weight function in the spatial variable to control the self consistence force and collision operator

dynamics of money and income distributions we formulate a model of personal income distributions within society our aim is to understand the origin of the natural law first proposed by pareto who suggested that the high end part of the income distribution follows a power law where f 1 w w 1 we generalise a model proposed the so called cc ccm model introducing the possibility of exchanges of money at random times unlike the cc ccm model this generalisation predicts tail exponents greater than unity in line with pareto s prediction and empirical observation

stability convergence to self similarity and elastic limit for the boltzmann equation for inelastic hard spheres we consider the spatially homogeneous boltzmann equation for inelastic hard spheres in the framework of so called constant normal restitution coefficients alpha in 0 1 in the physical regime of a small inelasticity that is alpha in alpha 1 for some constructive alpha in 0 1 we prove uniqueness of the self similar profile for given values of the restitution coefficient alpha in alpha 1 the mass and the momentum therefore we deduce the uniqueness of the self similar solution up to a time translation

probabilistic study of the speed of approach to equilibrium for an inelastic kac model this paper deals with a one dimensional model for granular materials which boils down to an inelastic version of the kac kinetic equation with inelasticity parameter p 0 in particular the paper provides bounds for certain distances such as specific weighted distances and the kolmogorov distance between the solution of that equation and the limit it is assumed that the even part of the initial datum which determines the asymptotic properties of the solution belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent 2 1 p with such initial data it turns out that the limit exists and is just the aforementioned stable distribution a necessary condition for the relaxation to equilibrium is also proved some bounds are obtained without introducing any extra condition sharper bounds of an exponential type are exhibited in the presence of additional assumptions concerning either the behaviour close to the origin of the initial characteristic function or the behaviour at infinity of the initial probability distribution function

boundedness of multidimensional hausdorff operators on h1 rn abstract in this paper we consider the h 1 boundedness of multidimensional hausdorff operators defined by h a f x r n u f a u x d u where l l o c 1 r n a u a i j u i j 1 n is an n n matrix and each a i j u is a measurable function of u let b i j 1 n b i j for the matrix b b i j u i j 1 n we prove that h a is bounded from the hardy space h 1 to itself if r n u det a 1 u ln 1 a 1 u n det a 1 u d u our result improves known results in addition we show that the above condition is optimal in the size condition

kinetic approach to long time behavior of linearized fast diffusion equations we show that the rate of convergence towards the self similar solution of certain linearized versions of the fast diffusion equation can be related to the number of moments of the initial datum that are equal to the moments of the self similar solution at a fixed time as a consequence we find an improved rate of convergence to self similarity in terms of a fourier based distance between two solutions the results are based on the asymptotic equivalence of a collisional kinetic model of boltzmann type with a linear fokker planck equation with nonconstant coefficients and make use of methods first applied to the reckoning of the rate of convergence towards equilibrium for the spatially homogeneous boltzmann equation for maxwell molecules

growth and structure of stochastic sequences we introduce a class of stochastic integer sequences in these sequences every element is a sum of two previous elements at least one of which is chosen randomly the interplay between randomness and memory underlying these sequences leads to a wide variety of behaviours ranging from stretched exponential to log normal to algebraic growth interestingly the set of all possible sequence values has an intricate structure

self similarity and power like tails in nonconservative kinetic models in this paper we discuss the large time behavior of solution of a simple kinetic model of boltzmann maxwell type such that the temperature is time decreasing and or time increasing we show that under the combined effects of the nonlinearity and of the time monotonicity of the temperature the kinetic model has non trivial quasi stationary states with power law tails in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution the same idea is applied to investigate the large time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and or decreasing of the mean wealth in this last case the large time behavior of the solution shows a pareto power law tail numerical results confirm the previous analysis

self similar asymptotics for a modified maxwell boltzmann equation in systems subject to deformations in this paper we study a generalized class of maxwell boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix a this class of equations arises for instance from the analysis of homoenergetic solutions for the boltzmann equation considered by many authors since 1950s our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix a the main result of this paper is that for sufficiently small norm of a any non negative solution with finite second moment tends to a self similar solution of relatively simple form for large values of time we also prove that the higher order moments of the self similar profile are finite under further smallness condition on the matrix a

on a kinetic model for a simple market economy in this paper we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading we show that the kinetic model admits non trivial quasi stationary states with power law tails of pareto type in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution of wealth among individuals for this equation the stationary state can be easily derived and shows a pareto power law tail numerical results confirm the previous analysis

stable equilibrium based on levy statistics a linear boltzmann equation approach to obtain further insight on possible power law generalizations of boltzmann equilibrium concepts we consider stochastic collision models the models are a generalization of the rayleigh collision model for a heavy one dimensional particle m interacting with ideal gas particles with a mass m m similar to previous approaches we assume elastic uncorrelated and impulsive collisions we let the bath particle velocity distribution function to be of general form namely we do not postulate a specific form of power law equilibrium we show under certain conditions that the velocity distribution function of the heavy particle is levy stable the maxwellian distribution being a special case we demonstrate our results with numerical examples the relation of the power law equilibrium obtained here to thermodynamics is discussed in particular we compare between two models a thermodynamic and an energy scaling approaches these models yield insight into questions like the meaning of temperature for power law equilibrium and into the issue of the universality of the equilibrium i e is the width of the generalized maxwellian distribution functions obtained here independent of coupling constant to the bath

on the relation between rates of relaxation and convergence of wild sums for solutions of the kac equation abstract in the case of maxwellian molecules the wild summation formula gives an expression for the solution of the spatially homogeneous boltzmann equation in terms of its initial data f as a sum f v t n 1 e t 1 e t n 1 q n f v here q n f is an average over n fold iterated wild convolutions of f if m denotes the maxwellian equilibrium corresponding to f then it is of interest to determine bounds on the rate at which q n f m l 1 r tends to zero in the case of the kac model we prove that for every e 0 if f has moments of every order and finite fisher information there is a constant c so that for all n q n f m l 1 r cn e where is the least negative eigenvalue for the linearized collision operator we show that is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of f t to m a key role in the analysis is played by a decomposition of q n f into a smooth part and a small part this depends in an essential way on a probabilistic construction of mckean it allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regularity of the initial data

stability and exponential convergence for the boltzmann equation we prove existence uniqueness and stability for solutions of the nonlinear boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function the results are given for a range of spaces including l1 and extend previous results in l for the non homogeneous equation as well as the more developed lp theory for the spatially homogeneous boltzmann equation

multivariate hausdorff operators on the spaces lp rn abstract the operators h indicated in the title are characterized by a finite borel measure on r n a borel measurable function c on r n and an n n a e nonsingular matrix a whose entries are also borel measurable functions on r n and h f is defined by means of a lebesgue stieltjes integral with respect to we give simple sufficient conditions in order that these operators be bounded on the lebesgue spaces l p r n for some 1 p these sufficient conditions are exact even in the well known special cases of the cesaro and copson operators we also determine the hausdorff operator h which is adjoint to h in a certain sense we reveal interrelations among these operators and the fourier transform of a function f in l 1 r n on closing we present further special hausdorff operators

entropy production estimates for boltzmann equations with physically realistic collision kernels we establish strict entropy production bounds for the boltzmann equation with the hard sphere collision kernel using these entropy production bounds we prove results asserting that the rate at which strongl 1 convergence to equilibrium occurs is uniform in wide classes of initial data this extends our previous results in this direction which applied only to a very special collision kernel moreover the present results provide computable lower bounds compactness arguments are entirely avoided the uniformity is an important ingredient in our study of scaling limits of solutions of the non spatially homogeneous boltzmann equation and is the main focus of this paper however the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous boltzmann equation with initial data far from equilibrium to reach any given small strongl 1 neighborhood of equilibrium

approach to self similarity in smoluchowski s coagulation equations we consider the approach to self similarity or dynamical scaling in smoluchowski s equations of coagulation for the solvable kernels k x y 2 x y and xy in addition to the known self similar solutions with exponential tails there are one parameter families of solutions with algebraic decay whose form is related to heavy tailed distributions well known in probability theory for k 2 the size distribution is mittag leffler and for k x y and k xyit is a power law rescaling of a maximally skewed stable levy distribution we characterize completely the domains of attraction of all self similar solutions under weak convergence of measures our results are analogous to the classical characterization of stable distributions in probability theory the proofs are simple relying on the laplace transform and a fundamental rigidity lemma for scaling limits c 2004 wiley periodicals inc

kinetic models for the trading of goods in this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents the leading idea is to describe the trading of these goods by means of some fundamental rules in price theory in particular by using cobb douglas utility functions for the binary exchange and the edgeworth box for the description of the common exchange area in which utility is increasing for both agents this leads to a boltzmann type equation in which the post interaction variables depend in a nonlinear way from the pre interaction ones other models will be derived by suitably linearizing this boltzmann equation in presence of uncertainty in the exchanges it is shown that the solution to some of the linearized kinetic equations develop pareto tails where the pareto index depends on the ratio between the gain and the variance of the uncertainty in particular the result holds true for the solution of a drift diffusion equation of fokker planck type obtained from the linear boltzmann equation as the limit of quasi invariant trades

self similarity in random collision processes kinetics of collision processes with linear mixing rules are investigated analytically the velocity distribution becomes self similar in the long time limit and the similarity functions have algebraic or stretched exponential tails the characteristic exponents are roots of transcendental equations and vary continuously with the mixing parameters in the presence of conservation laws the velocity distributions become universal

stable equilibrium based on levy statistics stochastic collision models approach we investigate equilibrium properties of two very different stochastic collision models i the rayleigh particle and ii the driven maxwell gas for both models the equilibrium velocity distribution is a levy distribution the maxwell distribution being a special case we show how these models are related to fractional kinetic equations our work demonstrates that a stable power law equilibrium which is independent of details of the underlying models is a natural generalization of maxwell s velocity distribution

a survey of numerical solutions to the coagulation equation we present the results of a systematic survey of numerical solutions to the coagulation equation for a rate coefficient of the form aij i j i j and monodisperse initial conditions the results confirm that there are three classes of rate coefficients with qualitatively different solutions for 1 and 1 the numerical solution evolves in an orderly fashion and tends towards a self similar solution at large time t the properties of the numerical solution in the scaling limit agree with the analytic predictions of van dongen and ernst in particular for the subset with 0 and 1 the numerical solution tends towards a self similar solution as t approaches a finite time t0 the mass spectrum nk develops at t0 a power law tail nk k at large masses that violates mass conservation and runaway growth gelation is expected to start at tcrit t0 in the limit the initial number of particles n0 the exponent is in general less than the analytic prediction 3 2 and t0 k 1 n0a11 with k 1 2 if 1 1 for 1 the behaviours of the numerical solution are similar to those found in a previous paper by us they strongly suggest that there are no self consistent solutions at any time and that runaway growth is instantaneous in the limit n0 they also indicate that the time tcrit for the onset of runaway growth decreases slowly towards zero with increasing n0

well posedness of smoluchowski s coagulation equation for a class of homogeneous kernels the uniqueness and existence of measure valued solutions to smoluchowski s coagulation equation are considered for a class of homogeneous kernels denoting by 2 0 the degree of homogeneity of the coagulation kernel a measure valued solutions are shown to be unique under the sole assumption that the moment of order of the initial datum is finite a similar result was already available for the kernels a x y 2 x y and xy and is extended here to a much wider class of kernels by a different approach the uniqueness result presented herein also seems to improve previous results for several explicit kernels furthermore a comparison principle and a contraction property are obtained for the constant kernel

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

on steady distributions of kinetic models of conservative economies we analyze the large time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years as specific examples we study models with fixed saving propensity introduced by chakraborti and chakrabarti eur phys j b 17 167 170 2000 as well as models involving both exchange between agents and speculative trading as considered by cordier et al j stat phys 120 253 277 2005 we derive a sufficient criterion under which a unique non trivial stationary state exists and provide criteria under which these steady states do or do not possess a pareto tail in particular we prove the absence of pareto tails in pointwise conservative models like the one in eur phys j b 17 167 170 2000 while models with speculative trades introduced in j stat phys 120 253 277 2005 develop fat tails if the market is risky enough the results are derived by a fourier based technique first developed for the maxwell boltzmann equation gabetta et al in j stat phys 81 901 934 1995 bisi et al in j stat phys 118 1 2 301 331 2005 pareschi and toscani in j stat phys 124 2 4 747 779 2006 and from a recursive relation which allows to calculate arbitrary moments of the stationary state

inelastically scattering particles and wealth distribution in an open economy using the analogy with inelastic granular gases we introduce a model for wealth exchange in society the dynamics is governed by a kinetic equation which allows for self similar solutions the scaling function has a power law tail the exponent being given by a transcendental equation in the limit of continuous trading a closed form of the wealth distribution is calculated analytically

on convergence to equilibrium for kac s caricature of a maxwell gas the trend towards equilibrium of the solution to the one dimensional kac s caricature of a maxwell gas is investigated when the initial value has finite energy and finite linnik s entropy it is proven that the solution converges in l1 l to the equilibrium maxwellian distribution

explicit equilibria in a kinetic model of gambling we introduce and discuss a nonlinear kinetic equation of boltzmann type which describes the evolution of wealth in a pure gambling process where the entire sum of wealths of two agents is up for gambling and randomly shared between the agents for this equation the analytical form of the steady states is found for various realizations of the random fraction of the sum which is shared to the agents among others the exponential distribution appears as steady state in case of a uniformly distributed random fraction while gamma distribution appears for a random fraction which is beta distributed the case in which the gambling game is only conservative in the mean is shown to lead to an explicit heavy tailed distribution

existence of self similar solutions to smoluchowski s coagulation equation the existence of self similar solutions to smoluchowski s coagulation equation has been conjectured for several years by physicists and numerical simulations have confirmed the validity of this conjecture still there was no existence result up to now except for the constant and additive kernels for which explicit formulae are available in this paper the existence of self similar solutions decaying rapidly at infinity is established for a wide class of homogeneous coagulation kernels

the kac equation with a thermostatted force field we consider the kac equation with a thermostatted force field and prove the existence of a global in time solution that converges weakly to a stationary state as there is no an obvious candidate for the entropy functional in this case the convergence result is obtained via fourier transform techniques

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

power law distributions from additive preferential redistributions we introduce a nongrowth model that generates the power law distribution with the zipf exponent there are elements each of which is characterized by a quantity and at each time step these quantities are redistributed through binary random interactions with a simple additive preferential rule while the sum of quantities is conserved the situation described by this model is similar to those of closed particle systems when conservative two body collisions are only allowed we obtain stationary distributions of these quantities both analytically and numerically while varying parameters of the model and find that the model exhibits the scaling behavior for some parameter ranges unlike well known growth models this alternative mechanism generates the power law distribution when the growth is not expected and the dynamics of the system is based on interactions between elements this model can be applied to some examples such as personal wealths city sizes and the generation of scale free networks when only rewiring is allowed

strong convergence towards self similarity for one dimensional dissipative maxwell models we prove the propagation of regularity uniformly in time for the scaled solutions of the one dimensional dissipative maxwell models introduced in d ben avraham e ben naim k lindenberg a rosas self similarity in random collision processes phys rev e 68 2003 r050103 this result together with the weak convergence towards the stationary state proven in l pareschi g toscani self similarity and power like tails in nonconservative kinetic models j stat phys 124 2 4 2006 747 779 implies the strong convergence in sobolev norms and in the l1 norm towards it depending on the regularity of the initial data as a consequence the original nonscaled solutions are also proved to be convergent in l1 towards the corresponding self similar homogeneous cooling state the proof is based on the uniform in time control of the tails of the fourier transform of the solution and it holds for a large range of values of the mixing parameters in particular in the case of the one dimensional inelastic boltzmann equation the result does not depend of the degree of inelasticity

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

central limit theorems for solutions of the kac equation speed of approach to equilibrium in weak metrics this paper is part of our efforts to show how direct application of probabilistic methods pertaining to central limit general theory can enlighten us about the convergence to equilibrium of the solutions of the kac equation here we consider convergence with respect to the following metrics kolmogorov s uniform metric 1 and 2 gini s dissimilarity indices widely known as 1 and 2 wasserstein metrics weighted metrics our main results provide new bounds or improvements on already well known ones for the corresponding distances between the solution of the kac equation and the limiting gaussian maxwellian distribution the study is conducted both under the necessary assumption that initial data have finite energy without assuming existence of moments of order greater than 2 and under the condition that the 2 moment of the initial distribution is finite for some 0

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

dynamical analysis of neural networks of subgradient system in this technical note we consider a class of neural network which is a generalization of neural network models considered in the optimization context under some mild assumptions this neural network can be translated into a negative subgradient dynamical system at first we study the existence and uniqueness of solution of this neural network then by nonsmooth ojasiewicz inequality we prove the convergence of the trajectories of this neural network in the end a constrained minimization problem is studied which can be associated with this neural network it is proved that the local constrained minimum of the cost function coincides with the stable equilibria point of this neural network

large time behavior of the solutions to rosenau type approximations to the heat equation in this article we study the large time behavior of the solution to a general rosenau type approximation to the heat equation p rosenau phys rev a 3 46 1992 pp 12 15 by showing that the solution to this approximation approaches the fundamental solution of the heat equation the heat kernel but at a slower rate than the usual heat equation this result is valid in particular for the central difference scheme approximation of the heat equation a property which to our knowledge has never been observed before

on the cauchy problem for the inelastic boltzmann equation with external force in this paper the cauchy problem for the inelastic boltzmann equation with external force is considered for near vacuum data under the assumptions on the bicharacteristic generated by external force which can be arbitrarily large we prove the global existence of mild solution for initial data small enough with respect to the sup norm with exponential weight by using the contraction mapping theorem furthermore we prove the uniform l1 stability of the mild solution following from the exponential decay estimate and the gronwall s inequality for the case of soft potentials

sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation we derive a new lower bound for the entropy dissipation associated with the spatially homogeneous boltzmann equation this bound is expressed in terms of the relative entropy with respect to the equilibrium and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy with an explicit rate our result gives a considerable refinement of the analogous estimate by carlen and carvalho 9 10 under very little additional assumptions our proof takes advantage of the structure of boltzmann s collision operator with respect to the tensor product and its links with fokker planck and landau equations several variants are discussed

regularity and long time behavior of the boltzmann equation for granular gases abstract this paper investigates regularity of solutions of the boltzmann equation with dissipative collisions in a thermal bath in the case of pseudo maxwellian approximation we prove that for any initial datum f 0 in the set of probability density with zero bulk velocity and finite temperature the unique solution of the equation satisfies f t h r 3 for all t 0 furthermore for any t 0 0 and s 0 the h s norm of f t is bounded for t t 0 as a consequence the exponential convergence to the unique steady state is also established under the same initial condition

decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model we quantify the long time behavior of solutions to the nonlinear boltzmann equation for spatially uniform freely cooling inelastic maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures existence uniqueness and precise estimates of overpopulated high energy tails of the self similar profile proved in ref 9 are revisited and derived from this new liapunov functional for general initial conditions the solutions of the boltzmann equation are then proved to converge with computable rate as t to the self similar solution in this distance which metrizes the weak convergence of measures moreover we can relate this fourier distance to the euclidean wasserstein distance or tanaka functional proving also its exponential convergence towards the homogeneous cooling states the findings are relevant in the understanding of the conjecture formulated by ernst and brito in refs 15 16 and complement and improve recent studies on the same problem of bobylev and cercignani 9 and bobylev cercignani and one of the authors 11

propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas we prove an inequality for the gain term in the boltzmann equation for maxwellian molecules that implies a uniform bound on sobolev norms of the solution provided the initial data has a finite norm in the corresponding sobolev space we then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm these results are then combined using interpolation inequalities to obtain the optimal rate of exponential convergence in the strong l 1 norm as well as various sobolev norms these results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non linear boltzmann equation even for initial data that is far from equilibrium

nontrivial velocity distributions in inelastic gases we study spatially homogeneous inelastic gases using the boltzmann equation we consider uniform collision rates and obtain analytical results valid for arbitrary spatial dimension d and arbitrary dissipation coefficient in the unforced case we find that the velocity distribution decays algebraically p v t v for sufficiently large velocities the exponent d exhibits nontrivial dependence on the spatial dimension and the dissipation coefficient

fast and slow convergence to equilibrium for maxwellian molecules via wild sums we consider the spatially homogeneous boltzmann equation for maxwellian molecules and general finite energy initial data positive borel measures with finite moments up to order 2 we show that the coefficients in the wild sum converge strongly to the equilibrium and quantitatively estimate the rate we show that this depends on the initial data f essentially only through on the behavior near r 0 of the function j f r v 1 r v 2 df v these estimates on the terms in the wild sum yield a quantitative estimate in the strongest physical norm on the rate at which the solution converges to equilibrium as well as a global stability estimate we show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds these are the first examples of solutions of the boltzmann equation that converge to equilibrium more slowly than exponentially

on the rate of explosion for infinite energy solutions of the spatially homogeneous boltzmann equation let 0 be a probability measure on 3 representing an initial velocity distribution for the spatially homogeneous boltzmann equation for pseudo maxwellian molecules as long as the initial energy is finite the solution t will tend to a maxwellian limit we show here that if int mathbb r 3 v 2 mu 0 mathrm d v infty then instead all of the mass explodes to infinity at a rate governed by the tail behavior of 0 specifically for l0 define eta l int v le l v 2 mathrm d mu 0 v let b r denote the centered ball of radius r then for every r lim t to infty int b r mathrm d mu t v 0 the explicit rate is estimated in terms of the rate of divergence of l for example if l const l s some s 0 int b r mathrm d mu t v is bounded by a multiple of e 3s 10 9s t where is the absolute value of the spectral gap in the linearized collision operator note that in this case letting b t denote the ball of radius e rt for any r s 10 9s we still have lim t to infty int b t mathrm d mu t v 0

tanaka theorem for inelastic maxwell models we show that the euclidean wasserstein distance is contractive for inelastic homogeneous boltzmann kinetic equations in the maxwellian approximation and its associated kac like caricature this property is as a generalization of the tanaka theorem to inelastic interactions even in the elastic classical boltzmann equation we give a simpler proof of the tanaka theorem than the ones in 29 31 consequences are drawn on the asymptotic behavior of solutions in terms only of the euclidean wasserstein distance

on the self similar asymptotics for generalized nonlinear kinetic maxwell models maxwell models for nonlinear kinetic equations have many applications in physics dynamics of granular gases economics etc in the present manuscript we consider such models from a very general point of view including those with arbitrary polynomial non linearities and in any dimension space it is shown that the whole class of generalized maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual lipschitz conditions this property allows to describe in detail a behavior of solutions to the corresponding initial value problem in particular we prove in the most general case an existence of self similar solutions and study the convergence in the sense of probability measures of dynamically scaled solutions to the cauchy problem to those self similar solutions as time goes to infinity a new application of multi linear models to economics and social dynamics is discussed

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

dynamical scaling in smoluchowski s coagulation equations uniform convergence smoluchowski s coagulation equation is a fundamental mean field model of clustering dynamics we consider the approach to self similarity or dynamical scaling of the cluster size distribution for the solvable rate kernels k x y 2 x y and xy in the case of continuous cluster size distributions we prove uniform convergence of densities to a self similar solution with exponential tail under the regularity hypothesis that a suitable moment have an integrable fourier transform for discrete size distributions we prove uniform convergence under optimal moment hypotheses our results are completely analogous to classical local convergence theorems for the normal law in probability theory the proofs rely on the fourier inversion formula and the solution for the laplace transform by the method of characteristics in the complex plane

boundedness of hausdorff operators on real hardy spaces h1 over locally compact groups abstract results of liflyand and collaborators on the boundedness of hausdorff operators on the hardy space h 1 over finite dimensional real space are generalized to the case of locally compact groups that are spaces of homogeneous type special cases and examples of compact lie groups homogeneous groups in particular the heisenberg group and finite dimensional spaces over division rings are considered in conclusion we solve for the space l 2 an open question on compactness of a hausdorff operator posed by liflyand

asymptotic properties of the inelastic kac model we discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of kac caricature of a maxwellian gas it is shown that global equilibria different from concentration are possible if the energy is not finite these equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction if the initial density is assumed of finite energy with higher moments bounded it is shown that the solution converges for large time to a profile with power law tails these tails are heavily dependent of the collision rule

the inelastic enskog equation with external force in this paper the cauchy problem for the inelastic enskog equation with external force is considered for near vacuum data under the assumptions on the bi characteristic we proved the global existence of mild solution