equilibrium fluctuations for a non gradient energy conserving stochastic model in this paper we study the equilibrium energy fluctuation field of a one dimensional reversible non gradient model we prove that the limit fluctuation process is governed by a generalized ornstein uhlenbeck process whose covariances are given in terms of the diffusion coefficient

kinetic hierarchy and propagation of chaos in biological swarm models we consider two models of biological swarm behavior in these models pairs of particles interact to adjust their velocities one to each other in the first process called bdg they join their average velocity up to some noise in the second process called cl one of the two particles tries to join the other one s velocity this paper establishes the master equations and bbgky hierarchies of these two processes it investigates the infinite particle limit of the hierarchies at large time scales it shows that the resulting kinetic hierarchy for the cl process does not satisfy propagation of chaos numerical simulations indicate that the bdg process has similar behavior to the cl process c 2012 elsevier b v all rights reserved

chaos ergodicity and equilibria in a quantum kac model abstract we introduce quantum versions of the kac master equation and the kac boltzmann equation we study the steady states of each of these equations and prove a propagation of chaos theorem that relates them the quantum kac master equation qkme describes a quantum markov semigroup p n t while the kac boltzmann equation describes a non linear evolution of density matrices on the single particle state space all of the steady states of the n particle quantum system described by the qkme are separable and thus the evolution described by the qkme is entanglement breaking the results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory and a quantitative study of the rate of destruction of entanglement in a class of quantum markov semigroups describing binary interactions

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

on the markov sequence problem for jacobi polynomials abstract we give a simple and entirely elementary proof of gasper s theorem on the markov sequence problem for jacobi polynomials it is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by marc kac and the methods developed here yield new estimates relevant to the collision model

the linear fokker planck equation for the ornstein uhlenbeck process as an almost nonlinear kinetic equation for an isolated n particle system it is long known that the fokker planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a brownian test particle immersed in a heat bath of fixed temperature apparently it is not so well known that the same partial differential equation but now with constant coefficients which are functionals of the solution itself rather than being prescribed describes the kinetic evolution in the n limit of an isolated n particle system with certain stochastic interactions here we discuss in detail this recently discovered interpretation

spectral gap for the kac model with hard sphere collisions abstract we prove the analog of the kac conjecture for hard sphere collisions giving a computable close estimate on the spectral gap that is independent of the number of particles previous work has focused on the case in which the collision rates are independent of the particle velocities the case of so called maxwellian molecules the new methods introduced here allow us to deal with collision rates that are not bounded from below we also obtain information on the structure of the gap eigenfunction

cut off phenomenon in the uniform plane kac walk we consider an analogue of the kac random walk on the special orthogonal group so n so n in which at each step a random rotation is performed in a randomly chosen 2 plane of rnrn we obtain sharp asymptotics for the rate of convergence in total variance distance establishing a cut off phenomenon in the large nn limit in the special case where the angle of rotation is deterministic this confirms a conjecture of rosenthal ann probab 22 1994 398 423 under mild conditions we also establish a cut off for convergence of the walk to stationarity under the l2l2 norm depending on the distribution of the randomly chosen angle of rotation several surprising features emerge for instance it is sometimes the case that the mixing times differ in the total variation and l2l2 norms our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

entropy dissipation estimates in a zero range dynamics we study the exponential decay of relative entropy functionals for zero range processes on the complete graph for the standard model with rates increasing at infinity we prove entropy dissipation estimates uniformly over the number of particles and the number of vertices

spectral gap for stochastic energy exchange model with nonuniformly positive rate function we give a lower bound on the spectral gap for a class of stochastic energy exchange models in 2011 grigo et al introduced the model and showed that for a class of stochastic energy exchange models with a uniformly positive rate function the spectral gap of an n component system is bounded from below by a function of order n 2 in this paper we consider the case where the rate function is not uniformly positive for this case the spectral gap depends not only on n but also on the averaged energy e which is the conserved quantity under the dynamics under some assumption we obtain a lower bound of the spectral gap which is of order c e n 2 where c e is a positive constant depending on e as a corollary of the result a lower bound of the spectral gap for the mesoscopic energy exchange process of billiard lattice studied by gaspard and gilbert j stat mech theory exp 2008 2008 p11021 j stat mech theory exp 2009 2009 p08020 and the stick process studied by feng et al stochastic process appl 66 1997 147 182 are obtained

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

hydrodynamic limit of a disordered lattice gas we consider a model of lattice gas dynamics in d in the presence of disorder if the particle interaction is only mutual exclusion and if the disorder field is given by i i d bounded random variables we prove the almost sure existence of the hydrodynamical limit in dimension d 3 the limit equation is a non linear diffusion equation with diffusion matrix characterized by a variational principle

on the optimal choice of coefficients in a truncated wild sum and approximate solutions for the kac equation we study an approximate solution of the boltzmann problem for kac s caricature of a maxwellian gas by using a truncated and modified expansion of wild type we choose the coefficients in the wild sum approximation using a criterion based on exactly reproducing the behavior of the leading modes

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

hyperbolic versus parabolic asymptotics in kinetic theory toward fluid dynamic models in this work we are interested in the hyperbolic limits in kinetic theory we propose a nonstandard scaling to be understood as a sort of intermediate hyperbolic limit which connects the macroscopic hyperbolic limiting behavior of the physical system with the microscopic properties usually obtained under parabolic scalings we present our main result by means of a general kinetic frame for the intermediate hyperbolic limit which covers some well known examples in kinetic theory vlasov poisson fokker planck systems and linear relaxation for boltzmann type equations in semiconductor theory among others we will also apply our methods to deal with the kac approach to boltzmann operators

logarithmic sobolev inequality for lattice gases with mixing conditions let mu lambda l lambda gc denote the grand canonical gibbs measure of a lattice gas in a cube of sizel with the chemical potential and a fixed boundary condition let mu lambda l n c be the corresponding canonical measure defined by conditioning mu lambda l lambda gc on sigma x in lambda eta x n consider the lattice gas dynamics for which each particle performs random walk with rates depending on near by particles the rates are chosen such that for everyn andl fixed mu lambda l n c is a reversible measure suppose that the dobrushin shlosman mixing conditions holds for mu l lambda gc forall chemical potentials we prove that smallint f log fd mu lambda l n c leqq const l 2 d sqrt f for any probability densityf with respect to mu lambda l n c here the constant is independent ofn orl andd denotes the dirichlet form of the dynamics the dependence onl is optimal

logarithmic sobolev inequality for generalized simple exclusion processes let be a probability measure on the set 0 1 r for some r and l a cube of width l in d denote by gc l the grand canonical product measure on the configuration space on l with as the marginal measure here the superscript indicates the grand canonical ensemble the canonical ensemble denoted by c l n is defined by conditioning gc l given the total number of particles to be n consider the exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site the rates are chosen such that for every n and l fixed the measure c l n is reversible we prove the logarithmic sobolev inequality in the sense that flogfd c l n for any probability density f with respect to c l n here the constant is independent of n or l and d denotes the dirichlet form of the dynamics the dependence on l is optimal

interval valued fuzzy graphs we define the cartesian product composition union and join on interval valued fuzzy graphs and investigate some of their properties we also introduce the notion of interval valued fuzzy complete graphs and present some properties of self complementary and self weak complementary interval valued fuzzy complete graphs

spectral gap for kac s model of boltzmann equation we consider a random walk on s n 1 the standard sphere of dimension n 1 generated by random rotations on randomly selected coordinate planes i j with 1 i j n this dynamic was used by marc kac as a model for the spatially homogeneous boltzmann equation we prove that the spectral gap on s n 1 is n 1 up to a constant independent of n

the fast johnson lindenstrauss transform and approximate nearest neighbors we introduce a new low distortion embedding of ell 2 d into ell p o log n p 1 2 called the fast johnson lindenstrauss transform fjlt the fjlt is faster than standard random projections and just as easy to implement it is based upon the preconditioning of a sparse projection matrix with a randomized fourier transform sparse random projections are unsuitable for low distortion embeddings we overcome this handicap by exploiting the heisenberg principle of the fourier transform i e its local global duality the fjlt can be used to speed up search algorithms based on low distortion embeddings in ell 1 and ell 2 we consider the case of approximate nearest neighbors in ell 2 d we provide a faster algorithm using classical projections which we then speed up further by plugging in the fjlt we also give a faster algorithm for searching over the hypercube

ornstein uhlenbeck limit for the velocity process of an n particle system interacting stochastically an n particle system with stochastic interactions is considered interactions are driven by a brownian noise term and total energy conservation is imposed the evolution of the system in velocity space is a diffusion on a 3n 1 dimensional sphere with radius fixed by the total energy in the n limit a finite number of velocity components are shown to evolve independently and according to an ornstein uhlenbeck process

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

the phonon boltzmann equation properties and link to weakly anharmonic lattice dynamics for low density gases the validity of the boltzmann transport equation is well established the central object is the one particle distribution function f which in the boltzmann grad limit satisfies the boltzmann equation grad and much refined cercignani argue for the existence of this limit on the basis of the bbgky hierarchy for hard spheres at least for a short kinetic time span the argument can be made mathematically precise following the seminal work of lanford in this article a corresponding program is undertaken for weakly nonlinear both discrete and continuum wave equations our working example is the harmonic lattice with a weakly nonquadratic on site potential we argue that the role of the boltzmann f function is taken over by the wigner function which is a very convenient device to filter the slow degrees of freedom the wigner function so to speak labels locally the covariances of dynamically almost stationary measures one route to the phonon boltzmann equation is a gaussian decoupling which is based on the fact that the purely harmonic dynamics has very good mixing properties as a further approach the expansion in terms of feynman diagrams is outlined both methods are extended to the quantized version of the weakly nonlinear wave equation the resulting phonon boltzmann equation has been hardly studied on a rigorous level as one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal wigner functions for three phonon processes such a result requires extra conditions on the dispersion law we also outline the reasoning leading to fourier s law for heat conduction

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

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

poincare and logarithmic sobolev inequality for ginzburg landau processes in random environment we consider reversible conservative ginzburg landau processes in a random environment whose potential are bounded perturbations of the gaussian potential evolving on a d dimensional cube of length l we prove in all dimensions that the spectral gap of the generator and the logarithmic sobolev constant are of order l 2 almost surely with respect to the environment

global spectral gap for dirichlet kac random motions we prove that the global spectral gap for any dirichlet kac random motion is equal to the convergence rate of the limit motion

uniform contractivity in wasserstein metric for the original 1d kac s model we study here a very popular 1d jump model introduced by kac it consists of n velocities encountering random binary collisions at which they randomly exchange energy we show the uniform in n exponential contractivity of the dynamics in a non standard monge kantorovich wasserstein precisely the mkw metric of order 2 on the energy the result is optimal in the sense that for each n the contractivity constant is equal to the l 2 spectral gap of the generator associated to kac s dynamic as a corollary we get an uniform but non optimal contractivity in the mkw metric of order 4 we use a simple coupling that works better that the parallel one the estimates are simple and new to the best of our knowledge