Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit

We consider a stochastic N-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when N⟶∞, with non-asymptotic estimates: for any time T>0, we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in a relevant Hilbert space. The control got is Gaussian, i.e. we prove that the distance is bigger than xN ^−1/2 with a probability of type O(e^-x2)O(e^{-x^{2}}). The two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of Lipschitz continuity for the Boltzmann collision kernel. We study more extensively the case where our Hilbert space is the homogeneous negative Sobolev space [(H)\dot]^-s\smash {\dot {H}}^{-s}. Then we are only able to give bounds for Maxwellian models; however, numerical computations tend to show that our results are useful in practice.     

A maxwellian lower bound for solutions to the Boltzmann equation

We prove that the solution of the spatially homogeneous Boltzmann equation is bounded pointwise from below by a Maxwellian, i.e. a function of the formc ₁ exp(-c ₂ v ²). This holds for any initial data with bounded mass, energy and entropy, and for any positive timet≧t ₀. The constantsc ₁, andc ₂, depend on the mass, energy and entropy of the initial data, and ont ₀>0 only. A similar result is obtained for the Kac caricature of the Boltzmann equation, where the proof is easier.     

Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv ²), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA ₁ exp[−B ₁(t)v ²] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.     

Convergence to SPDEs in Stratonovich Form

We consider the perturbation of parabolic operators of the form ∂ _t  + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems.     

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.On leave from School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.On leave from C.F.M.C. and Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700 Lisboa codex, Portugal.     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.     

Quantum-statistical kinetic equations

Considering a homogeneous normal quantum fluid consisting of identical interacting fermions or bosons, we derive an exact quantum-statistical generalized kinetic equation with a collision operator given as explicit cluster series where exchange effects are included through renormalized Liouville operators. This new result is obtained by applying a recently developed superoperator formalism (Liouville operators, cluster expansions, symmetrized projectors,P _q rule, etc.) to nonequilibrium systems described by a density operator(t) which obeys the von Neumann equation. By means of this formalism a factorization theorem is proven (being essential for obtaining closed equations), and partial resummations (leading to renormalized quantities) are performed. As an illustrative application, the quantum-statistical versions (including exchange effects due to Fermi-Dirac or Bose-Einstein statistics) of the homogeneous Boltzmann (binary collisions) and Choh-Uhlenbeck (triple collisions) equations are derived.     

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.     

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⁾     

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.     

Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials

In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in L^￥_l{L^\infty_\ell}. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of Dudyński and Ekiel-Jeżewska (Commun Math Phys 115(4):607–629, 1985); this resolves the open question of global existence for the soft potentials.     

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

We consider unitary random matrix ensembles on the space of Hermitian n × n matrices M, where the confining potential V _s,t is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P _I ² equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P _I ² equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights on . The special solution of the P _I ² equation pops up in the n ^−2/7-term of the asymptotics.     

On the Rate of Explosion for Infinite Energy Solutions of the Spatially Homogeneous Boltzmann Equation

Let μ ₀ be a probability measure on ℝ³ 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 μ _t will tend to a Maxwellian limit. We show here that if , then instead, all of the mass “explodes to infinity” at a rate governed by the tail behavior of μ ₀. Specifically, for L0, define

Discrete Painlevé equations and their appearance in quantum gravity

We discuss an algorithmic approach for both deriving discrete analogues of Painlevé equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painlevé I can be used to characterize similarity solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painlevé IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painlevé I in terms of the solution of discrete Painlevé I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure exp [–t ₁ z ² –t ₂ z ⁴ –t ₃ z ⁶].     

A New Integrable Hierarchy,Parametric Solutions and Traveling Wave Solutions

In this paper we give a new integrable hierarchy. In the hierarchy there are the following representatives:

Equations with Singular Diffusivity

Recently models of faceted crystal growth and of grain boundaries were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations u _t=(u _x/|u _x|) _x and u _t=(1/a)(au _x/|u _x|) _x , where both of the related energies include a |u _x| term of power one which is nondifferentiable at u _x=0. The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when a depends on x. These equations naturally express nonlocal interactions through their singular diffusivities (infinitely large diffusion constant), which make the profiles of the solutions completely flat. The mathematical basis for justifying and analyzing these equations is explained, and theoretical and numerical approaches show how the solutions of the equations evolve.     

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.     

Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

We consider random Schrödinger equations on \({\mathbb{R}^{d}}\) for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ _t the solution with initial data ψ₀. The space and time variables scale as \({x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}}\) with 0 < κ <  κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ _t converges weakly to the solution of a heat equation in the space variable x for arbitrary L ² initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.     

Oscillatory Phase of Inflaton and Power-Law Expansion in Bianchi Type-I Universe

A homogeneous massive scalar field, minimally coupled to the spatially homogeneous and anisotropic background metric, in the semiclassical theory of gravity is examined. In the oscillatory phase of inflaton, the approximate leading solution to the semiclassical Einstein equation for the Bianchi type-I universe shows, each scale factor in each direction obeys t ^2/3 power-law expansion. Further noted that the evolution of scale factors are mutually correlated.     

A comment on chemical Langevin equations

It is noted that the diffusion Langevin stochastic sources in chemical reaction-diffusion theories should really arise from a stochastic source term added to the deterministic form of Fick's law. This gives rise to results for correlation functions which agree with those from stochastic master equations provided parameters are appropriately chosen.Some authors use the term Langevin force. Sinceg _i(x,t) is dimensionally not a force, we shall eschew this dangerous terminology.