Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

http://dsec.pku.edu.cn/~tieli/notes/numer_anal/MCQMC_Caflisch.pdf

Monte Carlo and quasi-Monte Carlo methods

We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution. From this stability result we deduce global existence of a solution to the Cauchy problem. Our method relies upon recent compactness results for velocity averages, a new formulation of the Boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions. It allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations.

https://www.jstor.org/stable/pdfplus/1971423.pdf

On the Cauchy problem for Boltzmann equations: global existence and weak stability

This paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory. The first member of the hierarchy is the Euler system, which is based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions. The closure proceeds in two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit. The second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct Navier-Stokes behavior. This is achieved through the introduction of a generalization of the BGK collision operator. The simplest such system in three spatial dimensions is a “14-moment” closure, which also recovers the behavior of the Grad “13-moment” system when the velocity distributions lie near local Maxwellians. The closure procedure can be applied to a general class of kinetic theories.

Moment Closure Hierarchies for Kinetic Theories

Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory

We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1158328658

Stochastic Burgers and KPZ equations from particle systems

We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3,










For macroscopic times τ = t/ϵN, ϵ « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ϵN−1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t0, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2.



We prove that (*) has solutions for t ≦ t0 which are close, to O(ϵ2) in a suitable norm, to the local Maxwellian [p/(2πT)d/2]exp{−[v − ϵu(x,t)]2/2T} with constant density p and temperature T. This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.

/pdf/incompressible-navier-stokes-and-euler-limits-of-the-23b24sf8ty.pdf

Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation

In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the following steady problem:

 $$v \cdot \nabla _{x}F =\frac{1}{{\rm K}_{\rm n}}Q(F,F),\qquad (x,v)\in \Omega \times \mathbf{R}^{3}, \quad \quad (0.1) $$
 $$F(x,v)|_{n(x)\cdot v 0}F(x,v^{\prime})(n(x)\cdot v^{\prime})dv^{\prime},\quad x \in\partial \Omega,\quad \quad (0.2) $$
 where Ω is a bounded domain in $${\mathbf{R}^{d}, 1 \leq d \leq 3}$$
 , Kn is the Knudsen number and $${\mu _{\theta}=\frac{1}{2\pi \theta ^{2}(x)} {\rm exp} [-\frac{|v|^{2}}{2\theta (x)}]}$$
 is a Maxwellian with non-constant(non-isothermal) wall temperature θ(x). Based on new constructive coercivity estimates for both steady and dynamic cases, for $${|\theta -\theta_{0}|\leq \delta \ll 1}$$
 and any fixed value of Kn, we construct a unique non-negative solution F
 s
 to (0.1) and (0.2), continuous away from the grazing set and exponentially asymptotically stable. This solution is a genuine non-equilibrium stationary solution differing from a local equilibrium Maxwellian. As an application of our results we establish the expansion $${F_s=\mu_{\theta_0}+\delta F_{1}+O(\delta ^{2})}$$
 and we prove that, if the Fourier law holds, the temperature contribution associated to F
 1 must be linear, in the slab geometry.

Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law

We consider the time evolution of a one dimensional n-gradient continuum. Our aim is to construct and analyze discrete approximations in terms of physically realizable mechanical systems, referred to as microscopic because they are living on a smaller space scale. We validate our construction by proving a convergence theorem of the microscopic system to the given continuum, as the scale parameter goes to zero.

https://hal.archives-ouvertes.fr/hal-01155474/document

Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials

We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prove that when the force is sufficiently small there exists a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip boundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.

/pdf/hydrodynamic-limit-of-the-stationary-boltzmann-equation-in-a-pmbnjl1qm7.pdf

Hydrodynamic limit of the stationary Boltzmann equation in a slab

Despite its conceptual and practical importance, a rigorous derivation of the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative $$L^{2}$$
 –
 $$L^{\infty }$$
 approach with new $$L^{{6}}$$
 estimates for the hydrodynamic part $$\mathbf {P}f$$
 of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to an asymptotical stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier system.

Raffaele Esposito

Papers

Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation

Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law

Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials

Hydrodynamic limit of the stationary Boltzmann equation in a slab

Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit