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

Topics in propagation of chaos

Several recent works have shown that the one-dimensional fully asymmetric exclusion model, which describes a system of particles hopping in a preferred direction with hard core interactions, can be solved exactly in the case of open boundaries. Here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices. With this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules. They obtain several explicit forms for these non-commuting matrices which are, in the general case, infinite-dimensional. Their approach allows exact expressions to be derived for the current and density profiles. Finally they discuss briefly two possible generalizations of their results: the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles.

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Exact solution of a 1d asymmetric exclusion model using a matrix formulation

These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow us to calculate the fluctuations and large deviations of the density and the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example, non-Gaussian fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium.

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Non-equilibrium steady states: fluctuations and large deviations of the density and of the current

http://www.lps.ens.fr/~derrida/PAPIERS/1998/altenberg-1998.pdf

An exactly soluble non-equilibrium system : the asymmetric simple exclusion process

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.

An invariance principle for reversible Markov processes. Applications to random motions in random environments

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

We study interacting spin (particle) systems on a lattice under the combined influence of spin flip (Glauber) and simple exchange (Kawasaki) dynamics. We prove that when the particle-conserving exchanges (stirrings) occur on a fast time scale of order ɛ−2 the macroscopic density, defined on spatial scale ɛ−1, evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are found explicitly. They grow, with time, to become infinite when the deterministic solution is unstable.

/pdf/reaction-diffusion-equations-for-interacting-particle-5f9vejn13g.pdf

Reaction-diffusion equations for interacting particle systems

This is the first of three papers on the Glauber evolution of Ising spin systems with Kac potentials. We begin with the analysis of the mesoscopic limit, where space scales like the diverging range, gamma -1, of the interaction while time is kept finite: we prove that in this limit the magnetization density converges to the solution of a deterministic, nonlinear, nonlocal evolution equation. We also show that the long time behaviour of this equation describes correctly the evolution of the spin system till times which diverge as gamma to 0 but are small in units log gamma -1. In this time regime we can give a very precise description of the evolution and a sharp characterization of the spin trajectories. As an application of the general theory, we then prove that for ferromagnetic interactions, in the absence of external magnetic fields and below the critical temperature, on a suitable macroscopic limit, an interface between two stable phases moves by mean curvature. All the proofs are consequence of sharp estimates on special correlation functions, the v-functions, whose analysis is reminiscent of the cluster expansion in equilibrium statistical mechanics.

Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics

This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential \(\frac{1}{N} \). This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system. We show that, as the system size N diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.

A. De Masi

Papers

An invariance principle for reversible Markov processes. Applications to random motions in random environments

Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation

Reaction-diffusion equations for interacting particle systems

Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics

Hydrodynamic limit for interacting neurons