Ecole Normale Supérieure de Lyon,
The theory of stochastic processes has played a central role in the study of mesoscopic systems. For example, in Brownian motion a large number of degrees of freedom are represented by a random noise force. Many equilibrium and nonequilibrium phenomena are accurately described by a stochastic representation but often this formulation does not lend itself to analytic treatment. As such, numerical methods, including computer simulations, have played an important role in the field.
This tutorial covers stochastic algorithms used in the simulation of mesoscopic systems. The participants from a variety of disciplines (statistical mechanics, fluid mechanics, physical chemistry, etc.) with a background in scientific programming are encouraged to attend.
Overview (Garcia & Baras): The general ideas common to mesoscopic stochastic algorithms will be discussed and a spectrum of applications will be described.
Random number generation (Garcia): Stochastic algorithms use random numbers from many different distributions. General techniques for generating numbers from arbitrary distributions and specialized techniques for common distributions will be described.
Langevin equation (Baras): Numerical methods for Brownian motion will be outlined and illustrated in a variety of topics (e.g., velocity correlation function, fluctuation-dissipation theorem, Brownian motion of a harmonic oscillator, molecular motion of polymers in dilute solutions, tracers in complex flows, etc.) Stochastic partial differential equations will be discussed in the context of fluctuating hydrodynamics.
Master equation (Baras): The Gillespie method will be described and illustrated in the analysis of reactive systems (homogeneous systems with oscillations or multi steady-states; spatially extended systems, etc.) A brief overview on related Monte Carlo methods used in surface reactions will be presented.
Direct Simulation Monte Carlo (Garcia): Basic concepts from kinetic theory, such as binary collisions, Boltzmann equation, Chapman-Enskog expansion, will be reviewed . The algorithmic elements of DSMC (moving particles, boundary conditions, sorting, collisions, sampling, etc.) will be presented. A variety of advanced topics (e.g., granular flows, hybrid algorithms) will be discussed.