PHYS4213 : Simulations Numériques et Thermodynamique Statistique

The course introduces the methods employed for exploring the static and dynamic properties of particle based systems. Computational exercises, where these methods are applied to simple, but powerful models, form an integral part of the module.  The underlying idea is to inverse the order in which the subject is typically taught: starting from simulations allows to "discover" the laws of Statistical Mechanics and Thermodynamics in a very direct and intuitive manner.

Topics:

• Modern computers and Laplace’s demon: Just do it?!
  • Newton’s and Hamilton’s equations of motion: Exact solvable cases, conservation laws and collisions
  • So much from so little I: Multi-particle collision (MPC) dynamics 
    • Phase space and microstates
    • Irreversibility, the H-theorem and ergodicity
    • Equilibrium ensembles
    • Driven systems
  • So much from so little II: Lattice-spin and gas models
    • Phase transitions
•  Exploring emergent static properties
  • Exact enumeration of small systems: Reweighting and the exact evaluation of partition functions
  • Monte Carlo simulations
    • Simple Sampling: Statistical errors and the limits of reweighting
    • Importance Sampling: The Metropolis algorithm; Statistical errors, dynamical correlations, Glauber vs. Kawasaki for spin systems and lattice gas models.
  • Thermodynamic integration
• Exploring emergent dynamic properties
  • Molecular Dynamics Simulations
    • Integrating Newton’s equations of motion for continuous potentials: The secret behind the Verlet algorithm: Symplectic integrators
    • Data structures for running ensembles of statistically independent simulations and sweeping parameter space
  • Lattice-Boltzmann Simulations
    • The emergence of Navier-Stokes-like fluid flow