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Computational methods and numerical simulations offer a basic approach to one of the fundamental problems in statistical physics aiming at establishing the connection between the microscopic processes (at the molecular or atomic level) and the macroscopic description (given by the phenomenological equations) for equilibrium as well as transport properties, and even for reactive processes. The molecular dynamics approach is based on the microscopic equations of motion using "realistic" interaction potentials, and has become a classical tool of computational physics. In recent years, a different approach has been developed starting from a simple micro-world constructed as an automaton universe based, not an a realistic description of interacting particles, but merely on the basic laws of symmetry and invariance. This lattice dynamics approach and molecular dynamics can be considered as complementary methods, and sometimes they can be blended in hybrid methods in particular for some problems in the physics of fluids where stochastic processes play an important role.

Lattice Dynamics

Consider a Bravais lattice of vertices and bonds connecting the vertices. Now put point particles on the vertices, with velocities in the directions of the bonds, such that there can be at most one particle with a given velocity at each vertex. Next suppose that at discrete times, each particle moves from one vertex to the next along the direction of its velocity, and when it arrives at the next vertex, its velocity is adjusted according to a set of deterministic or stochastic "collision rules", depending on the number and velocities of the other particles arriving at the same vertex at the same time. After the velocities of all particles are adjusted, the process is repeated. This set of rules, when specified in detail, can be implemented on several kinds of computers using Boolean arithmetic, and the dynamical properties of this system of particles can be described by general methods of statistical physics, including methods familiar from the kinetic theory of gases. The systems so described are called Lattice Gas Automata (LGA).
(J.R. Dorfman, J.Stat.Phys., April 2002)

• A short introduction to automata
• Some examples of simulations.

Molecular Dynamics

Molecular Dynamics has become a standard tool for the computation of equilibrium properties of matter based on the modelling of basic microscopic processes. We have been working on extending the method to non-equilibrium states. The phenomena studied include shock wave propagation in fluids and solids, emergence of hydrodynamic flows and instabilities at the microscopic level, and, more recently, fluidized granular systems. Equilibrium systems are also being investigated in order to establish, at a fundamental level, the connection between chaos and transport, using Molecular Dynamics simulations and a kinetic model, explaining the existence of collective excitations in the so-called Lyapunov spectra, an indicator of chaotic motion in phase space. In the area of applied computational physics, we have recently developed a density functional theory based method (of the classical Car-Parrinello type) in order to analyze the properties of polar solvents: the method has been applied to the computation of solvation free energies of large biological molecules.

Stochastic Processes

A stochastic process is one involving an element of unpredictability. Whereas deterministic systems evolve necessarily in a unique way which is entirely predictable once a sufficient number of initial conditions are known, stochastic systems evolve in a way that can only be described in probabilistic terms. The sources of the randomness can be manifold, e.g. the number of particles in the system is so large that it precludes a perfect knowledge of the initial conditions, the system may continually be subjected to influence of the "outside" world which cannot be completely controlled,... We are particularly interested in the influence of stochasticity on the behaviour of physical and chemical systems in the vicinity of instabilities and bifurcation points (noise-induced phase transitions, stochastic limit cycles,...), as well as the influence of dimensionality ( low and/or fractal ) on non-linear chemical reaction kinetics. More abstract topics such as random recurrences and random-walk related problems are also being investigated.