Viscosity and Microscopic Chaos : The Helfand-moment Approach
  In this thesis, we first devote a section on the history of the concept of irreversibility; of the hydrodynamics, branch of physics in which the viscosity appears; of the kinetic theory of gases establishing relationships between the microscopic dynamics and macroscopic processes like viscosity; and, finally, the interest brought in statistical mechanics of irreversible processes by the theory of chaos, more precisely, the microscopic chaos. We propose a method based on the Helfand moment in order to calculate the viscosity properties in systems of particles with periodic boundary conditions. We apply this method to the simplest system in which viscosity already exists: the two-hard-disk model. The escape-rate formalism, establishing a direct relation between chaotic quantities of the microscopic dynamics (e.g. Lyapunov exponents, fractal dimensions, etc.), is applied in this system. The results are in excellent agreement with those obtained by our Helfand-moment method. We extend the calculation of the viscosity properties to systems with more than two hard balls. Finally, we compute viscosity as well as thermal conductivity thanks to our own method also based on the Helfand moment. 
Sebastien Viscardy