Viscosity from Newton to Nonequilibrium Statistical Mechanics∗

This paper is devoted to the transport process of momentum, the viscosity. We outline the historical development of hydrodynamics, the branch of physics in which the shear and bulk viscosities are expressed. We show the transition between the mechanical philosophy claiming that the conservation of energy is coupled to cyclic, reversible dynamics, and the birth of a new science, thermodynamics, in which the conservation of energy (the first law) is associated with the irreversibility (the second law). Through the 19th century, the kinetic theory of gases was developed and accumulated successes establishing relationships between the microscopic dynamics of the atoms and molecules, and macroscopic properties, among which the viscosity. In this context appeared the paradox of irreversibility to which Boltzmann was the first to provide an explanation based on a statistical interpretation of the entropy. We finally outline the history of the theory of chaos and show how the chaotic character of the microscopic dynamics provides a dynamical explanation of the irreversible processes such as transport, and in particular, the viscosity. In this context, we present three successfull approaches developed the last two decades: the thermostated-system method, the escape-rate formalism and the hydrodynamic-mode method. This paper consists in a revision of the introduction of the PhD thesis presented on the 21th of September, 2005, entitled : Viscosity and Microscopic Chaos : The Helfand-moment Approach (Viscardy, 2005). e-mail: [email protected]