The purpose of these lectures is to review some of the recent work devoted to understanding the microscopic foundations of irreversible behavior in fluid systems. We begin by considering the properties of systems whose microscopic dynamics is chaotic. Our goal is to show that, for simple model systems, one can understand the approach of a sufficiently smooth initial phase space distribution to an equilibrium state, or more properly, to a local equilibrium state, without the need to introduce stochastic elements into the description of the system's dynamics. To follow this argument one needs to understand some of the basic ideas of dynamical systems theory as applied to chaotic systems. We first consider the notions of ergodicity and mixing, and discuss the extent to which these ideas really might be applied to systems of large numbers of particles. Then we broaden the discussion to describe the behavior of hyperbolic dynamical systems with exponential separation of infinitesimally close phase space trajectories. These systems are characterized by stable and unstable manifolds, and nonzero Lyapunov exponents. We will briefly touch upon such topics as SRB measures, entropy production and the relations between transport coefficients and properties of the underlying microscopic chaotic behavior of the phase space trajectories of the system. We illustrate these ideas as well as their application to transport theory with several simple models, among them the baker and multi-baker maps. In the second part of these lectures we consider the fact that microscopic chaos is neither necessary nor sufficient for good transport properties. This will lead to a brief discussion of classical systems that are {ITALIC pseudochaotic}. These are systems with zero Lyapunov exponents, but with some microscopic properties that are similar to those of chaotic systems, including the separation, in time, of nearby phase space trajectories. However for pseudochaotic systems the separation is proportional to some power of time rather than exponential. The third and final part of these lectures is devoted to a consideration of transport in a quantum version of the simple model discussed in the first part, the multi-baker map. The quantum version shows quite different behavior and we conclude the lectures with a brief description of the transition from quantum to classical behavior in the semi-classical limit.

PACS numbers: 05.45.Df, 05.60.--k, 05.70.Ln