It is proposed to continue the work of the Principal Investigator and collaborators on the connections between non-equilibrium statistical mechanics and dynamical systems theory. The central idea of this work is to relate transport and other nonequilibrium properties, such as entropy production, of fluid systems to the underlying chaotic dynamics of the constituent particles of the fluid. Work by the PI and collaborators over the past few years has shown that Boltzmann equation techniques and related methods of nonequilibrium statistical mechanics can be fruitfully applied to calculate not only transport properties of fluids but their chaotic properties as well. This unified approach allows one to make the connections between transport and chaotic properties especially clear. Work on model systems has led to the development of advanced methods which allow detailed analytic calculations of the chaotic properties of more realistic systems where all of the particles can move. It has been possible, for example, to compute the Kolmogorov-Sinai entropy of a dilute gas of hard disks or of hard spheres. These systems are still quite complicated mathematically, and many interesting features of their chaotic behavior are related to the semi-dispersive nature of the particle collisions. It is proposed here: (a) to continue the work on analytic methods to compute the chaotic properties, of dense Lorentz gas systems; (b) to carry out similar calculations for more realistic systems, namely, for dilute and moderately dense gases of particles interacting with short ranged forces; (c) to continue the work of the PI and collaborators on the connection between the irreversible entropy production of irreversible thermodynamics and seemingly closely related quantities that appear naturally in the dynamical systems theory approach to transport; and (d) to extend these methods and results to the quantum case. The latter point is of major importance since: (a) nature is fundamentally quantum mechanical rather than classical, (b) many deep and basic questions need to be answered before one has any clear understanding of how classical chaos emerges from quantum systems, and (c) quantum systems such as quantum dots and quantum anti-dots are of great physical and practical interest.
