This grant is supported jointly by the Divisions of Materials Research, Physics and Mathematical Sciences. The aim of this research is a better understanding of the properties of macroscopic systems originating in the collective behavior of their microscopic constituents. The emphasis is on non-equilibrium systems and the methods used range from rigorous mathematical analysis to computer simulations. We are particularly interested in bridging the gap between rigorous results and applications. Intellectual Merit includes: 1) An extension of Boltzmann's entropy and H-theorem to systems not in local thermal equilibrium. It is proposed to generalize recent work for dense fluids to more complex systems, including nano and biological ones, where entropic considerations play an important role. This will require an appropriate generalization of Boltzmann's entropy to non-equilibrium quantum systems. 2) Recently-obtained exact large deviation functions, describing fluctuations in stationary non-equilibrium states of model systems are of an unexpected form. Their most striking feature, non-locality on the macroscopic scale, can be traced to the long-range correlations, measurable experimentally by neutron scattering, which exist in such systems. Extension and application of these results to more realistic systems is planned. 3) For open quantum systems, such as current-carrying nano-wires, the classical modeling of the reservoirs by stochastic interactions is problematic. The usual approach is to use free fields, or ideal gases, as reservoirs. This is unsatisfactory in many cases and we plan to continue our investigation of alternative approaches such as the use of strongly coupled systems, represented by random matrices, as reservoirs. 4) Techniques we developed for dealing with the response of quantum systems to strong time dependent external fields have yielded rigorous results about simple model systems. Applications to the optimal control of quantum transitions in molecules and to external fields in solids are planned. 5) We have obtained novel results about the structure of systems with reduced particle number fluctuations. These have applications to cosmology and to queuing theory, which will be explored further. 6) An important question in developing approximation schemes for fluids is the possibility of constructing particle distributions having specified densities and pair correlations. This can be achieved in some cases by the explicit construction of point processes via determinants and renewals. General existence criteria are being investigated. 7) We have applied statistical mechanical methods to the mathematical study of epidemics taking into account correlations as well as saturation effects on networks. New macroscopic equations for the description of the evolution and prevalence of an endemic infected state improve agreement with more detailed microscopic models. Extension of these techniques to models of population dynamics and ecology is planned. Broader Impact: The proposed research is highly interdisciplinary, bringing together physicists, mathematicians, chemists and those working in theoretical areas of the biological and social sciences. The expected applications are in material science, complex fluids and biological and social systems. Our program also includes the organization of two conferences every year in which both core subjects and new developments in statistical mechanics are discussed in a collegial atmosphere. Graduate students, postdocs and minority scientists are encouraged to present talks on their work and interact with leaders in the field. The conferences also serve as a clearing house for positions and often lead to new collaborations.
