Brownian Motion in a Non - Equilibrium Bath by Joan - Emma Shea

We start by considering the Brownian motion of a large spherical particle of mass M immersed in a non-equilibrium bath of N light spherical particles of mass m. A FokkerPlanck equation and a generalized Langevin equation for an arbitrary function of the position and momentum of the Brownian particle are derived from first principles of statistical mechanics using time dependent projection operators. The Fokker-Planck equation is obtained by eliminating the fast bath variables of the system, while the Langevin Equation is obtained using a projection operator which averages over these variables. The two methods yield equivalent results, valid to second order in the 1 small parameters E = (•) , A and A*, where A is a measure of the magnitude of the macroscopic gradients of the system and A* reflects the difference in mean bath and Brownian velocities. The treatment is generalized to an elastic system of several Brownian particles immersed in a non-equilibrium bath of light particles and the Fokker-Planck equation of the Brownian system is derived using techniques similar to those used for the one particle case. The Fokker-Planck equation contains the usual equilibrium streaming and dissipative terms as well as terms reflecting spatial variations in the bath pressure, temperature and velocity. We make use of the effective Liouvillian obtained from the Fokker-Planck equation and of time dependent projection operators involving properties of local equilibrium distribution functions to derive the exact non-linear hydrodynamic equations of the Brownian particles. The exact equations are simplified using the fact that the thermodynamic forces vary slowly on a molecular timescale. The resulting local transport equations are expressed in terms of homogeneous local equilibrium averages. The number density hydrodynamic equation remains unchanged from the case of a system of isolated Brownian particles, but the momentum and energy density expressions are no longer conserved. They contain additional terms accounting for the non-equilibrium nature of the bath and for the irreversible processes occurring in the system. Thesis Supervisor: Irwin Oppenheim Title: Professor Acknowledgments I would like to start by thanking my advisor Irwin Oppenheim. His careful and rigorous approach to science has greatly influenced my own outlook on science. He has encouraged me to think independently and his scholarly advice, given with patience and good humor, has been invaluable. I would like to thank the Chemistry Department for giving me the unique opportunity of "TAing" 5.60 for nine semesters. I would like to thank Professors Garland, Oppenheim, Silbey, Bawendi and Phillips, my fellow TAs and my students, too numerous to name, for making it such an enjoyable experience. I would like to thank the Canadian funding agencies NSERC and FCAR for financial support. I would like to thank the members of my thesis committee, Professors Brenner and Silbey, for taking the time to read a thesis with no pictures and more equations than words. I would also like to thank my zoo buddies, Cliff, Dave, Frank, Juan, Sophia and Welles, who make the zoo the unique place that it is. A special thanks to Welles, my zoo-mate and roommate, for many helpful discussions and for doing my laundry while I was writing my thesis. I would like to thank my favorite non-zoo friends Nitu, Nadine, Bibi, A.B. Wong and Goldorak. Finally, I would like to thank my family, Daddy, Mummy, Herby, Louisa (Gurguth Salt), Cecilia, Mikey, Oma, and Crunchy, whose unfaltering love and support have helped me through my graduate years and will help me in the years to come.