Halbert, James ThomasRecent advances in science have led to a better understanding of physical phenomena across a vast range of time and length scales. This has given the research community access to mathematical models for most scales in a given problem. A common strategy applied to Hamiltonian systems has been to select scales of interest and remove the others through the Zwanzig-Mori formalism. As long as the scales involved are strongly separated this approach works well. However, many problems in science and engineering involve processes in which there is no clear scale separation. It is still possible to use this procedure in some such cases but it has notably failed in many others (e.g. complex fluids). This failure has been blamed on the presence of poorly understood empirical closures and much current work is dedicated to eliminating the need for these or at least quantifying the errors they introduce. I have constructed a model system that possesses many of the features present in relevant problems and have used it as a testbed for investigating a modification of the Zwanzig-Mori formalism. The modified formalism I propose is applicable beyond the standard class of Hamiltonian systems: it is designed to work with damped, noise-driven, Hamiltonian systems. This thesis describes the modest first steps in understanding the underlying functional analytic structure of the new formalism. In particular, I have placed the model into a hierarchy of systems related to one another by a map between scales. The scale connection between the hierarchy elements is made evident by the construction of an intrinsic entropy-based fluid moment system — each element of the hierarchy is realized as a formal coarsening of this fluid moment system. What is more, I have formally constructed the "infinite particle'' limit for the fluid moment system and found that it too has an associated entropy. The existence of these entropies implies an amenability of the new formalism to analysis — this is the most useful and novel aspect of the work.en-USDissertationMathematicsDimensional ReductionPartial Differential EquationsRenormalizationStatistical PhysicsStochastic ProcessesZwanzig-Mori Formalism