Recent 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.