A Modified Zwanzig-Mori Formalism
A Modified Zwanzig-Mori Formalism
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Date
2009
Authors
Halbert, James Thomas
Advisor
Levermore, Charles D.
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Abstract
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.