Statistical Mechanics of Coarse-Graining
Open Access
- Author:
- Foley, Thomas Thomassen
- Graduate Program:
- Physics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 02, 2017
- Committee Members:
- William George Noid, Dissertation Advisor/Co-Advisor
Jorge Osvaldo Sofo, Committee Chair/Co-Chair
Reka Z Albert, Committee Member
Lu Bai, Committee Member
Lu Bai, Outside Member - Keywords:
- Coarse-graining
Modeling
Renormalization
Statistical Mechanics
Gaussian Network Model
Complex Networks
Community Detection
Model Selection
Phase Transitions - Abstract:
- The systems scientists study are becoming increasingly complex while the data collected grows in both magnitude and dimension. This relentless advancement necessitates the utilization of models, especially simplifying models, to make efficient use of the data. Of course, models have long served as essential tools in science, but the utility of simplifying, or coarse-grained, models is greater than ever before (even outside the sciences) owing to this explosion in data. However, their tremendous potential is hindered by outstanding challenges concerning their optimal construction and appropriate use. Among other challenges, it is generally unclear what effect the simplifications used – which gives the coarse-grained model many of its desirable features – have upon the model's ability to describe the phenomena of interest. As one specific example, coarse-grained modeling typically relies upon physical intuition to inform the choice of representation for the system under consideration, which has become untenable as researchers model more complex systems. In this thesis, I present progress on the fundamentals of modeling, in the equilibrium statistical mechanics setting, by focusing on the basic questions that arise when constructing coarse-grained models. Particular attention is given to the role of the representation (or mapping) in determining the behavior of the coarsened system, which may be the least well understood step in model construction. This work carefully considers the statistical mechanics of the coarse-graining procedure as applied to the simple Gaussian network model (GNM), a linear mass-and-spring system that can also be viewed as a complex network, for which an exact renormalization is derived. I present an exact thermodynamic decomposition of the potential of mean force (PMF), which is the the central object in coarse-grained models. The consequences of this decomposition into entropic and energetic terms (a free energy) are illustrated by considering the variation with resolution of these terms for the coarsening of the GNM. This work provides a clarifying and concrete exposition of some long-held notions among model builders, in addition to suggesting some candidate principles for optimal model construction. Based upon this foundational work, I develop a statistical mechanics framework for studying the relationship between model representation and quality. Utilizing Monte Carlo simulations and free energy reweighting, I present the picture of a `model-scape', in analogy to free energy landscape, which is composed of: representations, which serve as `coordinates'; models, which serve as `configurations'; and weights, which are induced by varying combinations of energy functions and artificial temperatures. Employing the full statistical mechanics machinery and drawing heavily on analogy to physical systems, I explore and characterize this space, thus revealing the properties of this space. Two of our findings concerning the role of representation in model construction are a bona-fide phase transition on the space of models and a surprising anti-correlation between information preservation and spectral fitness. Exploiting the GNM's dual role as a complex network, we consider a compelling analogy between community detection and representation optimization. A variety of other interesting behavior is uncovered through our study of this model-scape, including possible glassiness, emergent resolution-dependent features and scaling, and intriguing correlations between seemingly distinct features of CG models.