How can we ever make sense of what we observe? As a practical matter, most complex systems---that is, many-bodied systems with strongly interacting degrees of freedom---can only be observed through a time-series of relatively few functionals of their microstate. Nevertheless, because of the strong coupling, the many instantaneously-hidden degrees of freedom inject themselves over time into the observable time-series---giving us hope for inference.
This dissertation delivers new broadly-applicable results regarding the generation, prediction, and physical implication of such stochastic time-series with hidden structure. After the development of the general mathematical theory, the remainder of the dissertation can be subdivided into three parts. The first part addresses the fundamental limits of generation and predictability of structured stochastic processes. The second part identifies the possible correlation in and diffraction patterns of chaotic crystals. The third part establishes new relationships that constrain and elucidate the fluctuations and thermodynamics of nonequilibrium systems.
One of the predominant themes in this dissertation is the use of rather flexible mathematical structures called `hidden Markov models'. Indeed, much of this dissertation grew out of the recognition that---beyond their ability to simulate many sophisticated nonlinear and non-Markovian processes of interest---hidden Markov models enable an exact linear algebraic analysis of processes they represent. However, to proceed required the development of a generalized spectral theory for arbitrary functions of potentially nondiagonalizable operators, which is developed and utilized herein. Despite its long history, it appears (somewhat surprisingly) that not all of linear algebra had been worked out to the extent necessary to address the physics of complex systems. This extension of the more familiar spectral theory is of interest in its own right, and has created several new and rather independent directions of inquiry.
|Advisor:||Crutchfield, James P.|
|Commitee:||Cox, Daniel L., Rundle, John B.|
|School:||University of California, Davis|
|School Location:||United States -- California|
|Source:||DAI-B 78/07(E), Dissertation Abstracts International|
|Subjects:||Physics, Theoretical physics, Computer science|
|Keywords:||Complexity, Functional calculus, Nonequilibrium thermodynamics, Prediction, Spectral theory, Stochastic chaos|
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