Aspects of the Renormalization Group
The main part of this thesis is on the renormalization group (RG). We will explore the results of the RG in two ways. In the first part we use information geometry, in which the local distance between models measures their distinguishability from data, to quantify the flow of information under the renormalization group. We show that information about relevant parameters is preserved, with distances along relevant directions maintained under flow. By contrast, irrelevant parameters become less distinguishable under the flow, with distances along irrelevant directions contracting according to renormalization group exponents. We develop a covariant formalism to understand the contraction of the model manifold. We then apply our tools to understand the emergence of the diffusion equation and more general statistical systems described by a free energy. Our results give an information-theoretic justification of universality in terms of the flow of the model manifold under coarse graining. In the second part, we use dynamical systems theory to systematize the results of the RG. The results of the RG are commonly advertised as the existence of power law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis. We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables. We recover and explain the existing literature, predict the nonlinear generalization for universal homogeneous functions, and show that the procedure leads to a better handling of the singularity with several examples including the Random Field Ising model and the 4-d Ising model. The RG is useful not just for systems in physics but has found application in a surprising variety of fields. In dynamical systems, it provided an nice explanation of the universality observed in the period doubling transition. We show the equivalent of so-called redundant variables in period doubling and offer a new interpretation for them. We then examine the consequences for the Ising model. Finally, the last part of this thesis is on a very different topic. Here, we use an effective Hamiltonian to characterize particle dynamics and find escape rates in a periodically kicked Hamiltonian. We study a model of particles in storage rings that is described by a chaotic symplectic map. Ignoring the resonances, the dynamics typically has a finite region in phase space where it is stable. Inherent noise in the system leads to particle loss from this stable region. The competition of this noise with radiation damping, which increases stability, determines the escape rate. Determining this `aperture' and finding escape rates is therefore an important physical problem. We compare the results of two different perturbation theories and a variational method to estimate this stable region. Including noise, we derive analytical estimates for the steady-state populations (and the resulting beam emittance), for the escape rate in the small damping regime, and compare them with numerical simulations.
Sethna, James Patarasp
McEuen, Paul L.; Myers, Christopher R.
Ph. D., Physics
Doctor of Philosophy
dissertation or thesis