Kinetic Theory of Coupled Oscillators

I have recently published two papers applying ideas from the kinetic theory of plasmas and nonequilibrium statistical mechanics to coupled oscillators. The first is Hildebrand, Buice and Chow, PRL 98:054101 and the second is Buice and Chow, PRE 76:031118. The main concern of both papers is understanding the dynamics of large but not infinite networks of oscillators. Generally, coupled oscillators are studied either in the small network limit where explicit calculations can be performed or in the infinite size “mean field” limit where fluctuations can be ignored. However, many networks are in between, i.e. large enough to be complicated but not so large that the effects of individual oscillators are not felt. This is the regime we were interested in and where the ideas of kinetic theory are useful.

In a nutshell, kinetic theory strives to explain macroscopic phenomenon of a many body system in terms of (the moments of the distribution function governing) the microscopic dynamics of the constituent particles (oscillators). In the coupled oscillator case, we actually have a macroscopic theory and what we want to understand is how the microscopic dynamics gave rise to that theory. For example, in the Kuramoto model of coupled oscillators, there is a phase transition from asynchrony to synchrony if the coupling strength is sufficiently strong. In the infinite oscillator limit where fluctuations can be ignored, an order parameter measuring the synchrony in the network can be shown to bifurcate from zero at a critical coupling strength. However, for a finite number of oscillators, the order parameter fluctuates and there is no longer a sharp transition from asynchrony to synchrony but rather a crossover. We show in the first paper that a moment expansion analogous to the BBGKY hierarchy can be derived for the coupled oscillator system and using a Leonard-Balescu-like approximation the variance of the order parameter can be computed explicitly.

The second paper shows that the moment hierarchy can be equivalently expressed in terms of a generating functional of the oscillator density (i.e. a density of the density if you like). Once expressed in this form, diagrammatic methods of field theory can be used to do perturbative expansions. In particular, we perform a one-loop expansion to show marginal modes in the mean field theory are stabilized by finite-size fluctuations. This problem of marginal modes had been a puzzle in the field for a number of years.