Kinetic theory of coupled oscillators

Last week, I gave a physics colloquium at the Catholic University of America about recent work on using kinetic theory and field theory approaches to analyze finite-size corrections to networks of coupled oscillators.  My slides are here although they are converted from Keynote so the movies don’t work.   Coupled oscillators arise in  contexts as diverse as the brain, synchronized flashing of fireflies, coupled Josephson junctions, or unstable modes of the Millennium bridge in London.  Steve Strogatz’s book Sync gives a popular account of the field.  My talk considers the Kuramoto model

\dot{\theta}_i = \omega_i+\frac{K}{N}\sum_j\sin(\theta_j-\theta_i)   (1)

where the frequencies \omega are drawn from a fixed distribution g(\omega). The model describes the dynamics of the phases \theta of an all-to-all connected network of oscillators.  It can be considered to be the weak coupling limit of a set of nonlinear oscillators with different natural frequencies and a synchronizing phase response curve.

Kuramoto in the 1970’s showed in the mean field limit N\rightarrow \infty, that when the coupling strength K is below a critical value K_c, the osciilators will be in an incoherent state but for K>K_c a group of oscillators will phase lock. This transition could be observed with the order parameter

Z=\frac{1}{N}\sum_j e^{i\theta_j}= r e^{i\psi}

The bifurcation is reminiscent of a magnetization phase transition in equilibrium statistical mechanics.  In 1991, Steve Strogatz and Rennie Mirollo showed that the incoherent state loses stability at the critical point as expected but it is marginally stable below criticality.  This was surprising because simulations of the Kuramoto model showed that the magnitude of the order parameter decayed to zero below the critical point quite quickly.  This was partially resolved a year later by Strogatz, Mirollo and Matthews, who showed that the order parameter decays due to dephasing, an effect similar to Landau damping in plasma physics.

However, open questions about how the system behaved for a large but finite number of oscillators remained.  Did finite size corrections stabilize the marginal incoherent state and not just the order parameter?  Also for finite size, the phase transition is no longer sharp but smoothed over by fluctuations in the order parameter.  Hence, a calculation of the variance of the order parameter was necessary.  How could this be calculated? In two papers with Eric Hildebrand and Michael Buice (Hildebrand et al. PRL 98 , 054101 (2007) and Buice and Chow, PRE 76 031118 (2007), both available here) we showed that an application of methods from the kinetic theory of plasmas and field theory can answer these questions.

Kinetic theory was developed by Boltzmann and Maxwell to show how microscopic Hamiltonian dynamics of particles could account for the thermodynamic properties of gases.  We adapted a formalism previously applied to plasma physics.  The idea starts by encoding the microscopic dynamics of the oscillators into a single probability density function \eta(\theta,\omega)=\sum_{ij}\delta(\theta-\theta_i)\delta(\omega-\omega_j).  The density will then obey a continuity equation in oscillator phase space of the form

\frac{\partial \eta}{\partial t}+\frac{\partial}{\partial\theta}[v(\theta,\omega)\eta]=0

where the velocity v=\dot\theta_i is given by the oscillator equation (1).  The continuity equation is called the Klimontovich equation in plasma physics and is only defined in the weak sense of distributions.  It is entirely equivalent to the original oscillator system and is only made useful for calculations by taking averages.  However, the Klimontovich equation is nonlinear in \eta so when you take the average you’ll find that the equation for the first moment \langle \eta(\theta,\omega,t) \rangle will depend on the second moment \langle \eta(\theta,\omega,t)\eta(\theta',\omega',t')\rangle.  The second moment will in turn depend on the third and so on.  This is known as the BBGKY hierarchy.

The hierarchy is made tractable by truncating at some order with the idea that the higher  order moments (actually cumulants) will matter less and less as the number of oscillators increases.  The reason is that the cumulants represent direct oscillator interactions and the effect of a single oscillator goes to zero as the number of oscillators goes to infinity.  If two oscillator correlations are ignored then we end up with mean field theory, where there are no correlations or fluctuations.  We showed in Hildebrand et al. that the correlations could be computed by going to the next level in the hierarchy.  One thing that is important to note is that the mean field theory has the exact same form as the Klimontovich equation.  The only difference being that the mean field equation must have smooth solutions.  Almost everyone goes directly to mean field theory when analyzing coupled oscillators using this formalism and this distinction is glossed over.

Now, calculations using the moment hierarchy involve solving coupled integro-differential equations, which is still pretty unwieldy.  This can be simplified by noting that to compute the cumulants of \eta we can write down a generating functional.  In this case it will be an infinite dimensional path integral.  However, it can then be analyzed using the methods of field theory.  The generating functional, like any other finite dimensional integral, can be analyzed perturbatively using the methods of steepest descents and field theory is basically asymptotic analysis in infinite dimensions.  In our case, 1/N is the small parameter that allows us to use steepest descents, called a loop expansion in field theory.  We can keep track of the terms in the expansion (which multiply quite rapidly) by using Feynman diagrams.  The number of loops in the diagrams indicate the order in 1/N of the expansion.  In Buice and Chow, we show that the field theory formalism and the moment hierarchy are equivalent and a natural truncation or closure of the moment hierarchy is given by the loop expansion.  We also show that the incoherent state is strictly stable for finite N.

There is one sad story that is related to this project.  Eric was actually a grad student of J.D. Crawford at the University of Pittsburgh where I was on the math faculty from 1998 to 2004.  I had known about J.D.’s work in dynamical systems for several years prior to arriving at Pitt and was looking forward to finally meeting him.  However, he contracted cancer right around that time so I would miss him when I visited Pitt during my interview phase.  He then died the day I arrived to start my job so I never got to meet him.  He was only 44 and in his prime.  Eric eventually finished his PhD with me working on this project.

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