Correspondence between phase oscillator network and classical XY model with the same infinite-range interaction in statics

TL;DR

This paper establishes a theoretical correspondence between phase oscillator networks and classical XY models with infinite-range interactions, linking their self-consistent equations and analyzing specific interactions like uniform and Mexican-hat types.

Contribution

It demonstrates a one-to-one correspondence between solutions of self-consistent equations and saddle point equations, revealing new insights into the relation between oscillator networks and XY models.

Findings

Kernel correspondence between oscillator networks and XY models.

Proven conditions for frequency distribution and interaction types.

Discovery of a limit cycle oscillation in oscillator networks with associative memory interactions.

Abstract

We study the phase oscillator networks with distributed natural frequencies and classical XY models both of which have a class of infinite-range interactions in common. We find that the integral kernel of the self-consistent equations (SCEs) for oscillator networks correspond to that of the saddle point equations (SPEs) for XY models, and that the quenched randomness (distributed natural frequencies) corresponds to thermal noise. We find a sufficient condition that the probability density of natural frequency distributions is one-humped in order that the kernel in the oscillator network is strictly decreasing as that in the XY model. Furthermore, taking the uniform and Mexican-hat type interactions, we prove the one to one correspondence between the solutions of the SCEs and SPEs. As an application of the correspondence, we study the associative memory type interaction. In the XY model with this interaction, there exists a peculiar one-parameter family of solutions. For the oscillator network, we find a non-trivial solution, i.e., a limit cycle oscillation.