Statistical Physics of Synchronization

Synchronizing systems.-  Introduction.-  The oscillators and their interaction: A qualitative discussion.-  Oscillators as limit cycles.- Interacting limit-cycle oscillators.- Synchronizing systems as statistical mechanical systems.- The features of a statistical physical description.- Some results for noiseless interacting oscillators.- The oscillators with inertia.- Appendix 1: A two-dimensional dynamics with a limit-cycle attractor.- Appendix 2: The Lyapunov exponents.- Appendix 3: The one-body distribution function in an N-body system.- Oscillators with first-order dynamics.- The oscillators with distributed natural frequencies.- The Kuramoto model.- Unimodal symmetric g(w).- Nonunimodal g(w).- Appendix 1: An H-theorem for a particular simple case.- Appendix 2: Form of the function r(K) for symmetric and unimodal frequency distributions in the Kuramoto model.- Appendix 3: The numerical solution of Eq. (2.34).- Oscillators with second-order dynamics.-  Generalized Kuramoto model with inertia and noise.-  Nonequilibrium first-order synchronization phase transition: Simulation results.- Analysis in the continuum limit: The Kramers equation.- Phase diagram: Comparison with numeric.- Appendix 1: The noiseless Kuramoto model with inertia: Connection with electrical power distribution models.- Appendix 2: Proof that the dynamics (3.9) does not satisfy detailed balance.- Appendix 3: Simulation details for the dynamics (3.9).- Appendix 4: Derivation of the Kramers equation.- Appendix 5: Nature of solutions of Eq. (3.32).- Appendix 6: Solution of the system of equations (3.39).- Appendix 7: Convergence properties of the expansion (3.38).