Statistical Physics of Synchronization / by Shamik Gupta, Alessandro Campa, Stefano Ruffo.

Format:

Book

Author/Creator:

Gupta, Shamik., Author.

Campa, Alessandro., Author.

Ruffo, Stefano., Author.

Series:

SpringerBriefs in Complexity, 2191-5334

Language:

English

Subjects (All):

Mathematical physics.

Dynamics.

Nonlinear theories.

Theoretical, Mathematical and Computational Physics.

Applied Dynamical Systems.

Mathematical Physics.

Local Subjects:

Theoretical, Mathematical and Computational Physics.

Applied Dynamical Systems.

Mathematical Physics.

Physical Description:

1 online resource (135 pages)

Edition:

1st ed. 2018.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2018.

Summary:

This book introduces and discusses the analysis of interacting many-body complex systems exhibiting spontaneous synchronization from the perspective of nonequilibrium statistical physics. While such systems have been mostly studied using dynamical system theory, the book underlines the usefulness of the statistical physics approach to obtain insightful results in a number of representative dynamical settings. Although it is intractable to follow the dynamics of a particular initial condition, statistical physics allows to derive exact analytical results in the limit of an infinite number of interacting units. Chapter one discusses dynamical characterization of individual units of synchronizing systems as well as of their interaction and summarizes the relevant tools of statistical physics. The latter are then used in chapters two and three to discuss respectively synchronizing systems with either a first- or a second-order evolution in time. This book provides a timely introduction to the subject and is meant for the uninitiated as well as for experienced researchers working in areas of nonlinear dynamics and chaos, statistical physics, and complex systems.

Contents:

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).

ISBN:

3-319-96664-2