Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

Journal Article · Sun Nov 01 04:00:00 EST 1987 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01009349· OSTI ID:5300486

Kuramoto, Y; Nishikawa, I

A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.

• Research Organization: Kyoto Univ. (Japan)
• OSTI ID: 5300486
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 49:3/4; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

656002 -- Condensed Matter Physics-- General Techniques in Condensed Matter-- (1987-)
657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
COUPLING CONSTANTS
CRYSTAL MODELS
FLUCTUATIONS
HAMILTONIANS
INVARIANCE PRINCIPLES
LATTICE VIBRATIONS
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MEAN-FIELD THEORY
MECHANICS
ORDER PARAMETERS
ORDER-DISORDER TRANSFORMATIONS
OSCILLATIONS
PHASE TRANSFORMATIONS
QUANTUM MECHANICS
QUANTUM OPERATORS
STATISTICAL MECHANICS
STOCHASTIC PROCESSES
THERMODYNAMICS
VARIATIONS