Universal Critical Behavior of Noisy Coupled Oscillators
T. Risler, J. Prost, and F. Julicher. Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
We show that the synchronization transition of a large number of
noisy coupled oscillators is an example for a dynamic critical point
far from thermodynamic equilibrium. The universal behaviors of such
critical oscillators, arranged on a lattice in a $d$-dimensional
space and coupled by nearest neighbors interactions, can be studied
using field theoretical methods. The field theory associated with
the critical point of a homogeneous oscillatory instability (or Hopf
bifurcation of coupled oscillators) is the complex Ginzburg-Landau
equation with additive noise.
We perform a perturbative renormalization group (RG) study in a $4-\epsilon$ dimensional space. We develop an RG scheme that eliminates the phase and
frequency of the oscillations using a scale-dependent oscillating
reference frame. Within a Callan-Symanzik RG scheme to two-loop
order in perturbation theory, we find that the RG fixed point is
formally related to the one of the model A dynamics of the real
Ginzburg-Landau theory with an $O(2)$ symmetry of the order
parameter. Therefore, the dominant critical exponents for coupled
oscillators are the same as for this equilibrium field theory.
This formal connection with an equilibrium critical point imposes a
relation between the correlation and response functions of coupled
oscillators in the critical regime. Since the system operates far
from thermodynamic equilibrium, a strong violation of the
fluctuation-dissipation relation occurs and is characterized by a
universal divergence of an effective temperature. The formal
relation between critical oscillators and equilibrium critical
points suggests that long-range phase order exists in critical
oscillators above two dimensions.
%0 Book Section
%1 statphys23_1093
%A Risler, T.
%A Prost, J.
%A Julicher, F.
%B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics
%C Genova, Italy
%D 2007
%E Pietronero, Luciano
%E Loreto, Vittorio
%E Zapperi, Stefano
%K bifurcation complex coupled critical dynamic equation equilibrium ginzburg-landau group hopf noisy oscillators out point renormalization statphys23 topic-3
%T Universal Critical Behavior of Noisy Coupled Oscillators
%U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1093
%X We show that the synchronization transition of a large number of
noisy coupled oscillators is an example for a dynamic critical point
far from thermodynamic equilibrium. The universal behaviors of such
critical oscillators, arranged on a lattice in a $d$-dimensional
space and coupled by nearest neighbors interactions, can be studied
using field theoretical methods. The field theory associated with
the critical point of a homogeneous oscillatory instability (or Hopf
bifurcation of coupled oscillators) is the complex Ginzburg-Landau
equation with additive noise.
We perform a perturbative renormalization group (RG) study in a $4-\epsilon$ dimensional space. We develop an RG scheme that eliminates the phase and
frequency of the oscillations using a scale-dependent oscillating
reference frame. Within a Callan-Symanzik RG scheme to two-loop
order in perturbation theory, we find that the RG fixed point is
formally related to the one of the model A dynamics of the real
Ginzburg-Landau theory with an $O(2)$ symmetry of the order
parameter. Therefore, the dominant critical exponents for coupled
oscillators are the same as for this equilibrium field theory.
This formal connection with an equilibrium critical point imposes a
relation between the correlation and response functions of coupled
oscillators in the critical regime. Since the system operates far
from thermodynamic equilibrium, a strong violation of the
fluctuation-dissipation relation occurs and is characterized by a
universal divergence of an effective temperature. The formal
relation between critical oscillators and equilibrium critical
points suggests that long-range phase order exists in critical
oscillators above two dimensions.
@incollection{statphys23_1093,
abstract = {We show that the synchronization transition of a large number of
noisy coupled oscillators is an example for a dynamic critical point
far from thermodynamic equilibrium. The universal behaviors of such
critical oscillators, arranged on a lattice in a $d$-dimensional
space and coupled by nearest neighbors interactions, can be studied
using field theoretical methods. The field theory associated with
the critical point of a homogeneous oscillatory instability (or Hopf
bifurcation of coupled oscillators) is the complex Ginzburg-Landau
equation with additive noise.
We perform a perturbative renormalization group (RG) study in a $4-\epsilon$ dimensional space. We develop an RG scheme that eliminates the phase and
frequency of the oscillations using a scale-dependent oscillating
reference frame. Within a Callan-Symanzik RG scheme to two-loop
order in perturbation theory, we find that the RG fixed point is
formally related to the one of the model A dynamics of the real
Ginzburg-Landau theory with an $O(2)$ symmetry of the order
parameter. Therefore, the dominant critical exponents for coupled
oscillators are the same as for this equilibrium field theory.
This formal connection with an equilibrium critical point imposes a
relation between the correlation and response functions of coupled
oscillators in the critical regime. Since the system operates far
from thermodynamic equilibrium, a strong violation of the
fluctuation-dissipation relation occurs and is characterized by a
universal divergence of an effective temperature. The formal
relation between critical oscillators and equilibrium critical
points suggests that long-range phase order exists in critical
oscillators above two dimensions.},
added-at = {2007-06-20T10:16:09.000+0200},
address = {Genova, Italy},
author = {Risler, T. and Prost, J. and Julicher, F.},
biburl = {https://www.bibsonomy.org/bibtex/2b9a607a6083a46124947f9a2ef42f285/statphys23},
booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano},
interhash = {bf0debf570820d52bf75d9e105df33b8},
intrahash = {b9a607a6083a46124947f9a2ef42f285},
keywords = {bifurcation complex coupled critical dynamic equation equilibrium ginzburg-landau group hopf noisy oscillators out point renormalization statphys23 topic-3},
month = {9-13 July},
timestamp = {2007-06-20T10:16:38.000+0200},
title = {Universal Critical Behavior of Noisy Coupled Oscillators},
url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1093},
year = 2007
}