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Comportement critique d'oscillateurs couples ; Groupe de renormalisation et classe d'universalite

Abstract : The astonishing efficiency of the auditory organ of mammals is particularly
due to the generic properties of the coupled critical oscillators which make up
the system. This thesis presents a study of the generic critical properties of
spatially extended systems of coupled stochastic oscillators, operating in the
proximity of a uniform oscillatory instability or Hopf bifurcation. In this
context, this bifurcation constitutes an out of equilibrium critical point
with universal features, which are canonically described by the complex
Ginzburg-Landau equation in the presence of noise. The formulation of the
problem in terms of a non-Hamiltonian dynamical statistical field theory
allows us to study the critical behavior of the system by using perturbative
renormalization group techniques.

In a particular case, an exact analogy with the O(2) dynamical model allows
us to write a generalized fluctuation-dissipation relation and to deduce the
critical behavior directly from previous studies. In the general
case, we establish the structure of the renormalization group of the theory in
a 4-epsilon dimensional space, using adapted Wilson and Callan-Symanzik
schemes. The presence of a characteristic frequency in the system - the
frequency of the spontaneous oscillations at the transition - imposes to
perform a scale-dependant frame transformation during the renormalization
procedure. We perform two-loop order calculations in perturbation theory, and
show that the universality class of the model is described, in a suited
oscillating frame, by the fixed point of the dissipative O(2)
dynamics. Then, while the dynamics is highly out of equilibrium and breaks the
detailed-balance relations, a generalized fluctuation-dissipation relation is
asymptotically recovered at the transition. This relation imposes strong
constraints on the main experimental observables: the two-point correlation
function and the linear response function to an external sinusoidal stimulus.
Complete list of metadatas
Contributor : Thomas Risler <>
Submitted on : Monday, February 16, 2004 - 6:24:25 PM
Last modification on : Thursday, December 10, 2020 - 12:31:14 PM
Long-term archiving on: : Monday, September 20, 2010 - 11:46:14 AM


  • HAL Id : tel-00004449, version 2


Thomas Risler. Comportement critique d'oscillateurs couples ; Groupe de renormalisation et classe d'universalite. Biophysique []. Université Pierre et Marie Curie - Paris VI, 2003. Français. ⟨tel-00004449v2⟩



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