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A review of linear response theory for general differentiable dynamical systems

A review of linear response theory for general differentiable dynamical systems The classical theory of linear response applies to statistical mechanics close to equilibrium. Away from equilibrium, one may describe the microscopic time evolution by a general differentiable dynamical system, identify nonequilibrium steady states (NESS) and study how these vary under perturbations of the dynamics. Remarkably, it turns out that for uniformly hyperbolic dynamical systems (those satisfying the chaotic hypothesis), the linear response away from equilibrium is very similar to the linear response close to equilibrium: the KramersKronig dispersion relations hold, and the fluctuationdispersion theorem survives in a modified form (which takes into account the oscillations around the attractor corresponding to the NESS). If the chaotic hypothesis does not hold, two new phenomena may arise. The first is a violation of linear response in the sense that the NESS does not depend differentiably on parameters (but this nondifferentiability may be hard to see experimentally). The second phenomenon is a violation of the dispersion relations: the susceptibility has singularities in the upper half complex plane. These acausal singularities are actually due to energy nonconservation: for a small periodic perturbation of the system, the amplitude of the linear response is arbitrarily large. This means that the NESS of the dynamical system under study is not inert but can give energy to the outside world. An active NESS of this sort is very different from an equilibrium state, and it would be interesting to see what happens for active states to the GallavottiCohen fluctuation theorem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinearity IOP Publishing

A review of linear response theory for general differentiable dynamical systems

Nonlinearity , Volume 22 (4): 16 – Apr 1, 2009

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References (61)

Copyright
Copyright 2009 IOP Publishing Ltd and London Mathematical Society
ISSN
0951-7715
eISSN
1361-6544
DOI
10.1088/0951-7715/22/4/009
Publisher site
See Article on Publisher Site

Abstract

The classical theory of linear response applies to statistical mechanics close to equilibrium. Away from equilibrium, one may describe the microscopic time evolution by a general differentiable dynamical system, identify nonequilibrium steady states (NESS) and study how these vary under perturbations of the dynamics. Remarkably, it turns out that for uniformly hyperbolic dynamical systems (those satisfying the chaotic hypothesis), the linear response away from equilibrium is very similar to the linear response close to equilibrium: the KramersKronig dispersion relations hold, and the fluctuationdispersion theorem survives in a modified form (which takes into account the oscillations around the attractor corresponding to the NESS). If the chaotic hypothesis does not hold, two new phenomena may arise. The first is a violation of linear response in the sense that the NESS does not depend differentiably on parameters (but this nondifferentiability may be hard to see experimentally). The second phenomenon is a violation of the dispersion relations: the susceptibility has singularities in the upper half complex plane. These acausal singularities are actually due to energy nonconservation: for a small periodic perturbation of the system, the amplitude of the linear response is arbitrarily large. This means that the NESS of the dynamical system under study is not inert but can give energy to the outside world. An active NESS of this sort is very different from an equilibrium state, and it would be interesting to see what happens for active states to the GallavottiCohen fluctuation theorem.

Journal

NonlinearityIOP Publishing

Published: Apr 1, 2009

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