A review of linear response theory for general differentiable dynamical systems

By David Ruelle. 2009

258 citations recorded by the source corpus.

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 Kramers–Kronig dispersion relations hold, and the fluctuation–dispersion 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 Gallavotti–Cohen fluctuation theorem.

Subfields: Computer Networks and Communications, Economics and Econometrics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1bc6ef56e783864949860332fef1fa996a8f365c

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Citations in the corpus

Predicting Climate Change Using Response Theory: Global Averages and Spatial Patterns (2015, background): “Under suitable integrability conditions, the fact that the Green function G(t) is causal is equivalent to saying that its susceptibility obeys the so-called Kramers–Kronig relations [23,46], which provide integral constraints linking its real and imaginary part, so that χ(1)(ω) = iP(1/ω) χ(1)(ω), wh”
Stochastic climate theory and modeling (2015, unknown): “Recently, Wouters and Lucarini 177 have proposed to treat comprehensively the problem of model reduction in multi-scale systems by adapting the Ruelle response theory 136,137 for studying the eﬀect of the coupling between the fast and slow degrees of freedom of the system.”
Stochastic climate dynamics: Random attractors and time-dependent invariant measures (2011, methodology): “For hyperbolic deterministic systems, precise estimates exist for the response of their SRB measures to perturbations [53, 54].”
Frenesy: Time-symmetric dynamical activity in nonequilibria (2019, background): “in linear response theory around SRB-measures which has been investigated in much mathematical detail [131, 132], that identify or confirm that suggested frenetic contribution in physical examples.”
Gradients are Not All You Need (2021, background): “2 Preliminaries: Iterated Dynamical Systems Chaos emerges naturally in iterated maps [Bischof et al., 1991, Ruelle, 2009].”
The physics of climate variability and climate change (2019, background): “tions, the fact that the Green’s function G (t) Ψ,X is causal is equivalent to saying that its susceptibility obeys the so-called Kramers-Kronig relations (Lucarini and Sarno, 2011; Ruelle, 2009); these provide integral constraints”
Linear response, or else (2014, background): “) We refer to Ruelle’s articles [46, 48, 52] for motivation, applications to physics, and more conjectures.”
Mathematical and physical ideas for climate science (2013, background): “Ruelle (2009) shows that since the unperturbed invariant ρ(0)(dx) is singular, the response of the system contains two contributions, such that the first may be expressed in terms of a correlation function evaluated with respect to the unperturbed dynamics along the space tangent to the attractor…”