Fluctuation-dissipation relation for chaotic non-Hamiltonian systems

Abstract

In dissipative dynamical systems phase space volumes contract, on average. Therefore, the invariant measure on the attractor is singular with respect to the Lebesgue measure. As noted by Ruelle, a generic perturbation pushes the state out of the attractor, hence the statistical features of the perturbation and, in particular, of the relaxation, cannot be understood solely in terms of the unperturbed dynamics on the attractor. This remark seems to seriously limit the applicability of the standard fluctuation dissipation procedure in the statistical mechanics of nonequilibrium (dissipative) systems. In this paper we show that the singular character of the steady state does not constitute a serious limitation in the case of systems with many degrees of freedom. The reason is that one typically deals with projected dynamics, and these are associated with regular probability distributions in the corresponding lower dimensional spaces.

An update on the nonequilibrium linear response

“…The basic step from (9) is partial integration, which means that it is assumed here that ρ has a smooth density with respect to the reference volume element, ρ(dx) = ρ(x) dx. In many cases that appears to be a reasonable physical assumption when the level of description is mesoscopic to macroscopic, independent of whether the system is driven or not (see [9]).…”

Complexity, Chaos and Fluctuations

“…Much effort was needed, in particular, to identify the minimal mathematical ingredients as well as the physical mechanisms lying beneath the validity of such relations [23][24][25]. Another result, whose mathematical theory was originally dug out in the sixties of the last century [26,27] and was then gradually a e-mail: santoban@gmail.com b e-mail: matteo.colangeli1@univaq.it unravelled in the following decades, is represented by the Fluctuation-Dissipation Theorem [28][29][30][31][32][33], which allows to express the response of a system to an external perturbation in terms of a correlation function computed with respect to a reference measure.…”

Practical Approximations to Seasonal Fluctuation–Dissipation Operators Given a Limited Sample

“…For the purpose of producing FD operators, we also require that a unique inverse transformation from the feature space back to the original coordinates be available. Note also that a careful choice of projection of the state may also be seen as analogous to the coarse-graining approach proposed by Colangeli et al (2012).…”

“…Two immediate consequences of this are the presence of singularities in the system's probability density function (PDF) and the possibility that part of the response because of an external perturbation may be decoupled from internal variability (Ruelle 2009;Lucarini et al 2014). Attempts to circumvent these issues include, for example, accounting for aspects of the response that go beyond the unstable direction along the climate attractor (Lucarini and Sarno 2011;Ragone et al 2015), coarse graining of the variable at hand (Colangeli et al 2012), assuming the presence of a smoothing background noise (Gritsun andBranstator 2007, hereafter GB2007), and constructing a tangent model along the unstable direction of the climate attractor (Abramov and Majda 2007). The seriousness of these issues is yet to be fully clarified in the climate context.…”