Linear response for macroscopic observables in high-dimensional systems

Abstract

The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood.We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional weakly coupled deterministic dynamical systems, where the weak coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system, and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, linear response is induced via self-generated noise.We present examples in the thermodynamic limit where the macroscopic observable satisfies LRT, although the microscopic subsystems individually violate LRT, as well a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent non-trivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.

Response theory and phase transitions for the thermodynamic limit of interacting identical systems

“…9), and, more recently, rigorous results have been obtained in this direction [64,65]. An interesting link between response theory for deterministic and stochastic systems has been proposed in [66]. The results presented in [64,65] can be applied to the McKean–Vlasov equation in the absence of phase transitions to justify rigorously linear response theory and to establish fluctuation–dissipation results.…”

Predictors and predictands of linear response in spatially extended systems

“…The relationship between response theory in deterministic and stochastic systems has been thoroughly discussed in [33], where the authors also propose a very well-developed justification, not based on the chaotic hypothesis, for the broad pragmatic applicability of LRT in a vast class of deterministic chaotic systems.…”

Rigorous Computation of Linear Response for Intermittent Maps

“…Indeed, the work of Ruelle was refined in the uniformly hyperbolic setting [12,25], extended to the partially hyperbolic setting [15], and has been a topic of deep investigation for unimodal maps, see [8], the survey article [7], the recent works [3,9,14,39] and references therein. More recently, the topic of linear response was also studied in the context of random or extended systems [6,16,18,23,31,40,44]. Optimisation of statistichal properties through linear respone was develope in [1,2,22,30].…”