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Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig relations

We consider the general response theory proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions allows for writing a set of Kramers-Kronig relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of observable susceptibilities obey Kramers-Kronig relations. Specific results are provided for arbitrary order harmonic response, which allows for a very comprehensive Kramers-Kronig analysis and the establishment of sum rules connecting the asymptotic behavior of the susceptibility to the short-time response of the system. These results generalize previous findings on optical Hamiltonian systems and simple mechanical models, and shed light on the general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks for any experimental and model generated dataset. In order to connect the response theory for equilibrium and non equilibrium systems, we rewrite the classical results by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. We briefly discuss how these results, taking into account the chaotic hypothesis, might be relevant for climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, Kramers-Kronig relations might be more robust tools for the definition of a self-consistent theory of climate change.