 07232 Valerio Lucarini
 Response Theory for Equilibrium and NonEquilibrium Statistical Mechanics: Causality and Generalized KramersKronig relations
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Oct 4, 07

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Abstract. We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the nonequilibrium
steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of KramersKronig relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey KramersKronig relations. The apparent contradiction with the principle of causality is also clarified. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive KramersKronig analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the shorttime response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical Hamiltonian systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing selfconsistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. In order to gain a more complete picture, connecting the response theory for equilibrium and non equilibrium systems, we show how to rewrite the classical response theory by Kubo for systems close to equilibrium so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. Finally, we briefly discuss how the presented results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might have relevant implications for climate research. In particular, whereas the fluctuationdissipation theorem does not work for nonequilibrium systems, because of the nonequivalence between internal and external fluctuations, KramersKronig relations might be more robust tools for the definition of a selfconsistent theory of climate change.
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