Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig Relations |
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Authors: | Valerio Lucarini |
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Institution: | (1) Department of Physics, University of Bologna, Bologna, Italy |
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Abstract: | We consider the general response theory recently 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
entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders
of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results
are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the
establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time
response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for
optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality
for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and
model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations
to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations.
In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response
theory 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. These results, taking into account the chaotic hypothesis by Gallavotti and
Cohen, might be relevant in several fields, including 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,
K-K relations might be robust tools for the definition of a self-consistent theory of climate change. |
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Keywords: | Axiom A dynamical systems Non-equilibrium steady states Kubo response theory Ruelle response theory SRB measure Chaotic hypothesis Kramers-Kronig relations Harmonic generation Climate |
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