Response Theory for Equilibrium and Non-Equilibrium Statistical
Mechanics: Causality and Generalized Kramers-Kronig
relations
Valerio Lucarini
Department of Physics
Abstract:
We consider the general response theory
proposed by Ruelle for
describing the impact of small perturbations on the non-equilibrium
steady states of 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 use of the principle of
causality for testing self-consistency: the described dispersion
relations constitute solid benchmarks for any experimental or
model generated dataset. In order to connect the response theories for
equilibrium and non equilibrium systems, we rewrite the classical
results by Kubo so that response functions are obtained that are formally
identical to those proposed by Ruelle, apart from the
measure involved
in the phase space integration,. 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.