**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.