Valerio Lucarini, Matteo Colangeli
In this paper we re-examine the traditional problem of connecting the
internal fluctuations of a system to its response to external forcings and
extend the classical theory in order to be able to encompass also nonlinear
processes. With this goal, we try to join on the results by Kubo on statistical
mechanical systems close to equilibrium, i.e. whose unperturbed state can be
described by a canonical ensemble, the theory of dispersion relations, and the
response theory recently developed by Ruelle for non-equilibrium systems
equipped with an invariant SRB measure. Our derivations highlight the strong
link between causality and the possibility of connecting unambiguously
fluctuation and response, both at linear and nonlinear level. We first show in
a rather general setting how the formalism of the Ruelle response theory can be
used to derive in a novel way Kramers-Kronig relations connecting the real and
imaginary part of the linear and nonlinear response to external perturbations.
We then provide a formal extension at each order of nonlinearity of the
fluctuation-dissipation theorem (FDT) for general systems possessing a smooth
invariant measure. Finally, we focus on the physically relevant case of systems
close to equilibrium, for which we present explicit fluctuation-dissipation
relations linking the susceptibility describing the $n^{th}$ order response of
the system with the expectation value of suitably defined correlations of $n+1$
observables taken in the equilibrium ensemble. While the FDT has an especially
compact structure in the linear case, in the nonlinear case joining the
statistical properties of the fluctuations of the system to its response to
external perturbations requires linear changes of variables, simple algebraic
sums and multiplications, and a multiple convolution integral. These
operations, albeit cumbersome, can be easily implemented numerically.
View original:
http://arxiv.org/abs/1202.1073
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