CHAOS AND LINEAR-RESPONSE - ANALYSIS OF THE SHORT-TIME, INTERMEDIATE-TIME, AND LONG-TIME REGIME

We study the response of a classical Hamiltonian system to a weak perturbation in the regime where the dynamics is mixing, with the purpose of critically examining both the foundation of the Kubo linear response theory (LRT) and van Kampen's well known objections to LRT [Phys. Norv. 5, 279 (1971)]. Although the exactness of LRT for short times is not surprising, we prove that for the class of model studied here the LRT must also become accurate in the limit of long times, even for macroscopically large external perturbations. Hence, if the LRT breaks down, the breakdown occurs in the region of intermediate times. We also show that, for a given system, if any macroscopic linear response exists, it must coincide with Kubo LRT; thus, if a generic system responds nonlinearly to an external perturbation, this nonlinear response is observable only in an intermediate-time range. Numerical calculations carried out on some model systems with only a few degrees of freedom support these arguments.