Classical nonlinear response of a chaotic system. I. Collective resonances

Abstract

We consider the classical response in a chaotic system. In contrast to behavior in integrable or almost integrable systems, the nonlinear classical response in a chaotic system vanishes at long times. The response also reveals certain features of collective resonances which do not correspond to any periodic classical trajectories. The convergence of the response is shown to hold due to the exponential time dependence of the stability matrix. The growing exponentials corresponding to strong instability do not inhibit the convergence. We calculate both linear and second-order response in one of the simplest chaotic systems: free classical motion on a surface of constant negative curvature. We demonstrate the relevance of the model for applications to spectroscopic experiments.

Semiclassical mean-trajectory approximation for nonlinear spectroscopic response functions

“…As discussed in connection with Eq. ͑2.47͒, the classical response function at the echo condition displays unbounded temporal growth, 31,[37][38][39][43][44][45][46] while the quantum response function shows recurrences with a period that diverges as ប → 0. For the Morse oscillator, this period is 2 / ⌬, with the anharmonic frequency shift ⌬ defined in Eq.…”

“…96 This phenomenon of temporal divergences in classical response functions has been shown to hold more generally [43][44][45][46] for integrable systems. Recent work [37][38][39] has also shown that for a system with classically chaotic motion, the response function has damped time dependences, despite the exponential growth of the monodromy matrix. As a check on the approximations that led to Eq.…”

“…The difficulties associated with fully quantum calculations of vibrational response functions have motivated assessments of fully classical [27][28][29][30][31][32][33][34][35][36][37][38][39] and semiclassical [40][41][42][43][44][45][46][47][48] calculations. Classical calculations permit the determination of a response function by propagating classical trajectories and stability matrices, but in general these only agree with quantum calculations at short times, [43][44][45][46][47][48][49] which may indeed suffice to describe some ultrafast experiments.…”

Nonlinear response theory in chemical kinetics

“…Such stability derivatives can be unbounded in time and may lead to the problems of divergence of nonlinear dynamical responses in systems with periodic 21,22 and chaotic Hamiltonian dynamics. 23 In some cases the latter divergence problem can be eliminated by canonical averaging 24 or in strongly chaotic 25,26 systems. It is expected that stability derivatives of chemical kinetics do not diverge in time due to the generally stable dynamic behavior of mass-action systems, 27,28 and thus should not lead to unphysical time growth of their dynamical response.…”

How a solute-pump/solvent-probe spectroscopy can reveal structural dynamics: Polarizability response spectra as a two-dimensional solvation spectroscopy

“…28 This procedure has, in fact, been successfully applied in multidimensional spectroscopy simulations, 24, 29-31 but the practical difficulty it faces is that Poisson brackets are essentially fluctuation quantities, making them intrinsically noisy to simulate, and looking for their correlation with yet another fluctuation makes the numerical situation even worse. 30,32,33 One approach to circumventing these problems has been to avoid linear response altogether by simulating the non-equilibrium response in the presence of explicit applied fields. [34][35][36] There have also been more recent approaches combining equilibrium and non-equilibrium trajectories.…”