Probabilistic description of dissipative chaotic scattering
We investigate the extent to which the probabilistic properties of a chaotic scattering system with dissipation can be understood from the properties of the dissipation-free system. For large energies $E$, a fully chaotic scattering leads to an exponential decay of the survival probability $P(t) \sim e^{-κt}$ with an escape rate $κ$ that decreases with $E$. Dissipation $γ>0$ leads to the appearance of different finite-time regimes in $P(t)$. We show how these different regimes can be understood for small $γ\ll 1$ and $t\gg 1/κ_0$ from the effective escape rate $κ_γ(t)=κ_0(E(t))$ (including the non-hyperbolic regime) until the energy reaches a critical value $E_c$ at which no escape is possible. More generally, we argue that for small dissipation $γ$ and long times $t$ the surviving trajectories in the dissipative system are distributed according to the conditionally invariant measure of the conservative system at the corresponding energy $E(t)<E(0)$. Quantitative predictions of our general theory are compared with numerical simulations in the Henon-Heiles model.