Scarring in classical chaotic dynamics with noise

Quantum-classical correspondence in quantum channels

Quantum channels describe subsystem or open-system evolution. Using the classical Koopman operator that evolves functions on phase space, four classical Koopman channels are identified as analogs of the four possible quantum channels in a bipartite setting. Thus, when the complete evolution has a quantum-classical correspondence, the correspondence at the level of the subunitary channels can be studied. The channels, both classical and quantum, can be interpreted as noisy single-particle systems. Having parallel classical and quantum operators gives us new access to study the fine details of these major limiting theories. Using a coupled kicked rotor as a generic example, we contrast and compare spectra of the quantum and classical channels. The largest nontrivial mode of the quantum channel is seen to be mostly determined by the stable parts of the classical phase space, even those that are surprisingly small relative to the scale of an effective ℏ. In cases where the dynamics have a significant fraction of chaos, the spectrum exhibits a prominent annular density, which is approximately described by the single-ring theorem of random matrix theory. The ring shrinks in size when the classical limit is approached. However, the eigenvalues and modes that survive the classical limit appear to be scarred by unstable manifolds or, if they exist, stable periodic orbits.

Thermodynamics of chaotic relaxation processes

The established thermodynamic formalism of chaotic dynamics, valid at statistical equilibrium, is here generalized to systems out of equilibrium that have yet to relax to a steady state. A relation between information, escape rate, and the phase-space average of an integrated observable (e.g., Lyapunov exponent, diffusion coefficient) is obtained for finite time. Most notably, the thermodynamic treatment may predict the phase-space profile of any integrated observable for finite time, from the leading and subleading eigenfunctions of the Perron-Frobenius or Koopman transfer operator. Examples of that equivalence are shown, and the theory is tested analytically on the Bernoulli map while numerically on the perturbed cat map, the Hénon map, and the Ikeda map, all paradigms of chaos.