Stability of Quantum Dynamics under Constant Hamiltonian Perturbations

Concepts like typicality' and the eigenstate thermalization hypothesis’ aim at explaining the apparent equilibration of quantum systems, possibly after a very long time. However, these concepts are not concerned with the specific way in which this equilibrium is approached. Our point of departure is the (evident) observation that some forms of the approach to equilibrium, such as, e.g., exponential decay of observables, are much more common then others. We suggest to trace this dominance of certain decay dynamics back to a larger stability with respect to generic Hamiltonian perturbations. A numerical study of a number of examples in which both, the unperturbed Hamiltonians as well as the perturbations are modeled by partially random matrices is presented. We furthermore develop a simple heuristic, mathematical scheme that describes the result of the numerical investigations remarkably well. According to those investigations the exponential decay indeed appears to be most stable. Dynamics that are in a certain sense at odds with the arrow of time are found to be very unstable.