Decoherence Entails Exponential Forgetting in Systems Complying with the Eigenstate Thermalization Hypothesis

According to the eigenstate thermalization ansatz, matrices representing generic few body observables take on a specific form when displayed in the eigenbasis of a chaotic Hamiltonian. We examine the effect of environmental induced decoherence on the dynamics of observables that conform with said eigenstate thermalization ansatz. The obtained result refers to a description of the dynamics in terms of an integro-differential equation of motion of the Nakajima-Zwanzig form. We find that environmental decoherence is equivalent to an exponential damping of the respective memory kernel. This statement is formulated as rigorous theorem. Furthermore the implications of the theorem on the stability of exponential dynamics against decoherence and the transition towards Zeno-Freezing are discussed.