A scaling perspective on quantum energy flow in molecules

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Abstract
We present an analysis of quantum energy flow and localization in molecules from the point of view of a scaling approach. This scaling approach is based on earlier scaling theories of Anderson localization in disordered metals. The picture provides a simple general framework for describing both energy flow and the localization of eigenstates. This framework can be applied to molecules whose Hamiltonians are described by local random matrices, in which transitions between states nearby in quantum space occur more easily than those between distant states. In this theory the central quantity of interest is the survival probability. It is the squared overlap of a wave function at a later time with its initial condition. Its long time behavior varies depending on whether the eigenstates are localized or not. In the scaling description the survival probability approaches its long time limit varying inversely with time raised to some power. Different power laws are valid in different time regimes, depending on the degree of localization and the size of the molecule’s phase space. Near the localization transition we find that the survival probability decays as t−1. This decay is much slower than when energy flows easily, for a large number of dimensions participating in the energy flow.