Long-term relaxation of a composite system in partial contact with a heat bath

We study relaxational behavior from a highly excited state for a composite system in partial contact with a heat bath, motivated by an experimental report of long-term energy storage in protein molecules. The system consists of two coupled elements: The first element is in direct contact with a heat bath, while the second element interacts only with the first element, Due to this indirect contact with the heat bath, energy injected into the second element dissipates very slowly, according to a power law, whereas that injected into the first one exhibits exponential dissipation. The relaxation equation describing this dissipation is obtained analytically for both the underdamped and overdamped limits. Numerical confirmation is given for both cases.