Logarithmic Subdiffusion from a Damped Bath Model

A damped oscillator heat bath model is a modification of the standard heat bath model, wherein each bath oscillator itself has a Markovian coupling to its own heat bath [1]. We modify such a model to one where the resulting damping of the oscillators is linear in their frequency rather than being a constant. We find that this generates a memory kernel which behaves like $k(t) \sim 1/t$ as $t \to \infty$, which is a boundary case not considered in previous works. As the memory kernel does not have a finite integral, the reduced system is subdiffusive, and we numerically show that diffusion goes as $\langle \Delta Q^{2}(t)\rangle \sim t/\log(t)$ as $t \to \infty$. We also numerically calculate the velocity correlation function in the asymptotic regime and use it to confirm the aforementioned subdiffusion.