Logarithmic Subdiffusion from a Damped Bath Model

A damped heat bath is a modification of the standard oscillator heat bath model, wherein each bath oscillator itself has a Markovian coupling to its own heat bath. We modify such a model as described in $\textit{Plyukhin (2019)}$ to one where the oscillators undergo linear rather than constant damping, and find that this generates a memory kernel which behaves like $k(t) \sim 1/t$ as $t\to \infty$ when the spectral density is Ohmic. This is a boundary case not considered in previous works. As the memory kernel does not have a finite integral, it is subdiffusive, and we numerically show the diffusion to go 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.
Comment: 6 pages, 5 figures