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 [A. V. Plyukhin, Phys. Rev. E 99, 052125 (2019)]. 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 $k\left(t\right)\sim 1/t$ 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 $\langle \mathrm{\Delta }{Q}^2\left(t\right)\rangle \sim t/ln\left(t\right)$ as $t\to \infty$. We also numerically calculate the velocity correlation function in the asymptotic regime and use it to confirm the aforementioned subdiffusion.

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• Brownian motion
• Diffusion
• Open quantum systems