Diffusing diffusivity: Fractional Brownian oscillator model for subdiffusion and its solution

Our earlier model for diffusing diffusivity [R. Jain and K. L. Sebastian, J. Phys. Chem. B 120, 3988 (2016) and Phys. Rev. E 95, 032135 (2017)], is extended to cover the case of subdiffusing diffusivity. In this, the diffusion coefficient is taken to be the square of a random process $z$, that fluctuates with a mean square variation which is proportional to ${(\mathit{time})}^{(2\ensuremath{\alpha}\ensuremath{-}1)}$, with $1/2<\ensuremath{\alpha}\ensuremath{\le}1$. (The previous analysis was only of the case $\ensuremath{\alpha}=1$.) We model the diffusivity as the square of the position coordinate of a fractional Brownian oscillator. As in our earlier papers, the probability distribution of the displacement of a particle with diffusing diffusivity is given as the Fourier transform of the survival probability of the fractional oscillator, in the presence of an absorbing term that depends quadratically on its position. To derive the expression for the survival probability, we use phase-space path integration, which is surprisingly easy to use. An easy numerical method is used to calculate the survival probability. Also, we find analytical expressions for the short time and long time survival probabilities for arbitrary $\ensuremath{\alpha}$. Long time survival probability is found to be exponential in time. Using these to analyze the problem of diffusing diffusivity, we find that the results are similar to those obtained in our earlier model of diffusivity, i.e., $\ensuremath{\alpha}=1$ case. The short time probability distribution of the particle can be found exactly, and is non-Gaussian, while the long time result is Gaussian. Finally, exact numerical results are presented for probability distributions for different values of $\ensuremath{\alpha}$ and the time $T$.