I am trying to understand whether a slightly modified version of the Fokker-Planck equation is suitable or not to study some subdiffusive processes.
The original Fokker-Planck equation can be written as:
$$\partial_t P(x,t)= D \left(\partial_x^2 - \partial_x \frac{F(x)}{k_b T} \right) P(x,t)$$
I am interested in processes where there is no external force, so it can be written as a regular diffusion equation:
$$\partial_t P(x,t)= D \partial_x^2 P(x,t)$$
The solution of this equation with initial condition $P(x,0)=\delta(x)$ is a Gaussian distribution:
$$P(x,t)=\frac{1}{\sqrt{4\pi D t}} e^{-x^2/(4Dt)}$$
I was wondering if I could adapt this reasoning to obtain the probability distribution of a free subdiffusive particle without having to use a fractional Fokker-Planck equation. Would it be licit to write the Fokker-Planck equation for a "subdiffusive" time $\tau=t^\alpha$ as
$$\partial_\tau P(x,\tau)= D \partial_x^2 P(x,\tau)$$
and obtain the PDF as a function of $\tau$? I would obtain in that case a PDF that looks like
$$P(x,t)=\frac{1}{\sqrt{4\pi D t^\alpha}} e^{-x^2/(4Dt^\alpha)}$$
In case this is valid, why is this method not commonly used? And in case it is not, why is my above reasoning not valid?
