Fractional Brownian motion approach to polymer translocation: the governing equation of motion

We suggest a governing equation which describes the process of polymer chain translocation through a narrow pore and reconciles the seemingly contradictory features of such dynamics: (i) a Gaussian probability distribution of the translocated number of polymer segments at time $t$ after the process has begun, and (ii) a sub-diffusive increase of the distribution variance $\Delta (t)$ with elapsed time, $\Delta(t) \propto t^{\alpha}$. The latter quantity measures the mean-squared number $s$ of polymer segments which have passed through the pore, $\Delta(t) = <[s(t)-s(t=0)]^2>$, and is known to grow with an anomalous diffusion exponent $\alpha < 1$. Our main assumption - a Gaussian distribution of the translocation velocity $v(t)$ - and some important theoretical results, derived recently, are shown to be supported by extensive Brownian dynamics simulation which we performed in $3D$. We also numerically confirm the predictions made in ref.\cite{Kantor_3}, that the exponent $\alpha$ changes from $0.91$ to $0.55$, to $0.91$, for short, intermediate and long time regimes, respectively.