The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

Abstract

By collecting from literature data experimental evidence of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live Escherichia coli cells, we obtain the probability density function of molecules’ displacement and we derive the corresponding Fokker–Planck equation. Molecules’ distribution emerges to be related to the Krätzel function and its Fokker–Planck equation to be a fractional diffusion equation in the Erdélyi–Kober sense. The irreducibility of the derived Fokker–Planck equation to those of other literature models is also discussed.

Conditional entropy and weak fluctuation correlation in nonequilibrium complex systems

“…The role of spatiotemporal fluctuations in heterogeneous diffusion is a basic issue in connection with time evolution within the framework of so-called superstatistics [32], e.g., in Refs. [33][34][35][36][37][38][39][40][41]. A similar issue has also been mentioned in Ref.…”

Subordination and memory dependent kinetics in diffusion and relaxation phenomena

“…Here the power-like shape of the memory functions, sitting as kernels in the Caputo derivatives, turns out to be decisive to judge nonnegativity of obtained solutions and show the stable character of the subordination. Among another paths to generalize diffusion equations we mention giving up the constant character of the diffusion coefficient [67] or modeling diffusive processes using evolution equations with more than one fractional time derivative, including also the case of so-called distributed order fractional derivatives [20,21,37]. Models using the time dependent diffusion coefficient, inspired by experimental data, have been used to explain differences in behavior of MSDs for short and long times -relevant examples are provided by the scaled Brownian motion with diffusion coefficient D(t) ∝ t α−1 , where t ∈ R + and α > 0, for which the MSD grows linearly at short times, x 2 (t) ∝ t, whereas at long times behaves as x 2 (t) ∝ t α [17,42,68,69,81].…”