First passage time for subdiffusion: The nonextensive entropy approach versus the fractional model

TL;DR

This paper compares different subdiffusion models derived from nonextensive entropies and fractional calculus, analyzing their first passage time distributions and establishing conditions under which they are asymptotically equivalent.

Contribution

It derives a specific relationship between model parameters ensuring the asymptotic equivalence of FPT distributions from fractional and Sharma-Mittal models.

Findings

FPT distributions from different models are generally not equivalent.

The fractional model's FPT matches Sharma-Mittal's model asymptotically under a specific parameter condition.

Greens' functions from Sharma-Mittal and fractional models become similar when the condition is satisfied.

Abstract

We study the similarities and differences between different models concerning subdiffusion. More particularly, we calculate first passage time (FPT) distributions for subdiffusion, derived from Greens' functions of nonlinear equations obtained from Sharma-Mittal's, Tsallis's and Gauss's nonadditive entropies. Then we compare these with FPT distributions calculated from a fractional model using a subdiffusion equation with a fractional time derivative. All of Greens' functions give us exactly the same standard relation $<(\Delta x)^2> =2 D_\alpha t^\alpha$ which characterizes subdiffusion ($0<\alpha<1$), but generally FPT's are not equivalent to one another. We will show here that the FPT distribution for the fractional model is asymptotically equal to the Sharma--Mittal model over the long time limit only if in the latter case one of the three parameters describing Sharma--Mittal entropy $r$ depends on $\alpha$, and satisfies the specific equation derived in this paper, whereas the other two models mentioned above give different FTPs with the fractional model. Greens' functions obtained from the Sharma-Mittal and fractional models - for $r$ obtained from this particular equation - are very similar to each other. We will also discuss the interpretation of subdiffusion models based on nonadditive entropies and the possibilities of experimental measurement of subdiffusion models parameters.