Fractional Calculus as a Macroscopic Manifestation of Randomness

TL;DR

This paper extends Van Hove's method to non-ordinary statistical mechanics, showing that lack of time-scale separation leads to macroscopic dynamics best described by fractional calculus, revealing a link between microscopic randomness and macroscopic behavior.

Contribution

It introduces a generalized Van Hove method applicable to systems without time-scale separation, demonstrating the emergence of fractional calculus in macroscopic descriptions.

Findings

In systems with no time-scale separation, fractional calculus accurately models macroscopic dynamics.

The generalized Van Hove method transmits microscopic randomness to the macroscopic level.

Normal calculus suffices for systems with time-scale separation, despite microscopic randomness.

Abstract

We generalize the method of Van Hove so as to deal with the case of non-ordinary statistical mechanics, that being phenomena with no time-scale separation. We show that in the case of ordinary statistical mechanics, even if the adoption of the Van Hove method imposes randomness upon Hamiltonian dynamics, the resulting statistical process is described using normal calculus techniques. On the other hand, in the case where there is no time-scale separation, this generalized version of Van Hove's method not only imposes randomness upon the microscopic dynamics, but it also transmits randomness to the macroscopic level. As a result, the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus.