Fractional generalization of Fick's law: a microscopic approach

TL;DR

This paper introduces a generalized form of Fick's law for transport in complex systems lacking characteristic scales, based on a microscopic approach that maintains global reversibility.

Contribution

It develops a fractional generalization of Fick's law applicable to systems without finite characteristic scales, extending the microscopic understanding of transport phenomena.

Findings

The generalized law does not require finite characteristic scales.

It preserves the principle of global reversibility.

Applicable to complex systems with anomalous transport behaviors.

Abstract

In the study of transport in inhomogeneous systems it is common to construct transport equations invoking the inhomogeneous Fick law. The validity of this approach requires that at least two ingredients be present in the system. First, finite characteristic length and time scales associated to the dominant transport process must exist. Secondly, the transport mechanism must satisfy a microscopic symmetry: global reversibility. Global reversibility is often satisfied in nature. However, many complex systems exhibit a lack of finite characteristic scales. In this Letter we show how to construct a generalization of the inhomogeneous Fick law that does not require the existence of characteristic scales while still satisfying global reversibility.