Soft matter and fractional mathematics: insights into mesoscopic quantum and time-space structures

TL;DR

This paper explores how fractional mathematics can provide new insights into the mesoscopic quantum and time-space structures of soft matter, addressing fundamental physical laws behind fractal mesostructures.

Contribution

It introduces fractional calculus and related concepts to analyze mesoscopic phenomena in soft matter, offering novel relationships like fractional Planck energy and scaling transforms.

Findings

Fractional Planck quantum energy relationship established.

Identification of fractional phonons in soft matter.

Development of time-space scaling transform for mesoscopic analysis.

Abstract

Recent years have witnessed a great research boom in soft matter physics. by now, most advances, however, are of either empirical results or purely mathematical extensions. The major obstacle is lacking of insights into fundamental phsysical laws underlying fractal mesostructures of soft matter. This study will use fractional mathematics, which consists of fractal, fractional calculus, fractional Brownian motion, and Levy stable distribution, to examine mesoscopic quantum mechanics and time-space structures governing "anomalous" behaviors of soft matter. Our major results include fractional Planck quantum energy relationship, fractional phonon, and time-space scaling transform.