Anomalous diffusion and Noether's second theorem

TL;DR

This paper links anomalous diffusion and heat transport phenomena to non-local gauge transformations derived from Noether's second theorem, revealing how fractional equations of motion emerge from symmetry violations.

Contribution

It demonstrates that anomalous transport behaviors result from Lorentz-violating gauge transformations connected to Noether's second theorem, providing a systematic framework for fractional diffusion equations.

Findings

Anomalous diffusion characterized by $L\,\propto t^\gamma$ with $\gamma \neq 1/2$.

Heat conductivity diverges as a power law with system size.

Fractional equations of motion arise from non-local gauge transformations.

Abstract

Despite the fact that conserved currents have dimensions that are determined solely by dimensional analysis (and hence no anomalous dimensions), Nature abounds in examples of anomalous diffusion in which $L\propto t^\gamma$, with $\gamma\ne 1/2$, and heat transport in which the thermal conductivity diverges as $L^\alpha$. Aside from breaking of Lorentz invariance, the true common link in such problems is an anomalous dimension for the underlying conserved current, thereby violating the basic tenet of field theory. We show here that the phenomenological non-local equations of motion that are used to describe such anomalies all follow from Lorentz-violating gauge transformations arising from N\"other's second theorem. The generalizations lead to a family of diffusion and heat transport equations that systematize how non-local gauge transformations generate more general forms of Fick's and Fourier's laws for diffusive and heat transport, respectively. In particular, the associated Goldstone modes of the form $\omega\propto k^\alpha$, $\alpha\in \mathbb R$ are direct consequences of fractional equations of motion.