Anomalous diffusion and Noether's second theorem

Matteo Baggioli, Gabriele La Nave, Philip W. Phillips

Research output: Contribution to journalArticlepeer-review


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 Ltγ, with γ≠1/2, and heat transport in which the thermal conductivity diverges as Lα. 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 nonlocal equations of motion that are used to describe such anomalies all follow from Lorentz-violating gauge transformations arising from Noether's second theorem. The generalizations lead to a family of diffusion and heat transport equations that systematize how nonlocal 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 ωkα, αR are direct consequences of fractional equations of motion.

Original languageEnglish (US)
Article number032115
JournalPhysical Review E
Issue number3
StatePublished - Mar 2021

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Statistics and Probability


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