Baggioli Matteo, La Nave Gabriele, Phillips Philip W
Instituto de Fisica Teorica UAM/CSIC, c/ Nicolas Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain.
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA.
Phys Rev E. 2021 Mar;103(3-1):032115. doi: 10.1103/PhysRevE.103.032115.
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∝t^{γ}, with γ≠1/2, and heat transport in which the thermal conductivity diverges as L^{α}. 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 ω∝k^{α}, α∈R are direct consequences of fractional equations of motion.
尽管守恒流具有仅由量纲分析确定的维度（因此没有反常维度），但自然界中存在大量反常扩散的例子，其中L∝t^γ，γ≠1/2，以及热传导的例子，其中热导率随L^α发散。除了洛伦兹不变性的破缺，此类问题真正的共同联系是基础守恒流的反常维度，从而违反了场论的基本原理。我们在此表明，用于描述此类反常现象的唯象非局部运动方程均源自诺特定理第二定理产生的违反洛伦兹规范变换。这些推广导致了一族扩散和热传导方程，它们系统地阐述了非局部规范变换如何分别为扩散和热传导生成更一般形式的菲克定律和傅里叶定律。特别地，形式为ω∝k^α，α∈R的相关戈德斯通模式是分数运动方程的直接结果。
