Anomalous diffusion and Noether's second theorem
Anomalous diffusion and Noether's second theorem
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DOI：
10.1103/physreve.103.032115
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发表时间：
2021-03-12
影响因子：
2.4
通讯作者：
Phillips, Philip W.
中科院分区：
文献类型：
--
作者：
Baggioli, Matteo;La Nave, Gabriele;Phillips, Philip W.
尽管守恒电流的维度仅由维度分析确定（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 与 t(gamma) 成正比，且 gamma 不等于 1/2，以及热传输，其中热导率发散为 L-alpha。除了破坏洛伦兹不变性之外，此类问题的真正共同联系是潜在守恒流的反常维度，从而违反了场论的基本原理。我们在这里表明，用于描述此类异常的现象学非局部运动方程都遵循诺特第二定理产生的洛伦兹破坏规范变换。这些概括导致了一系列扩散和热传输方程，这些方程系统化了非局部规范变换如何分别生成扩散和热传输的菲克定律和傅里叶定律的更一般形式。特别是，与 k(alpha),alpha epsilon R 成比例的 omega 形式的相关戈德斯通模式是分数运动方程的直接结果。
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 proportional to t(gamma), with gamma not equal 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 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 omega proportional to k(alpha),alpha epsilon R are direct consequences of fractional equations of motion.