反常扩散与Noether第二定理
Anomalous diffusion and Noether's second theorem
摘要：
尽管守恒流的维数仅由维数分析确定（因此没有反常维数），但自然界中有大量反常扩散的例子，其中$L\propto t^γ$与$γ\ne 1/2$，以及热传导率发散为$L^α$的热传输。除了破坏洛仑兹不变性之外，这类问题的真正共同点是潜在守恒电流的反常维度，从而违反了场论的基本原则。我们在这里表明，用于描述这种异常现象的唯象非定域运动方程都遵循诺瑟第二定理产生的洛伦兹破坏规范变换。推广导致家庭的扩散和热输运方程，系统化如何非本地规范变换生成更一般形式的菲克和傅立叶定律的扩散和热输运，分别。特别地，形式为$ω\propto k^α$、$α\in \mathbb R$的相关Goldstone模式是分数运动方程的直接结果。
英文摘要：
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 $L\propto t^γ$, with $γ\ne 1/2$, 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 non-local equations of motion that are used to describe such anomalies all follow from Lorentz-violating gauge transformations arising from Nöther'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 $ω\propto k^α$, $α\in \mathbb R$ are direct consequences of fractional equations of motion.
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