Spatiotemporal flux memory in nondiffusive transport
  Anomalous diffusion constitutes a relation between tracer flux and tracer density gradient that is inherently nonlocal in space and/or time. Previous studies emphasize the non-Gaussian character of the tracer distribution that arises from adjusted constitutive relations but did not investigate the flux-gradient memory itself. Here, we present a universal analytic framework that enables systematic characterisation of nonlocality in a wide variety of transport regimes. A generalised diffusivity kernel $D^{\ast}$ fully embodies the spatiotemporal flux memory with respect to the gradient. An extension of the flux-gradient relation for subdiffusive transport is also proposed. Several conservation and invariance properties can be deduced, including that Poissonian flight processes have no flux memory in time while fractional time diffusion has no flux memory in space. We derive analytical expressions for $D^{\ast}(x,t)$ in several types of anomalous transport dynamics that are commonly encountered in practice, being fractional diffusion equations, tempered L\'evy superdiffusion, and tempered fractional time diffusion. This detailed knowledge of the shape and nature of the flux memory, and the corresponding length and time scales over which nonlocal effects are physically important, remain completely hidden in conventional analyses based on tracer distributions or flux-gradient diagrams. Practical capabilities include the interpretation of microscale heat superdiffusion experiments. Overall, the theory can serve as a valuable framework for anomalous transport dynamics across multiple disciplines. 
Bjorn Vermeersch, Ali Shakouri
cond-mat.stat-mech