A Paradigm for Anomalous Transport in Disordered Media
Nature as well as modern technology presents us a variety of disordered materials ranging from composites over gels to the inner structure of biological cells. We have shown that the transport properties of microscopic particles in such materials are directly connected to strong structural heterogeneities resulting from the presence of a broad range of length scales. It is known that these heterogeneities lead to a dramatic slowing down of transport processes due to a fractal dynamic behavior which has to be contrasted to normal diffusion. The latter being tightly bound to Brownian motion, is the dominant transport process in homogeneous materials and can be characterized by single length and time scales. Heterogeneous materials, however, lack such a single length scale, and the new fractal transport law depends on non-integer, i.e. fractal, powers of time and length.
Our analysis is based on investigations of the three-dimensional Lorentz model, a paradigm for transport in disordered media. Its connection to continuum percolation has been a longstanding open question. Our Molecular Dynamics simulation provide the first unambiguous evidence for an intimate connection between these models. Contrary to a similar scenario of anomalous transport in percolating lattices, the dynamic behavior can directly be inferred from the underlying geometric structures. It is demonstrated that the transport properties are dictated by the broad distribution of channel widths in the medium where the particles have to squeeze through. Our data corroborate a hyperscaling relation that connects dynamic and geometric critical exponents. In particular, we show the validity of a generalized dynamic scaling theory with two divergent length scales, and discuss corrections to scaling.