Transport and Diffusion in Disordered Media
Oskari H. Ajanki · Työväentutkimus Vuosikirja · 2012
The spreading of energy and other locally conserved quantities, such as momentum, can in principle be described in terms of fundamental physics at the level of atoms. On the other hand, these phenomena can also be often described by much more simple phenomenological transport equations at the macroscopic scale. When available, these two alternative descriptions are usually mathematically very different, and appear to be even contradictory in certain aspects. In this thesis mathematical methods which are aimed at deriving transport equations from microscopic descriptions are presented. In particular, these methods help to understand the effects of microscopic disorder, such as impurities and imperfect lattice structures, on the formation of transport equations. The new results concern two disordered microscopic models, the presentations and proofs of which, form two separate parts of the thesis. In the first part, the heat conductance of a chain formed by mechanical oscillators attached to their respective nearest neighbors by harmonic springs is studied. The key result is the derivation of an exact scaling law for the stationary energy current through the system as a function of the length of the chain, when both ends of the chain are kept at constant but different temperatures. This proves the numerically observed anomalous scaling law of the so-called Casher-Lebowitz model to be valid, as has been conjectured already in 1970s. The theorem requires harmonic springs, but is otherwise universal: it does not depend on the exact law of the random masses, etc. The proof itself clearly exposes the mechanisms by which even a slightest randomness of the masses cause the heat conductance to vanish as the length of the chain is increased. In the second part of the thesis a different class of models in which 'energy' spreads by independent random walkers is studied. In these simple systems, the disorder is modeled by assuming that the space- and time-dependent transition probabilities of the walkers are also random. The main result concerning these models is that the energy, or equivalently the probability distribution of the location of the random walkers, spreads diffusively, provided the transition probabilities are sufficiently close to their space- and time-independent averages. In particular, the transition probabilities can be strongly correlated in space, as long as the correlations in time decay in an integrable manner. The result relies on novel renormalization group techniques, which can be applied for more complicated microscopic models as well From a purely mathematical point of view, the thesis concerns spatially extended stochastic processes and their universal scaling limits. Roughly speaking, the results can be considered as alternatives and/or extensions of well known central limit theorems.