Path-integral approach to diffusion in random media

Random walk through a fertile site

We study the dynamics of random walks hopping on homogeneous hypercubic lattices and multiplying at a fertile site. In one and two dimensions, the total number N(t) of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate μ at the fertile site. When d>d_{c}=2, the number of walkers may remain finite forever for any μ; it surely remains finite when μ≤μ_{d}. We determine μ_{d} and show that 〈N(t)〉 grows exponentially if μ>μ_{d}. The distribution of the total number of walkers remains broad when d≤2, and also when d>2 and μ>μ_{d}. We compute 〈N^{m}〉 explicitly for small m, and show how to determine higher moments. In the critical regime, 〈N〉 grows as sqrt[t] for d=3, t/lnt for d=4, and t for d>4. Higher moments grow anomalously, 〈N^{m}〉∼〈N〉^{2m-1}, in the critical regime; the growth is normal, 〈N^{m}〉∼〈N〉^{m}, in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz., it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if d>2, there is again a critical multiplication rate, above which 〈N(t)〉 grows linearly (not exponentially) in time; when d≤d_{c}=2, the leading behavior is independent on μ and 〈N(t)〉 exhibits a sublinear growth.

Dynamics of perturbations in disordered chaotic systems

We study the time evolution of perturbations in spatially extended chaotic systems in the presence of quenched disorder. We find that initially random perturbations tend to exponentially localize in space around static pinning centers that are selected by the particular configuration of disorder. The spatiotemporal behavior of typical perturbations deltau(x,t) is analyzed in terms of the Hopf-Cole transform h(x,t) identical withlnmid R:deltau(x,t)mid R: . Our analysis shows that the associated surface h(x,t) self-organizes into a faceted structure with scale-invariant correlations. Scaling analysis of critical roughening exponents reveals that there are three different universality classes for error propagation in disordered chaotic systems that correspond to different symmetries of the underlying disorder. Our conclusions are based on numerical simulations of disordered lattices of coupled chaotic elements and equations for diffusion in random potentials. We propose a phenomenological stochastic field theory that gives some insights on the path for a generalization of these results for a broad class of disordered extended systems exhibiting space-time chaos.

Localization in disordered media, anomalous roughening, and coarsening dynamics of faceted surfaces

We study a surface growth model related to the Kardar-Parisi-Zhang equation for nonequilibrium kinetic roughening, but where the thermal noise is replaced by a static columnar disorder eta(x) . This model is one of the many representations of the problem of particle diffusion in trapping or amplifying disordered media. We find that probability localization in the latter translates into facet formation in the equivalent surface growth problem. Coarsening of the pattern can therefore be identified with the diffusion of the localization center. The emergent faceted structure gives rise to nontrivial scaling properties, including anomalous surface roughening in excellent agreement with an existing conjecture for kinetic roughening of faceted surfaces. In a wider context, our study sheds light onto the scaling properties in other systems displaying this kind of patterned surface.