Anomalous diffusion with under- and overshooting subordination: a competition between the very large jumps in physical and operational times

Stochastic tools hidden behind the empirical dielectric relaxation laws

The paper is devoted to recent advances in stochastic modeling of anomalous kinetic processes observed in dielectric materials which are prominent examples of disordered (complex) systems. Theoretical studies of dynamical properties of 'structures with variations' (Goldenfield and Kadanoff 1999 Science 284 87-9) require application of such mathematical tools-by means of which their random nature can be analyzed and, independently of the details distinguishing various systems (dipolar materials, glasses, semiconductors, liquid crystals, polymers, etc), the empirical universal kinetic patterns can be derived. We begin with a brief survey of the historical background of the dielectric relaxation study. After a short outline of the theoretical ideas providing the random tools applicable to modeling of relaxation phenomena, we present probabilistic implications for the study of the relaxation-rate distribution models. In the framework of the probability distribution of relaxation rates we consider description of complex systems, in which relaxing entities form random clusters interacting with each other and single entities. Then we focus on stochastic mechanisms of the relaxation phenomenon. We discuss the diffusion approach and its usefulness for understanding of anomalous dynamics of relaxing systems. We also discuss extensions of the diffusive approach to systems under tempered random processes. Useful relationships among different stochastic approaches to the anomalous dynamics of complex systems allow us to get a fresh look at this subject. The paper closes with a final discussion on achievements of stochastic tools describing the anomalous time evolution of complex systems.

Clustered continuous-time random walks: diffusion and relaxation consequences

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

Subdiffusion of a sticky particle on a surface

Conventional diffusion (ΔR2(t))=2Dt gives way to subdiffusion (ΔR2(t))∼t(μ), 0<μ<1 when the waiting time distribution φ(τ) is nonintegrable. We have studied a model system, colloidal particles functionalized with DNA "sticky ends" diffusing on a complementary coated surface. We observe a crossover from subdiffusive to conventional behavior for (ΔR2(t)) and φ(τ) as temperature is increased near the particle-surface melting temperature consistent with a simple Gaussian distribution of sticky ends. Our results suggest that any system with randomness in its binding energy should exhibit subdiffusive behavior as it unbinds. This will strongly affect the kinetics of self-assembly.