https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

Size-Distribution Analysis of Macromolecules by Sedimentation Velocity Ultracentrifugation and Lamm Equation Modeling

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

A persistent random walk can be regarded as a multidimensional Markov process. The bias-free telegraphers equation is ∂ 2 p ∂t 2 + 1 T ∂p ∂t =v 2 ∇ 2 p . It can be regarded as interpolating between the wave equation (T→∞) and the diffusion equation (T→0). Previously, it has found application in thermodynamics (cf. the review in Rev. Mod. Phys. 61 (1989) 41; 62 (1990) 375). More recent applications are reviewed in the present article.

Some applications of persistent random walks and the telegrapher's equation

Stochastic Processes in Chemical Physics: The Master Equation

There is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences. In this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the National Institutes of Health on May 6–8, 1985. This review will focus on (1) the Smoluchowski model for reaction rates together with its extension by Onsager, (2) first passage time formalism for discrete and continuous master equations and Fokker-Planck equations, (3) the Kramers model and its extensions, (4) diffusion in the presence of trapping centers.

Overview of theoretical models for reaction rates

Spectroscopic parameters for computation of van der waals forces

We analyze transport properties of a random walk on a comb structure, which serves as a model for a random walk on the backbone of a percolation cluster. It is shown that the random walk along the x axis, which is the analog of the backbone, exhibits anomalous diffusion in that 〈 x 2 ( n )〉 ∼ n 1 2 , and the expected number of x sites visited is proportional to n 1 4 for large n . The distribution function is found to be a two-dimensional Gaussian. If a field in the x direction, so that diffusion is asymmetric, the expected displacement is found to be asymptotically proportional to n 1 2 .

George H. Weiss

Papers

Some applications of persistent random walks and the telegrapher's equation

Stochastic Processes in Chemical Physics: The Master Equation

Overview of theoretical models for reaction rates

Spectroscopic parameters for computation of van der waals forces

Some properties of a random walk on a comb structure