# Brownian motion in disordered media

## Abstract

The Brownian motion in quenched disordered media is studied using random walk models and a Langevin approach. Focus is on highly ramified media that exhibits self-similarity over a wide range of length scales. Accurate estimates are obtained for the dynamical exponents for the blind and myopic ant random walks on precolation clusters in two and three dimensions at criticality and DLA (diffusion limited aggregation) clusters in two dimensions. The method exactly enumerates all Brownian paths from all starting points in a cluster. The scaling properties in the probability density for the position of the random walker are exploited to develop an approximation scheme for the mean squared displacement. A simple connection between the anisotropic properties of the diffusion process to the geometric structure of the clusters is established. Regular oscillations persist in the step correlation function for the myopic ant on bipartite clusters. In the regime of self-similarity, these oscillations decay as a power-law with an exponent given by half the spectral dimension. A formal mapping from a discrete time random walk to a continuous time master equation applicable to quenched disordered media is developed. The frequency dependent diffusion coefficient is expressed in terms of the step correlation function. Amore »

- Authors:

- Publication Date:

- Research Org.:
- Purdue Univ., Lafayette, IN (United States)

- OSTI Identifier:
- 7042373

- Resource Type:
- Miscellaneous

- Resource Relation:
- Other Information: Thesis (Ph.D.)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BROWNIAN MOVEMENT; CORRELATION FUNCTIONS; STATISTICAL MODELS; PARTICLE MOBILITY; RANDOMNESS; SCALING LAWS; STOCHASTIC PROCESSES; FUNCTIONS; MATHEMATICAL MODELS; MOBILITY; 661300* - Other Aspects of Physical Science- (1992-)

### Citation Formats

```
Jacobs, D J.
```*Brownian motion in disordered media*. United States: N. p., 1992.
Web.

```
Jacobs, D J.
```*Brownian motion in disordered media*. United States.

```
Jacobs, D J. Wed .
"Brownian motion in disordered media". United States.
```

```
@article{osti_7042373,
```

title = {Brownian motion in disordered media},

author = {Jacobs, D J},

abstractNote = {The Brownian motion in quenched disordered media is studied using random walk models and a Langevin approach. Focus is on highly ramified media that exhibits self-similarity over a wide range of length scales. Accurate estimates are obtained for the dynamical exponents for the blind and myopic ant random walks on precolation clusters in two and three dimensions at criticality and DLA (diffusion limited aggregation) clusters in two dimensions. The method exactly enumerates all Brownian paths from all starting points in a cluster. The scaling properties in the probability density for the position of the random walker are exploited to develop an approximation scheme for the mean squared displacement. A simple connection between the anisotropic properties of the diffusion process to the geometric structure of the clusters is established. Regular oscillations persist in the step correlation function for the myopic ant on bipartite clusters. In the regime of self-similarity, these oscillations decay as a power-law with an exponent given by half the spectral dimension. A formal mapping from a discrete time random walk to a continuous time master equation applicable to quenched disordered media is developed. The frequency dependent diffusion coefficient is expressed in terms of the step correlation function. A generalized Langevin equation with a power-law friction kernel adequately describes the anomalous diffusion. A random walk model with persistence is introduced for analysis of the electrical conductivity of disordered composite materials. Features of the model include a phenomenological parameterization with a high frequency cut off, a Drude-like mobility in homogeneous media, and relative carrier concentrations and mobilities are treated independently. The interplay between the mean free paths of each constituent, the disorder length scale, and the correlation length of the disorder is investigated. Scaling theories utilizing only conductivities are inadequate.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1992},

month = {1}

}