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Characterizing

*N*-dimensional anisotropic Brownian motion by the distribution of diffusivities

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10.1063/1.4828860

### Abstract

Anisotropic diffusion processes emerge in various fields such as transport in biological tissue and diffusion in liquid crystals. In such systems, the motion is described by a diffusion tensor. For a proper characterization of processes with more than one diffusion coefficient, an average description by the mean squared displacement is often not sufficient. Hence, in this paper, we use the distribution of diffusivities to study diffusion in a homogeneous anisotropic environment. We derive analytical expressions of the distribution and relate its properties to an anisotropy measure based on the mean diffusivity and the asymptotic decay of the distribution. Both quantities are easy to determine from experimental data and reveal the existence of more than one diffusion coefficient, which allows the distinction between isotropic and anisotropic processes. We further discuss the influence on the analysis of projected trajectories, which are typically accessible in experiments. For the experimentally most relevant cases of two- and three-dimensional anisotropic diffusion, we derive specific expressions, determine the diffusion tensor, characterize the anisotropy, and demonstrate the applicability for simulated trajectories.

© 2013 AIP Publishing LLC

Received 29 August 2013
Accepted 19 October 2013
Published online 11 November 2013

Acknowledgments: We thank Sven Schubert for stimulating discussions and valuable suggestions. We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) for funding of the research unit FOR 877 “From Local Constraints to Macroscopic Transport.” We also appreciated valuable suggestions of the anonymous referees, which helped to improve the paper considerably.

Article outline:

I. INTRODUCTION

II. DEFINITIONS

A. Anisotropic diffusion

B. Distribution of diffusivities

C. Moments

III. PROPERTIES OF THE DISTRIBUTION OF DIFFUSIVITIES FOR HOMOGENEOUS ANISOTROPIC BROWNIAN MOTION

A. Distribution of diffusivities

B. Characteristic function, cumulants, and moments

C. Asymptotic decay

D. Reconstruction of the diffusion tensor

IV. PROJECTION TO A *M*-DIMENSIONAL SUBSPACE

V. SPECIFIC SYSTEMS

A. Two-dimensional systems

1. Limiting cases

2. Reconstruction of **D**

B. Three-dimensional systems

1. Limiting cases

2. Curvature of the distribution of diffusivities

3. Reconstruction of **D**

VI. CONCLUSIONS

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2013-11-11

2014-04-17

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