Random walks and diffusion

This section discusses the concept of diffusion at the microscopic and macroscopic levels. It includes applications of random walks to foraging behaviors and to bacterial motion.

• Diffusion at the microscopic level
• Diffusion at the macroscopic level
• Bacterial motion
• Foraging behaviors

Diffusion at the microscopic level

Brownian motion

The motion of molecules in a fluid, such as molecules of dye in water, is, at non-zero temperature, typically random. It is an example of Brownian motion. What we call diffusion at the macroscopic level is the consequence of random motion at the microscopic level.

Random walk on the plane

To make this more intuitive, consider a particle undergoing a random walk on the plane: each step has a given length l, but can be taken in a random, uniformly distributed, direction. After N steps, or equivalently a time T = N δt, where δt is the (constant) time elapsed between any two consecutive steps, the particle will be at a distance L from its original position, such that L² ∝ T. This relationship should be understood in a probabilistic sense: the expected value of L² is proportional to T.

Simulation

The MATLAB GUI Diffusion simulates the random motion of M non-interacting particles on a two-dimensional grid - so that each particle can only go up, down, left or right, with equal probability. It illustrates how the relationship L² ∝ T, where T = N δt, depends on the number N of steps taken, and on the number M of particles.

Web resources on Brownian motion

The first two references below are historical perspectives on Einstein's work on Brownian motion. One of the links at the end of D. Cassidy's article points to a simulation of Brownian motion. The last reference is a course module that considers applications of Brownian motion to nanotechnology.

• Einstein's random walk, by M. Haw, University of Edinburgh (posted by Physicsworld.com)
• Einstein on Brownian motion, by D. Cassidy (posted by the Center for History of Physics)
• Brownian motion module, by J. Holden and K. Kelly, Rice University

Just-in-time mathematics

• Elementary probability theory
• Wiener process

Diffusion at the macroscopic level

The heat equation

The heat equation is a partial differential equation that describes the evolution, in space and time, of a diffusive macroscopic quantity. If Q is diffusing, the heat equation for Q is phenomenologically derived by expressing the conservation of Q and applying Fick's law, which assumes that the flow of Q is in a direction opposite to that of its gradient. The scaling properties of the heat equations are such that X² ∝ T, where X is a characteristic length and T is a characteristic time.

Reaction-diffusion equations

The same derivation may be applied to quantities (such as concentrations of chemicals) that not only diffuse, but also change in time through local dynamical processes. The resulting models are reaction-diffusion equations, which are used to describe Turing patterns as well as the dynamics of unstable diffusive interfaces.

Simulations

The MATLAB GUI Heat Equation on the Whole Line solves this equation in one dimension, using the Fourier transform.

Web resources on reaction-diffusion equations

There is a vast literature on reaction-diffusion equations, including research papers and textbooks. Below, we only mention Turing's original paper and an article by D. Kessler and H. Levine on the instability of diffusive fronts.

• The chemical basis of morphogenesis, by A.M. Turing
• Fluctuation-induced instabilities, by D.A. Kessler and H. Levine

Just-in-time mathematics

• Partial differential equations: diffusion

Bacterial motion

Random walk motion of flagellated bacteria

Flagellated bacteria move by a succession of runs and tumbles. This dynamics, which happens at a mesoscopic scale (smaller than our macroscopic scale but much larger than the atomic scale), is analogous in nature to a random walk, and can therefore be described as a diffusive process.

Macroscopic description

Because of the above, it is thus natural to describe bacterial colonies in terms of reaction-diffusion equations at the macroscopic level. However, whether normal or anomalous diffusion applies depends on the microscopic characteristics of the bacterial random walk.

Chemotaxis

If the bacteria are chemotactic (for instance to food), then the outcome is a biased random walk, which is modeled, at the macroscopic level, by adding an advection term to the corresponding reaction-diffusion equation.

Web resources on bacterial motion

The first three links point to experimental articles discussing the motion of flagellated bacteria. The next two concern chemotaxis. The last article discusses microscopic (at the level of individual bacteria) and macroscopic (at the level of populations of bacteria) aspects of bacterial motility.

• Motile Behavior of Bacteria, by H. Berg
• Movie: Runs and tumbles, from H. Berg's web site, Harvard University
• Marvels of bacterial behavior, by H. Berg
• The gradient-sensing mechanism in bacterial chemotaxis, by R.M. Macnab and D.E. Koshland
• Biased random walk by stochastic fluctuations of chemoattractant-receptor interactions at the lower limit of detection, by P.J.M. van Haastert and M. Postma
• Random motility of swimming bacteria: Single cells compared to cell populations, by B.R. Phillips, J.A. Quinn, and H. Goldfine

Just-in-time mathematics

• Partial differential equations: advection

Foraging behaviors

Anomalous diffusion

What would happen if a particle performing a random walk got trapped, from time to time, in some of the regions it is visiting? One would expect the mean square displacement after time T to be shorter than what would correspond to a purely diffusive process. This is called subdiffusion, since L² grows more slowly, as a function of T, than a linear function.

Similarly, if the length of each step has a broad distribution (or equivalently if particles are likely to move in the same direction for consecutive steps of constant duration δt), then L² will grow faster than a linear function of T, and superdiffusion will be observed.

Anomalous (sub- or super-) diffusion is now commonly seen in nature (as for instance discussed in the news article by J. Klafter and I. Sokolov listed below).

Applications to animal foraging behaviors

How do animals look for food? Is it simple diffusion, or should they alternate local random walks with longer excursions that would take them from their current location to more distant places? Once they have found a good patch, should they stay put for a while, or keep moving? Could these patterns be described in terms of anomalous diffusive processes?

Do animal search patterns provide the most effective ways to look for resources? If so, could their approach be used to improve spatial search techniques?

Some of these questions are addressed in the references listed below. It is important to note how ideas from physics combine with mathematical concepts to describe ecological behaviors, and how what is learned in ecology can, in turn, pose interesting mathematical questions and lead to technological applications.

Web resources on anomalous diffusion and foraging behaviors

The first three links point to news articles describing our understanding of animal foraging techniques. In particular, it is explained how incomplete data sets led to debatable conclusions on the importance of Lévy flights in ecology.

• Anomalous diffusion spreads its wings, by J. Klafter and I.M. Sokolov
• How animals find things
• Do wandering albatrosses care about math?, by J. Travis

The three references below are research papers. O. Bénichou et al. describe optimal ways of foraging for food and possible applications to search and rescue techniques; A. Edwards et al. critique previous published work on Lévy flights and present a revised analysis of the corresponding data; C. Rhodes and A. Reynolds discuss situations where they believe that Lévy flights still apply to ecology.

• Optimal search strategies for hidden targets, by O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez
• Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer, by A.M. Edwards et al.
• The influence of search strategies and homogeneous isotropic turbulence on planktonic contact rates, by C.J. Rhodes and A.M. Reynolds

Just-in-time mathematics

• Anomalous diffusion
• Partial differential equations: fractional derivatives