# Difference between revisions of "Random walk"

A random walk is a trajectory that is created from successive random steps. Depending on the dimension of the space the walk is performed in and the definition of randomness used for each step, different kinds of walks can be formulated.

Different spaces for a random walk include the one-dimensional space of integers (e.g. successively flipping a coin with values of $\pm 1$ and adding these up or the Fermi estimation), the plane with real coordinates (e.g. the drunkard's walk), or 3D euclidean space (e.g. Levy Flight or the Wiener process). More uncommon spaces include graphs (see the overview here) or groups in the mathematical sense (a short introduction can be found here).

Steps can be performed at defined time intervals or at random times, with a defined or random step length and each random component can be modified based on previous steps.

In the context of soft matter, the two most important applications of the concept of random walks are diffusion limited aggregation and the modeling of polymers as freely-jointed chains.

For a polymer chain undergoing a completely ramdom walk the end-to-end length $R$ scales with the number of segments $N$ like

$R \propto \sqrt{N}$

## Examples

### Perturbation Spreading in Many-Particle Systems: A Random Walk Approach

V. Zaburdaev, et al. showed that a perturbation traveling through a collection of many interacting particles in a non-dissipative medium (energy is conserved and not lost due to friction) propagates via a random walk. Zaburdaev specifically applied the "continuous-time random walk formalism" (CTRW) to the particles. The CTRW implies that particles travel at a constant speed $v_o$ and randomly change direction at "turning points."

## Physical Mechanisms for Chemotactic Pattern Formation by Bacteria

M. Brener et al. demonstrated that E. coli bacteria motion, with running mode and tumbling mode, can be modeled by simple equations. With this approach the author explained the swarm ring pattern and aggregates that bacteria form in different environmental conditions.