Random walk

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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 <math>\pm 1</math> 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 <math>R</math> scales with the number of segments <math>N</math> like

<math>R \propto \sqrt{N}</math>


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 <math>v_o</math> 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.

Keyword in references:

Cationic Nanoparticles Stabilize Zwitterionic Liposomes Better than Anionic Ones

Physical Mechanisms for Chemotactic Pattern Formation by Bacteria

Relationship between cellular response and behavioral variability in bacterial chemotaxis

Fine-tuning of chemotactic response in E. coli determined by high-throughput capillary assay

Sensorimotor control during isothermal tracking in Caenorhabditis elegans

Statistical dynamics of flowing red blood cells by morphological image processing

Asymmetric network connectivity using weighted harmonic averages

Fast Microtubule Dynamics in Meiotic Spindles Measured by Single Molecule Imaging: Evidence that the Spindle Environment does not Stabilize Microtubules

Rise of the Source-Sink Model

Non-Hermitian localization and population biology

Single molecule statistics and the polynucleotide unzipping transition

Intracellular transport by active diffusion

Non-random-coil behavior as a consequence of extensive PPII structure in the denatured state

Shear Unzipping of DNA

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

Langevin Dynamics Deciphers the Motility Pattern of Swimming Parasites

Mechanical Response of Cytoskeletal Networks

Origin of de-swelling and dynamics of dense ionic microgel suspensions