Perturbation Spreading in Many-Particle Systems: A Random Walk Approach
Original entry by Bryan Weinstein, Fall 2012
Authors: V. Zaburdaev, S. Denisov, and P. Hanggi
The authors of this paper examine the propagation dynamics of a perturbation traveling through a non-dissipative medium (energy is conserved and not lost due to friction). They modeled the medium as a collection of many interacting particles. As energy was conserved, they were able to apply Hamiltonian dynamics to analyze the system.
The authors knew that a propagation would travel through the material at approximately a finite velocity. Typical diffusion equations, however, imply infinite propagation speeds; the authors consequently knew this would not accurately describe their system's behavior. Consequently, the authors decided to model the perturbation kinetics by treating the particles as if they were undergoing a random walk.
Specifically, the authors applied the "continuous-time random walk formalism" (CTRW) to the particles. This implied that a particle would travel at a constant speed <math>v_o</math> and at turning points would randomly change the direction of its motion. The time between collisions was drawn from a Probability Density Function (PDF) described by