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 a power law.
However, they soon realized that this model of the particle's behavior did not provide acceptable results; the shape of the propagation fronts were not correct. They made an extension to the model that assumed that the medium was "active:" the particle interacted with the medium it was moving through, causing fluctuations of the particle's velocity. This behavior can be seen in Figure 1. Note that the particle could both gain and lose energy from its environment. Both the energy gain and loss processes were in balance so on average, the particles traveled at the specified speed <math>v_o</math>.
After making this change, the authors were able to obtain physical results for perturbations through the material. They found the perturbation fronts displayed "smooth, Gaussian-like profiles." They also applied their theory to various theoretical systems that provided reasonable results. Their assumption that they could model a perturbation through a non-dissipative material as a collection of particles undergoing a random walk was justified.