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
Keywords: Random Walk
The authors of this paper examined 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 it.
The authors knew that a propagation would travel through the material at a finite velocity. Typical diffusion equations, however, imply infinite propagation speeds; the authors consequently knew this would not accurately describe the system's behavior. Consequently, they 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 would randomly change the direction of its motion at "turning points." 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 physical systems that provided reasonable results (i.e. a hard point gas). Their assumption that a perturbation traveling through a non-dissipative material could be described as a collection of particles undergoing a random walk was justified.
I personally thought this work was quite interesting. I have often wondered how information would travel through a many body system where energy is conserved. I now know that it obeys a random walk. This paper is particularly powerful because it provides insight into the dynamics of a large number of interacting particles, a topic we encounter all the time in physics. I hope that in the future the authors apply this work to a real experimental system.