Dynamics of Fronts, discrete models

Instructions

This Java applet (If you don't see anything above, please install
Java 1.4) models the dynamics of a front moving through a noisy medium. It contains two discrete models, that simulate respectively the KPZ (Karder-Parisi-Zhang) equation and the Sneppen model (PRL, 1992). The discrete KPZ simulation for an interface h(x,t) is simulated by at each time-step:
  1. select a random point x along the interface.
  2. increase the height of the interface at this point h(x)->h(x)+1
  3. increase h subsequently at neighbor points y=x+1, y=x+2... such that h(y)-h(y+1) is never larger than 1. Do the same for y=x-1,x-2, etc.
The Sneppen model concerns an interface h(x,t) and a random underlying medium modeled by random numbers eta(x,h). The model is by definition discrete, at each time-step:
  1. select the point x along the interface which have the smalles random number.
  2. increase the height of the interface at this point h(x)->h(x)+1
  3. increase h subsequently at neighbor points y=x+1, y=x+2... such that h(y)-h(y+1) is never larger than 1. Do the same for y=x-1,x-2, etc. For all points where h is changed, the associated random number is changed to a new one in the interval [0:1]
Philosophy
Dynamics of fronts present the simplest spatio-temporal model systems. The dynamics describes a non-equilibrium process where old states (represented by a height variable h(x) are replaced with new fronts. There is no consideration of detailed balance, and as a result the system reaches steady state dynamical state where particular modes of excitation across the system. In the applet we show the two, philosophically quite different approaches to front dynamics: One based on annealed noise, and another based on quenched noise and extremal dynamics.

In the annealed case (KPZ), the noise of the update changes from time to time, even at the same point (x,h(x)). As a result the front always move a little on all points.

In the quenched case, then when there is no movemenet at a given point (x,h(x)) at some time, then there is typically no movement either at the subsequent timestep. The fronts gets pinned. This is in particular captured by the exremal dynamics of the Sneppen interface model.

References:

M. kardar, G. Parisi and Y.-C. Zhang, Physical Review Letter 56 889 (1986).

K. Sneppen, Physical Review Letter 70 3833 (1992).

M. Paczuski, S. Maslov and P. Bak, Physical Review E53 414 (1996).