Distance matrix approach applied to a kinetically constrained lattice gas

This page serves as an archive for accompanying movies to the recent publication 'Dynamic Facilitation Explains Democratic Particle Motion of Metabasin Transitions', L.O. Hedges and J.P. Garrahan arXiv:0706.0902. Please refer to the paper for specific details of each movie.

Distance Matrix

The distance matrix (DM) provides a convenient means of exploring the microscopic nature of particle motion within a sub-region of an extended system and is defined as follows

where k_i(t) specifies the number of events, or "kinks", at site i after time t. Shown below are two example DMs for the (2)-TLG at a density of 0.79. The left hand panel illustrates a particular matrix for a 5 x 5 sub-lattice, the right a larger system from the same run, 10 x 10 sites in size. Both images illustrate the heterogeneous nature of dynamics in the (2)-TLG model. For extendend periods of time the system stays close to one region of configuration space, indicated by the dark blue square-like regions, before eventually finding a pathway to a new region. The transitions are less well defined for a larger system since multiple independent particle rearrangements obscure the observation of a single transition.

Click on each figure to view a full size image.

Sub-size = 5 x 5

Sub-size = 10 x 10

Particle Trajectories

To further illustrate the dynamical feature highlighted using the distance matrix approach it is useful to directly analyse the particle motion along the trajectory. The following movies indicate the position of particles within the sub-region at 200 time points. Comparing the motion to the corresponding DM above it is clear that the blue square-like regions correspond to periods when the majority of particles within the region are blocked and remain frozen in position. A rapid burst of motion results from a series of unlocking events that temporarily allows a significant number of particles to move a substantial amount in a relatively small amount of time. As mobility passes the system is left in another fully or partially blocked configuration.

Click on a movie to view or right click to download.

Sub-size = 5 x 5

Sub-size = 10 x 10

Average squared kinks

Consider the distance matrix along a diagonal. For a given super (or sub) diagonal at a chosen offset the average squared kinks shows extended minima of little activity and sharp peaks of sudden motion. At the peaks the fraction of sites within the sub-region that experience a high degree of activity (particles entering or leaving) is substantial, i.e. a rapid rearrangement of a large number of particles. This is illustrated in the following figure (corresponding to the 10 x 10 subsystem), please refer to the reference for further information.

Click on the figure to view a full size image.

Average squared kinks

Particle field

The right hand panel shows snapshots of the particle field at the times inidcated in the left hand figure. Coloured black are those particles which have moved in the time interval. At a time corresponding to one of the peaks in the average squared kinks there is a large rearrangement of a significant number of particles within the sub-region.