Results for 2D OFC model

For the 2D OFC model the parameters we are using are K_C = 100, K_L = 1, \sigma_F = 1, and \sigma_R is random distributed in the range 0.0 to 0.5 . These parameters are the same as M. Anghel et al. (cond-mat/0002459).

Results for the 2D OFC model with zero loading velocity

• Cluster size distribution for N = 256 x 256 at R = 30 (see plot) after decomposing into three ranges which are similar as M. Anghel's. The exponents of the first two scaling range are close to 1.5. More events need to get for good statistics for the third range corresponding large events.
• The evolution of corresponding quantities for two different kind of events are shown in the plots in the table below. The center of mass is calculated from all sites failed already at that time.

  size of event	s = 3096	s = 46736	s = 50315
  graphic stress distribution at (t = 0)	plot	plot	plot
  position of initiator (t =0)	x =61, y = 95	x =207, y = 70	x = 44, y = 195
  failed sites from t = 1 to t = 10	plot	plot	plot
  number of failed sites at time t	plot	plot	plot
  total # of failed sites for t = 0 to t	plot	plot	plot
  radius of gyration	plot	plot	plot
  center of mass (all failed sites till t)	plot	plot	plot
  avrg Stress at t = 0	plot	plot	plot
  avrg Stress of system at time t	plot	plot	plot
  avrg Stress within a radius (t =1)  of center of mass (all failed sites till t)	plot	plot	plot
  avrg Stress within a radius (t =2)	plot	plot	plot
  avrg Stress within a radius (t =3)	plot	plot	plot
  avrg Stress within a radius (t =4)	plot	plot	plot
  avrg Stress within a radius (t =5)	plot	plot	plot
  avrg Stress within a radius (t =6)	plot	plot	plot
  avrg Stress within a radius (t =7)	plot	plot	plot
  avrg Stress within a radius (t =8)	plot	plot	plot
  avrg Stress within a radius (t =9)	plot	plot	plot
  avrg Stress within a radius (t =10)	plot	plot	plot
  avrg Stress within a radius (t =11)	plot	plot	plot
  avrg Stress within a radius (t =12)	plot	plot	plot
  avrg Stress within a radius (t =13)	plot	plot	plot
  avrg Stress within a radius (t =14)	plot	plot	plot
  avrg Stress within a radius (t =15)	plot	plot	plot
  avrg Stress within a radius (t =16)	plot	plot	plot
  avrg Stress within a radius (t =17)	plot	plot	plot
  avrg Stress within a radius (t =18)	plot	plot	plot
  avrg Stress within a radius (t =19)	plot	plot	plot
  avrg Stress within a radius (t =20)	plot	plot	plot
  avrg Stress within a radius (t =30)	plot	plot	plot
  avrg Stress within a radius (t =40)	plot	plot	plot
  avrg Stress within a radius (t =50)	plot	plot	plot
  avrg Stress within a radius (t =60)	plot	plot	plot
  avrg Stress within a radius (t =70)	plot	plot	plot
  avrg Stress within a radius (t =80)	plot	plot	plot
  avrg Stress within a radius (t =90)	plot	plot	plot
  avrg Stress within a radius (t =100)	plot	plot	plot
  avrg Stress within a radius (t =110)	plot	plot	plot
  avrg Stress within a radius (t =120)	last plot at t =118	plot	plot
  avrg Stress within a radius (t =130)		plot	plot
  avrg Stress within a radius (t =140)		plot	plot
  avrg Stress within a radius (t =150)		plot	plot
  avrg Stress within a radius (t =200)		plot	plot
  avrg Stress within a radius (t =250)		plot	plot
  avrg Stress within a radius (t =300)		plot	plot
  avrg Stress within a radius (t =350)		plot	plot
  avrg Stress within a radius (t =400)		plot	plot
  avrg Stress within a radius (t =450)		plot	plot
  avrg Stress within a radius (t =500)		last plot at t = 465	plot
  avrg Stress within a radius (t =533)			last plot at t = 533

• I also calculated the connectedness length (\xi) of clusters as a function of interation range R and found \xi \sim R^1.5. The connectedness length is a kind of measurement of correlation length of a system. It is defined by \xi^2 = \sum_s{s (s-1) n_s Rg_s^2}/ \sum_s{s(s-1) n_s}, where Rg_s is the radius of gyration of cluster with size s, and n_s is the cluster size distribution.  Note that I only use the clusters within the 1.5 exponent region at the cluster size distribution.  The plot shows that the connectedness length of clusters scales as the interaction range R with exponent of 1.49 for a system with N = 256 x 256. The next plot also shows similar behavior for a bigger 512 x 512 system. From the plot of N = 512 x 512 we can see that rescaling the amplitude of random noise has no obvious difference. I rescale the amplitude of random noise by fixed the random_amp*R to 15. 
• According to Olami, Feder, and Christensen (PRL, 68, 1244 (1992) ), we can define a parameter \alpha , which is the fraction of stress drop of a site that can be transfered to its neighbors. Hence, 1 - \alpha of stress drop is dissipated after a site failed. I calculated the variance of stress added, stress dissipated, and stress after an event as a function of dissipated amount 1-\alpha, as shown in the table as follows.
  

  	N = 256 x 256, R = 20
  variance of stress added	plot , no scaling.
  variance of stress dissipated	plot, a scaling region with an exponent 0.5 roughly.
  variance of stress before and after an event	plot, no obvious scaling region.
  mean stress before and after an event	plot , no obvious scaling region.
  connectedness length	plot, a scaling region with 0.5 exponent.

• According to spinodal nucleation theory, there might be four different kinds of clusters. I can verify there four different kinds of clusters as M. Anghel in long -range  RJB  model.  I also studied how these different kinds of clusters grow by plotting the total number of failed sites as  a function of  radius of gyration of cluster as shown in table below for the system with N = 256 x 256, R = 30 and 30 x 10^6 events. We can find that there is no apparent difference at early times, while at the late time there is a much quicker filling in process involved for the arrested nucleation events and  breakout events. This filling-in process results in almost compact clusters and produces the deviation from 1.5 scaling at large events end of the frequency-size distribution.  We also checked the evolution of the average stress of system during these different kinds of events as shown in the plot of average stress evolution . It seems that the system is in a higher stress state before a larger event happens and a breakout event will make the system drop to a much lower stress state.
  

  breakout clusters	arrested nucleation clusters	failing nucleation clusters	fluctuation clusters
  s = 50315	s = 10184	s = 3096	s = 250
  s = 49934	s = 10115	s = 2003	s = 224
  s = 49183	s = 10079	s = 1973	s = 100
  s = 48520	s = 10006	s = 1077	s = 80

• The table below shows the inverse metric of the stress just after an event at different coarse-graining levels as a function of loading updates for N = 256x256 and 1024x1024 with R = 20 and different kinds of percentage of defect sites. A defect site means that it never fails with zero stress and hence never accepts stress from or transfers stress to its neighbors.
  

  	0%	2%	10%
  N = 256 x 256	see Fig.	see Fig.	see Fig.
  N = 1024 x 1024			see Fig.

Results for the 2D OFC model with nonzero loading velocity (to be added)

Updated on 16 November, 2005.