November 01 2002

 Flatness of 603 atoms at different temperature levels (continued)

According to Dr. Dellago's suggestion, the global minimum  of flatness can be solved directly by applying undetermined Lagrange multipiler

d/dn( sum( (A(x[i]-px) + B(y[i]-py) + C(z[i]-pz) )2  - lamda * (A2+B2+C2) ) = 0

where n is the vector orthogonal to the plane, n = (A,B,C). Take the derivatives, we have three linear functions with the variables of (A,B,C) and a multiplier of lamba. It's a three dimensional eigenvalue problem which can be solved by Jacobi Transformations (Numerical Recipies in C, second edition, W.H.Press, Page 463).

The last configuration of coolings at different temperature went through Andersen Thermostat for 106 steps and 100 configurations have been saved for each temperature. The statistics of the flatness is based on those configurations. The averages and standard deviations of the flatness have also been calculated.

The histograms include 100 bins.

Averages and standard deviations of flatness


Flatness distribution of 100K configurations quenched by conjugate gradient method

Flatness distribution of 1500K configurations quenched by conjugate gradient method

Comparison of flatness distributions at 100K, 600K and 1000K

Flatness distributions at different temperatures

The data files of all temperature levels can be downloaded here to achieve more comparisons on the same plot.

1) 100K

2) 200K

3) 300K

4) 400K

5) 500K

6) 600K

7) 700K

8) 800K

9) 900K

10) 1000K

11) 1500K