Velocity participation factors in lowest 4 modes (2x triple)

Calculation of shear modulus for 2D, dense, bi-disperse, soft sphere granular mixture

On making an affine shear transformation of a square cell of granular, bi-disperse particles at equilibrium we get non-affine displacements to preserve equilibrium.

I divided the domain into many pieces. Below is an example of the domain divided into 4 pieces and their individual non-affine corrections plotted together for convenience.



In the following picture, the domain is divided into 400 pieces and the non-affine correction to the shear modulus is computed for each sub-domain and plotted.


NOTES: (i) these are computed using the formulas using the formulas in Craig's PRE of 2006. (ii) there were 2 critical bugs in the coding the potential & I had confused the diameter as the radius making the system super dense. (iii) the code that does this stuff is sitting on research-1 at:
/second_drive/asad/granularShearData/

Lattice averaged Green's function from the covariance matrix

Using the covariance matrix I computed the average value of the following components of the green's function for each reciprocal lattice vector (k): Grr, Gnn, Grn.
NOTE: (i) 'r' is along 'k' while 'n' is normal to 'r'.
(ii) name of the module: computeGreensFunction.py (in ..../disordered/2d-22/ on navier).

Plots of Grr, Gnn, Grn (respectively) follow:




NOTE: this program hasn't been thoroughly debugged.

2D hex, soft sphere, disordered system density of states

For my most disordered system (gamma=1.0), I assembled the covariance matrix for 4 million steps and diagonalized it for its eigenvalues. This system has 4200 particles. The plot above shows the convergence of 1/eigenvalues for various sampling intervals.

The plot below shows the convergence of histograms of the eigenvalues plotted above.



In the plot above, I have shown the DOS of various window sizes (shown in the legend). All eigenvalues were obtained by diagonalizing the covariance matrix obtained after sampling 4 million steps. For small frequencies, we expect linear DOS. Note the bimodal nature of the DOS. For comparison, below is a plot of the ordered (gamma=0.0) system's DOS.

100 plane, FCC 3D, mean square amplitude of displacement field FT onto plane waves

The plane waves are identical to those shown in the previous post.

The SLOPE of the linear part in the above plots was measured to be nearly -1.2 which was close to the expected 2*(2/3).

Fourier transfrom of 2D, disordered, soft-sphere, NVT displacement field

For my most disordered 2D system (gamma=1.0) I computed the average projection of displacement fields onto transverse plane wave traveling along the x direction. Essentially i computed the quantity:
U' = sum(n: 0-->N-1) {exp(2*pi*i*(n/N)*x)*Uy}

Here is an example of the real part of the plane wave with n=5 & N=60:The above is the 'cos' part, the 'sin' is merely shifted by half a fringe-width.

Now the following is a plot of , averaged over 4 million timesteps at T=0.0025.

The different curves are for different window sizes, the number in the legend denotes the number of particles along the horizontal axis (radius=1.0). On the vertical axis, the square amplitude is measured in units of particle size, while on the hor axis the wavevector is in unit on 1/particleSize. For comparison, above is the corresponding plot for the ordered system, gamma=0.0.

Melting in FCC, 3d, soft sphere crystals

I found that using LAMMPS' Nose-Hoover NVT setup the melting temperature was very low ( below 1E-5). To see this I simply tracked a particle for some time after starting the simulation. Why this is the case is unexplained especially since in 2D i have been getting reasonable statistics at T=0.0025.

Switching to NVE, after some short sampling at various T, it seemed that melting seemed occurred between T=0.05-0.1. I ran 2 simulations at T=0.01 and T=0.005. The following plots are the Y versus X coordinate of a SINGLE particle sampled at 10,000 timesteps from 0-4 million steps.
The MSD of the particle in the upper plot was HUGE ~27.5 !! The main contribution was ~= 26.2. It seems that T=0.005 is suitable for analysis.

NOTE: the good config (lower picture) is in the: fcc-3d/dumpfiles3/ folder and its 100 plane is extracted to the folder: fcc-3d/extractDump3/planeDump3/.