Given Answer in Python Code: Pls given full answer not code outline or guide. Investigate the difference in the time dependent autocorrelation function of velocities, $($t$),$ of $3$ Lennard$-$Jones systems with N$=32$ with periodic boundary conditions at $3$ densities. The system are initiated with the atoms placed in the sites of a face$-$centered$-$cubic $($fcc$)$ lattice consistent with a$=1\mathrm{.}59,$ a$=1\mathrm{.}7,$ a$=1\mathrm{.}8.$ The temperature to equilibrate the systems will be around T$=0\mathrm{.}7.$ Follow the same requirements as the image concerning the system parameters. Consider the system of homework $4$ with $N=32$ atoms interacting via the Lennard$-$Jones

potential with mass $===1,$ and Boltzmann's constant $k_B=1.$ The potential energy of the

system is:

$U=4{\displaystyle \underset{i=1}{\overset{31}{}}}{\displaystyle \underset{j>i}{\overset{32}{}}}(\frac{1}{r^{12}}-\frac{1}{r^6})$

Generate the positions of $32$ atoms in a cubic box accommodated in fcc sites by creating

a new short code that enables to have a computationally generated file with such

positions. For that, the fcc unit cell $($contains the position of $4$ atoms$)$ needs to be

translated twice in each direction $x,y,z($in $3$D$).$ Unit cell coordinates for the $4$ atoms with

$a=1\mathrm{.}65$ are:

$([x_1],[y_1],[z_2])=([0\mathrm{.}0],[0\mathrm{.}0],[0\mathrm{.}0]),([x_2],[y_2],[z_2])=([0\mathrm{.}5a],[0\mathrm{.}5a],[0\mathrm{.}0]),([x_3],[y_3],[z_3])=([0\mathrm{.}0],[0\mathrm{.}5a],[0\mathrm{.}5a]),([x_4],[y_4],[z_4])=([0\mathrm{.}5a],[0\mathrm{.}0],[0\mathrm{.}5a])$

Note that the $32$ atoms will be inside a cubic box with edge length $L=2a.$

The coordinates of the $32$ atoms are obtained in $(1)$ and will be the initial positions for

an MD simulation. Generate with normally distributed random numbers the initial

velocities $(v_x,v_y,v_z)$ for each of the $32$ atom. These velocities should be consistent with a

temperature $T=0\mathrm{.}3.$ See subroutine "Init" of Fortran code added to the online materials

of February $20.$

With $(1)$ and $(2)$ the initial conditions are set. In your MD computer implementation

impose periodic boundary conditions using the minimum image convention, a cut$-$off

distance $r$ cut$=1\mathrm{.}64,$ and shift the potential $($review lecture of February ${13}^{th},$ part$2).$ Do a

MD simulation for $50,000$ steps, collecting production results over the last $25,000$ and

calculate Kin : and corresponding standard deviations. Visualize where the

particles of your final system are located in $3-$D $($use any of these modelers Avogadro,

VMD$,$ Chimera, Ovito$).$ Include the atomic picture in your report and compare it with a

picture of the system with atoms located in initial fcc positions.