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Intro. Scientific Computing, HW10 - Due Friday, April 22.

1) Write a complete code to simulate the Lennard-Jones gas in a 2D periodic box
   with box length L = 14 and n=100 particles.  Start them off on a 10x10 grid
   with Gaussian distributed velocities.  Ignore units and assume beta = m = 1.
   The Hamiltonian is given by H = sum_j m v_j^2/2 + 1/2 sum_{i != j} u_ij^2 - u_ij
   where u_ij = |x_i - x_j|**-6
   The force on each particle, i, is therefore
   F_i = sum_{j != i} 6 (x_i - x_j) / |x_i - x_j|^2 ( 2 u_ij^2 - u_ij )

2) Run the simulation for 100 steps, and create a plot showing the locations of the atoms
   every 10 steps.

3) For every timestep, calculate the kinetic and potential energies.  What do you observe
   about the behavior of the potential energy?

4) Make a plot of the total energy vs. time for your 100 step simulation.  Overlay these plots
   for several different values of the numerical timestep, dt.

Hints: <source lang="python">

  1. Wrap all coordinates in an array to the range [0,L)

print x - L*floor(x/L)

  1. Find the closest distance between two points, r_ij

y = x[i] - x[j] print y - L*floor(y/L + 0.5)

  1. Calculate the LJ force on an atom at point z

r = closest_distance(z, x) # matrix of closest distances (n x 3) r2 = sum(r*r, 1) # vector of squared distances (n) u = r2**-3 print 6 * sum(r * ((2*u*u - u)/r2)[:,newaxis], 0) # sum over other atoms </source>

A code that allows you to visualize your (steps x atoms x 2) trajectory is shown below:

<source lang="python">

  1. !/usr/bin/env python

from numpy import * import matplotlib.animation as anim import pylab as plt

dt = 0.01

def show_trj(x, dt):

   fig = plt.figure()
   ax = fig.add_subplot(111, autoscale_on=False,
                        xlim=(0.0, 10.0), ylim=(0.0, 10.0))
   line, = ax.plot([], [], 'o')
   template = "time %.1f"
   time_text = ax.text(0.05, 0.9, , transform=ax.transAxes)
   def animate(i):
       line.set_data(x[i,:,0], x[i,:,1])
       time_text.set_text(template % (i*dt))
       return line, time_text
   def init():
       line.set_data([], [])
       return line, time_text
   ani = anim.FuncAnimation(fig, animate, \
               arange(len(x)), interval=25, blit=True, \
  1. Replace this with your actual dynamics code!

trj = random.random((100,10,2))*10.0

show_trj(trj, dt) </source>