Physics 171.636: Modeling Across Multiple Length and Time Scales
Spring Semester, 2005

Week 1: Due Feb. 17

1) Consider an infinite line of atoms in one dimension along the x-axis. Suppose they interact with a Lennard-Jones interaction that is cut off abruptly for r > 2.5s.

a) Calculate the equilibrium lattice constant, i.e the spacing between atoms that minimizes the energy.

b) Suppose that all atoms for x>0 are removed. What is the equilibrium spacing of the outer atom from its nearest-neighbor?

2) Not everyone may manage to get this done, but try to at least give it a shot and let me know if you have trouble.

Download a MD program from one of the sites below and try running a simulation of 1000 particles interacting with a Lennard-Jones potential. To speed things up you should truncate the potential at the minimum 2^1/6 s.

Start all particles on a three dimensional cubic lattice with separation 1.1s. You’ll want to use periodic boundary conditions so you’ll have to have the period in each direction be 10*1.1s =11.0s.

Give all particles a speed of unity in Lennard-Jones units (make sure you understand where the time comes from) but with random directions. If you have trouble with random directions choose an axis (x,-x,y,-y,z,-z) at random and have speed along this axis.

Integrate for 5000 to 10000 steps using a time step of 0.005 in Lennard-Jones units and don’t use any thermostat (this is called the microcanonical ensemble).

Write out all the velocities every 100 or 200 steps.

For each time calculate the probability of having a given value of the speed.

Initially all atoms have the same speed, but the distribution will approach the square of the speed times a Gaussian- see how long this takes.

You can also look at the distribution of each component of the speed (vx, vy, vz). They should all be symmetric, so you can average the results for the three directions. You should find a Gaussian.

3) If you’re still interested, try playing with other initial conditions and try to figure out whether the final state is a liquid or crystal.

4) Try a time step of 0.0025, 0.0075, 0.010, .... What happens when dt gets too big? Does the Gaussian distribution change with dt?

Possible programs:

LAMMPS

http://www.cs.sandia.gov/~sjplimp/lammps.html

This is a cool parallel MD code that our group uses. The one problem is that you’ll need mpi on your machine in order to compile it. It is in C++. You can contact robhoy@pha.jhu.edu about this.

http://www.ccp5.ac.uk/librar.shtml has several codes including “moldy” and the programs from Allen and Tildesley.

Moldy has a variety of flavors for unix and windows. It seems to be a fairly complete code and is in C.

ftp://ftp.dl.ac.uk/ccp5/ALLEN_TILDESLEY/F.03 is a very simple and dumb code for Lennard-Jones that works well

for this first assignment. Note that you have to scale all the particles to a unit cube. It can go from 0 to 1 in each direction

or from -0.5 to 0.5. You then give it the density and the code maps it back to real units. The code is in fortran or basic