2011 Fall : ESS 582200 MD simulations

Homework #2 Due : Nov 18, 2011

1. In MD simulations, the integrating timestep Dt is chosen to be a value of order of magnitude smaller than the smallest 1/w in the system, where w is the oscillating frequency of an interaction potential.

(a) For a Lennard-Jones potential , what is the effective w near the bottom of the potential well?

(b) Given the parameters for an argon liquid: /kB=120K, =3.405Å, m=39.95g/mol, which value of timestep (in second) should be chosen in the simulation?

(c) A system comprises three carbon-carbon bonds which can be described by the potential . The parameters for the single, double, and triple bonds are listed in the table:

Give an estimation of Dt to run a MD simulation for the system.

2. Show that Verlet algorithm is time-reversible.

3. The velocity distribution of a three-dimensional system in equilibrium satisfies the Maxwell-Boltzmann distribution: . Using this information, calculate the limit value of Boltzmann H-function for the three-dimensional system.

4. Consider a Lenard-Jones (LJ) fluid system of N particles in a three dimensional cubic box under periodic boundary condition. We describe the system in reduced unit. The cutoff of the LJ interaction is set at rc*=2.5 and the number density of the system is r*=0.05. Temperature is controlled at T*=1.0.

(a) Perform simulations for N= 32, 256, 864, 2048 using all-pairs method to calculate force and energy. Show that the simulation time grows up as a square of N.

(b) Same as (a) but using Verlet neighbor list method. How does the simulation time depend on N?

(c) Modify the codes (on the website) to the method which uses cell subdivisions in the calculation of force and energy. Verify if the potential energy and pressure obtained from the modified code are consistent with the results of (a) and (b), when the system reaches an equilibrium state. Show that the simulation time depends linearly on N.

(d) Set N=864. Show that Boltzmann H-function is a decreasing function of time. How much time is needed to bring the system into an equilibrium state if (1) the initial velocity is set to zero, and (2) the initial velocity is chosen randomly which follows the Boltzmann distribution?