Computational Methods Junior Honours : Checkpoint 6

This checkpoint investigates many particle gas simulations.

It is expected that you will need to work rather more independently on this checkpoint.

Boltzmann distribution and the Ideal Gas Law

You will set up a simulation of a number of particles in a box at a given temperature. This means microscopically that the distribution
of velocities must display the Boltzmann distribution as a set of initial conditions.

The particles will be run initially without interactions (ideal gas). The particles must be reflected at the walls of the box, and the momentum
impulse contributes to the pressure on the wall. Later, inter-particle forces will be introduced, and deviations from the ideal gas law measured.

Code 1: Extend a class ParticlesForceBox from ParticlesForce.

This will add members such as

Box size
Accumulated impulse on all faces
Time elapsed

It will also have methods to

Read and write the state of an array of particles from/to a file
You will have problems using StreamTokenizer with scientific input unless your getDouble is updated in the style of

scientific.java

note that the output of Vector3d.toString() can be read back using your readFromTokens code

Calculate the area, impulse and pressure
A leapPositionBox method that implements the box boundary conditions described below.

Code 2: Implementing initial conditions:

Write a seperate program to generate a Boltzmann distribution as the initial particle state, with uniformly distributed random locations.

Variable length arrays of particles (length is an integer num) can be allocated with:

Particle3d ps[] = new Particle3d[num];

The Boltzmann distribution can be formed by first producing a relevant

Gaussian random distribution of appropriate width for each of the initial x,y,z velocities
Uniform random distribution in [0,L] for each of the initial x,y,z positions.

See http://java.sun.com/j2se/1.4.2/docs/api/java/util/Random.html

Random.nextGaussian()
Random.nextDouble()

Code 3: Implementing the simulation:

You should write a programme to use your ParticlesForceBox class that reads its initial particle state, runs it for some time,
and, when finished, both prints the average pressure over this time interval and prints the final particle state to a new file.

Calculating the pressure

Every time a particle tries to move out of the box in the x-direction to L+D, reflect it to L-D and flip it's velocity

This imparts an impulse of : 2 m |Vx| on the face of the box

Similarly every time a particle tries to move out of the box in the x-direction to -D reflect it to +D and flip its velocity

This imparts an impulse of: 2 m |Vx| on the face of the box

This boundary condition should be applied to x,y,z

Optional:

Create an option to apply periodic boundary conditions to the system (instead of hard walls)
When calculating the inter-particle force between two particles in the periodic case use the "minimum image convention" (search the web) to take account of the periodicity

Calculation 1: Verify ideal gas law

Set up a non-interacting system with a Boltzmann distribution and random initial positions for 250 particles.
Run this for some time to acquire statistics and verify that P V = n K T
Plot the accumulated pressure versus time for two different seeds of the random number generator.

how long does it need to settle down? Why?

Optional:

make your programme run the system N times with different starting conditions
Calculate the error on the resulting pressure using the standard error on the mean

Plot Pressure versus Volume for two different temperatures (these are Isotherms).

Congratulations : you've just verified that the ideal gas satisfies Boyle's law

Calculation 2: Deviations from Ideal Gas Law

Now put in a small Lennard-Jones interaction term.

The range should be much smaller than the box size

Produce two isotherms at the same temperatures as the ideal case

Adjust the interaction strength until small deviations are shown

Where do the deviations first occur? Why?