### Bouncing billiard balls: modelling the dynamics of hard sphere systems

#### (supervisor: Edwin Flikkema)

Nature of project: software, data analysis

Available to full-time physicists or joint students.

#### Project description and methodology

Hard spheres are idealised objects that can be used to model a variety of systems. The hard spheres can represent various physical objects ranging from atoms to colloids to billiard balls.

In this project, the dynamics and thermodynamics of hard spheres confined to a container will be studied using computer simulation. In particular, the dynamics will be simulated, based on elastic collisions between spheres and between a sphere and the walls of the container. This is an event-based type of simulation, each collision representing an event. The method is based on determining (after each collision event) what (and when) the next collision event is going to be and subsequently handling the transfer of momentum between the colliding entities.

An important part of the project is to determine time-averaged values of various physical quantities. These include the density profile (as a function of the distance to the container wall), the radial distribution function and the pressure. Different densities can be studied by changing the size of the container or changing the number of spheres inside the container.

A number of phenomena can be studied. A starting point would be to study the equilibration of the (kinetic) energy and check whether the Maxwell-Boltzmann distribution is achieved. The next step is to determine density profiles. From this the pressure can be inferred and the equation of state can be calculated (i.e. the pressure as a function of density).

A successful project will develop beyond the above in one/some of the following directions:
To develop the project further, radial distribution functions can be calculated. The results can be compared to Monte Carlo simulations [3] (to check the equivalence of ensembles).

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

Additional scope or challenge if taken as a Year-4 project: For an MPhys project, phase transitions can also be studied. Alternatively, polydisperse systems can be considered (i.e. hard spheres with varying radii). A phenomenon that can be studied is the so-called "depletion interaction" between large spheres in a solution of smaller spheres.

Initial literature for students:

1. M. P. Allen and D. J. Tildesley, Computer simulation of liquids, Oxford University Press, (1987)
2. D. Frenkel and B. Smit, Understanding molecular simulation, Academic Press (second edition, 2002)
3. N. Metropolis, A. W. Roenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys., 21:1087, 1953.

#### Novelty, degree of difficulty and amount of assistance required

The supervisor has already written a code that simulates the hard sphere system (in 2D in a square container) by determining the sequence of collision events. The student can use this code as a starting point. The student will need to add code for calculating physical quantities and for computing their time-averages. The code is written in C, but this should not be much of a problem as C and Fortran/Python are quite similar. In the coding stage, probably a lot of assistance by the supervisor is required.

#### Project milestones and deliverables (including timescale)

milestoneto be completed by
Equations for collision detection and momentum transfer rederived.end of November
Familiarisation with existing code.Christmas
Coding of physical quantities completed (rdf, density profiles, etc.).mid-March
Production runs completed.Easter