This was written after hours within that torture chamber known as the exam hall.....
Consider a 2-dimensional lattice of n x n candidates, spaced equally to form a 'simple cubic' pattern. The lattice contains a number J of free moving particles, such that J = n2/30. What are the factors affecting the free particle motion ?
It could be assumed that the lattice points exert no force on the free moving particles and that the classical Drude model of electron motion in metals may apply. An ideal gas reasoning would be appropriate only if we assume the lattice points to be infinitessimally small (ie not there) in which case J free moving particles would probably be found congregating in other areas, unconstrained by The Laws of Invigilation. The Drude model does collapse on consideration that the lattice points have similar dimensions to the free moving particles, as does the lattice spacing (the fundamental Separation Constant for lattice points is 1m). As this is the case a multi-collision theory may need to be developed (based on the arguments of statistical mechanics and the individual states of each of the J particles); however, unless a free moving particle is in a particularly unusual state (spin up/down/all around), then collisions are not, in practice, seen. All directional changes in the movement seem to be at right angles to the original path. This would imply some sort of force exerted by some of the lattice points. This appears attractive at times when several of the free moving particles are seen to gravitate to one particular lattice point. Here the time-variance emerges and a four dimensional vector-space must be used to fully examine the motion. There are also repulsive forces, at times, when other factors enter the system - possibly much smaller but as yet unidentified particles that seem to diffuse through the lattice causing much disruption as they pass.
Often there are greater than average concentrations of free-moving particles in certain areas: certainly an attraction to the outer edges of the lattice (known as the Lean against the Wall boundary condition); there is also an attraction of free-moving particles to each other (similar to Van de Waals forces which enables the exchange of information between the particles). Controlling factors that influence the location of the free moving particles seem to be any oscillations of the lattice points (which should be fixed) - some areas are often more disturbed than others (a phenomenon known as tiering).
When the condition J>>n2/30 occurs, the classical model breaks down and quantum effects dominate. Sometimes the particle and lattice points remain fixed for many hours but this state is uncertain (Heisenberg). If there is a 'one particle in a box' situation with one lattice point and J=1 then the paths become totally unpredictable and the free-moving particle is best considered in k-space as a probability wave. It is important that the system is undisturbed during the lattice lifetime - any attempt from outside to examine the wave functions and 'see what is going on' would lead to total collapse of the wave possibly with invigilator and candidate alive, dead or suspended for ever in external exam conditions.
copyright Karin Parker 1995