CITS4403 | The Hardy–de Pazzis–Pomeau (HPP) lattice gas model

CITS4403, Computational Modelling
The Hardy–de Pazzis–Pomeau (HPP) lattice gas model”

Task
Implement the ‘HPP’ cellular automaton rule, which model a gas of colliding particles.
The HPP lattice gas automata is defined on a 2D square lattice. Particles
can move along the orthogonal directions of the lattice. Particles are associated
with both a position on the lattice (lattice site), and a discrete velocity (four cardinal
directions), i.e. the velocity with which the particle is assumed to have entered
the site (see Figure). An exclusion principle is assumed, which prevents more than
one particle to be at a same position with the same velocity. However, more than
one particle can be found at a lattice site if the particles have different velocities.
Four bits of information in each site are enough to describe the system during
its evolution. For instance, if at time t the lattice site at r has the following state
s(t; r) = [1011], it means that three particles are entering the site along directions
1, 3, and 4, respectively.
The CA rule describing the evolution of s(t; r) is usually split into two steps:
collision and motion. The collision phase specifies how particles entering the
same site will interact and change their trajectories. During the motion phase,
or propagation, the particles are actually moved to the nearest neighbor site they
were traveling to.
The figure (right) illustrates the HPP rules:
(a) a single particle has a ballistic motion until it experiences a collision
(b) and (c) the two nontrivial collisions of the model: two particles experiencing
a head-on collision are deflected in the perpendicular direction. In the other
situations, the motion is ballistic, that is the particles are transparent to each
other when they cross the same site.
1
According to the boolean representation of the particles at each site, the collision
part for a two-particle head-on collision is expressed as:
[1010] ! [0101]; [0101] ! [1010]
all the other configuration being unchanged. During the propagation phase, the
first bit of the state is shifted to the east neighbor cell, the second bit to the north,
and so on.
Be sure to start from an initial condition in which the exclusion principle is
satisfied.
Write a rule for a collision with a hard wall, at which the particle ‘bounces
back’
Implement the HPP CA model
Start with periodic boundary conditions and a concentrated density of cells
around the middle of the lattice. Does the gas spread to a homogeneous
distribution?
Modify the periodic boundary conditions into hard walls. Measure the pressure
on one wall (i.e. the number of particles colliding with the wall during
a time increment divided by the length of the wall) and make a plot of how
pressure varies with the initial density of particles (i.e. number of particles
divided by the area of the lattice).
Add a wall separating the left hand side from the right hand side. Include
a small opening in the wall. Show the evolution of pressure on the wall on
the right hand side and the wall on the left hand side.
2
Do you observe the phenomenon of “relaxation”? Does the system evolve
to a state that is uniform in a certain sense? Discuss this in light of the
microscopic reversibility of the system and the 2nd law of thermodynamics.
Imagine new situations that could be modelled with the HPP CA model and
implement them (e.g. obstacles, flow collisions etc.) Exemplify and discuss
your results.
Background
The purpose of the HPP rule is to model a gas of colliding particles. The essential
features that are borrowed from the real microscopic interactions are the conservation
laws, namely local conservation of momentum and of particle number.
References
[1] Hardy J, de Pazzis O and Pomeau Y, Molecular dynamics of a classical lattice
gas: transport properties and time correlation functions, Phys. Rev. A,
13, 1949–1961