Homework assignments (4,5 credits) in Computational Physics

9 Write a Monte Carlo program to simulate the one
component plasma (OCP)
Instructor: Olle Edholm
Use periodic boundary conditions and make the simulations at fixed N and Γ (see definition
below). Calculate the energy and heat capacity per particle. Read first the second
(classical) reference below (http://prola.aps.org/abstract/PRA/v8/i6/p3096_1). Try
to reproduce some of the results of that article. You may first start with a cutoff method
but you have to use the Ewald method (see the first reference) in the end. (Reference:
Allen and Tildesley, and J.P. Hansen Phys. Rev A. 8 (1973) p. 3096-3109)
The properties of the OCP do not depend separately on density and temperature,
but these can be combined into one dimensionless parameter:
Γ = 1
4πɛ 0
(Ze) 2
k B T (4π 3 ρ)1/3
If you use the dimensionless coordinates: y i = x i (4πρ/3) 1/3 , the Coulomb energy between
two particles may be written:
u(r)
k B T = Γ r ,
with r being the inter particle distance in the dimensionless units defined above. If Z = 1
(protons) and T=300K, Γ = 1 corresponds to a low number density (about 10 21 protons
per m 3 ) while Γ = 200 corresponds to more the kind of densities one has in ordinary
matter (10 28 protons per m 3 ). Vary the parameter Γ in this range. In a typical plasma
the temperature is usually much higher, but also the density. Since the density is taken
to the power one third, even typical conditions inside a white dwarf star correspond to
quite small values of Γ.
Observe, finally that you have to include the attractive interaction with the electrons
as a uniform attractive background. This may be justified since at typical plasma conditions
the electrons can be treated as a quantum mechanical highly degenerate Fermi-Dirac
gas. Using a cutoff, you may add this contribution (which can be calculated analytically)
in the end. Using Ewald summation you have to include this when you calculate the electrostatic
repulsion between the protons. If you do not do this each part of the interactions
will diverge separately.
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