Course 443, Problem Set, Michaelmas Term, 2005
• Given the partition function for a (non-relativistic) ideal gas in the
classical canonical ensemble
Z=
1 V N
.
N ! λ3
(1)
Derive expressions for the internal energy, U , the specific heat at constant volume, cV and the entropy, S.
• Sketch the phase diagram (external magnetic field as a function of temperature) for the 1D Ising model.
Sketch the isotherms (lines of constant temperature) on a plot of of
mean magnetisation as a function of external field for 3 cases: T <
Tc , T = T c , T > T c .
Draw a sketch of the 2D Ising model and write down its Hamiltonian
in the presence of an external field and with nearest neighbour spin
interactions.
• Consider the cluster expansion of the partition function in the canonical
ensemble
1
Z=
QN
(2)
N !λ3N
with
Z
Y
QN = d3 x (1 + fij ).
(3)
i<j
Write an expression for the contribution from Q4 . Write explicitly (in
terms of fij ) the contribution from S(1, 0, 1, 0). Calculate the symmetry
factor for this term and show that it agrees with that derived from an
explicit writing of the term.
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