Homework 8

Physics 504: Statistical Mechanics and Kinetic Theory
HOMEWORK SHEET 8
Due 6pm Mon May 4 2015 in the 504 box.
Question 8–1.
Consider a fluid system described by the grand partition function
V
Ξ(λ, V ) = (1 + λ) 1 + λαV ,
where α > 0 and V is the volume of the system in some units, such as the hard core radius,
and λ is the fugacity. This form of Ξ is chosen to illustrate some points of principle; I do
not know if there is a real system with this form of Ξ.
(a) Write down the equation of state in parametric form (i.e. in terms of λ), and sketch
the graphs of p and v −1 versus λ, in the thermodynamic limit. Show that there is a
first order transition, and find the specific volumes of the two phases at coexistence.
(b) Find the roots of Ξ(λ, V ) = 0 in the complex λ plane at fixed V . Show that they
converge to the point on the real λ axis at which the phase transition in (a) occured,
in the thermodynamic limit.
(c) Find the equation of state in the gas phase, and notice that it exhibits no signature
of the phase transition as the volume per particle is decreased beyond that where the
phase transition occurs according to (a). Explain.
(d) Check that the form of the grand partition function, which we have assumed, is
physically acceptable, by calculating the isothermal compressibility, and explain the
significance of this calculation.
Question 8–2.
This question concerns the virial theorem, which we proved for a system in thermal
equilibrium. Here, we are concerned with the virial theorem for an isolated classical system
that is not necessarily in thermal equilibrium: we shall prove it by considering the dynamics
of the system. Thus, thermal equilibrium is a sufficient
but not a necessary condition for
P
the theorem to apply. We define the virial V ≡ i fi · ri , where fi is the force on the ith
particle whose position is ri . We consider the time average of a quantity A by
Z
1 T
dt A(t).
hAi ≡ lim
T →∞ T 0
(a) Using Newton’s second law, show that hVi = −2 hKi, where K is the total kinetic
energy of the system. In your proof, you will need to show that the average of the
time derivative of a certain quantity is zero; a hint is that this quantity is bounded
above and below. Say when this is true, and try to prove the theorem as carefully as
you can, using this fact. If you are unable to obtain a proof algebraically, or even if
you are successful, argue graphically why the theorem holds. If you can do this, you
will still get full credit (even if an algebraic proof eludes you). Explain why thermal
equilibrium is not necessary for the virial theorem to be valid.
(b) Consider an isolated cluster of N galaxies. Show that the mean potential energy
is minus twice the mean kinetic energy. Suggest how this result might be useful to
astronomers.
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Question 8–3.
This question concerns the mean field theory for the nearest neighbour Ising model
on a d-dimensional hypercubic lattice with coordination number z = 2d, where d is the
spatial dimensionality.
(a) Let M be the magnetization per spin. By writing each spin variable σi = M +(σi −M )
and ignoring terms in the Hamiltonian which are quadratic in the fluctuations, show
that the free energy per spin f (H, M, T ) = JzM 2 /2−kB T ln[2 cosh((Jzm+H)/kB T ).
For H = 0 and M 1 expand the free energy/spin up to and including O(M 4 ). What
happens to the quadratic term at Tc ? Plot the functional form of f (0, M, T ) for the
separate cases of T > Tc and T < Tc .
(b) Hence explain why the solution M = 0 is not physically acceptable in the mean
field theory of the zero field Ising ferromagnet below Tc . How is this reflected in the
behavior of the isothermal susceptibility as calculated about the M = 0 state above
and below Tc ?
(c) Calculate the isothermal susceptibility χT for zero external field H in the critical
region for T < Tc and verify that the way it diverges very close to Tc is of the form
χT ∼
A−
,
Tc − T
and determine A− .
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as T → Tc−