Statistical Mechanics - Physics at Oregon State University

64 CHAPTER 3. VARIABLE NUMBER OF PARTICLES
the next chapter we will see that Bose-Einstein condensation is an example of
a natural phenomenon that would not be described at all if we ignored such an
error analysis. In fact, in the description of many phase transitions one needs
to be careful!
3.8 Problems for chapter 3
Problem 1.
The Helmholtz free energy of a system at an appropriate temperature is given
by F (V, N) = N log( N
V ) − N.
1. Calculate the pressure and chemical potential.
M such systems, all with volume V, are in diffusive contact and in equilibrium.
In addition, there is a potential energy per particle Φi in system i. The total
number of particles in the M combined systems is N.
2. Find the number of particles in each of the subsystems.
Problem 2.
The quantum states of a given system have a variable volume. The energy of
state n is ɛn, while the volume of state n is νn. This system is at a given pressure
p.
a) Show that the probability of finding the system in state n is proportional to
e−β(ɛn+pνn) .
b). In this case we define a partition function Z(T, p, N) =
e−β(ɛn+pνn) . Show
that the Gibbs energy G = U − T S + pV is related to Z by G = −kBT log(Z).
c) Find a formula relating the fluctuations in volume to the isothermal compressibility.
Problem 3.
A system contains an ideal gas of atoms with spin 1
2 in a magnetic field
B(r). The concentration of the spin up (down) particles is n↑(r) ( n↓(r) ). The
temperature is T.
(A) Evaluate the total chemical potential for the spin up and down particles.
(B) These two chemical potentials have to be the same and independent of r.
Explain why.
(C) Calculate the magnetic moment of this gas as a function of position.
(D) Show that the concentration of magnetic particles is high in regions with
a large magnetic field.
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