Problem Set 6

Problem Set 6

DQ 6 -- Due Tuesday, March 16, by 5pm

Suppose we have a system of states labeled by the index i, such that E_i is the energy in state i, and N_i is the number of particles in state i. Suppose we know that the average energy is <E> and we know that the average number of particles is <N>.
Using the methods in Notes 2-15, maximize the entropy S = -k_BΣ_i P_i ln P_i, subject to the constraint of fixed <E> and fixed <N>, to determine the probability P_i that the system will be found in state i.

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Problems -- Due Thursday, March 18, by 5pm.

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Problem 1 [20 points total]
Consider a classical ideal gas of indistinguishable, non-interacting, non-relativistic, particles confined to a region of three dimensional space by a harmonic potential U(r) = (1/2)mω₀²|r|² rather than the walls of a box. This might be a model for a gas of atoms in a magnetic trap (we will talk more about this later in the semester). The single particle Hamiltonian is then,

H⁽¹⁾(r, p) = |p|²
2m + 1
2 mω_o²|r|²

Working in the canonical ensemble for a gas of N particles,
a) Compute the average energy E of the gas as a function of temperature T.
b) Compute the density of particles n(r) at position r. Since the potential is spherically symmetric, this density should depend only on the radial distance r=|r| from the origin. n(r) should be normalized so that ∫d³r n(r) = N.
c) What is the average radial distance <r> of particles from the origin?
d) What is the pressure of the gas p(r) at a radial distance r from the origin? To do this part, consider a small volume of the system ΔV centered at a position r in the gas. The length of this volume should be small on the length scale that the potential U(r) varies, but ΔV should should be big enough to contain lots of particles, ΔN = ΔV n(r). You can then compute the partition function for this small region ΔV of the gas and use that to find the pressure p(r) in the gas at position r.
Problem 2 [20 points]
We have discussed the canonical ensemble, in which the temperature T, volume V, and number of particle N of a system are fixed, while the energy E is allowed to fluctuate. Suppose now that you wish to describe a system in which the temperature T, number of particles N, and pressure p are fixed, while the volume V is allowed to fluctuate. This would describe a system in contact with a thermal and mechanical reservior, in which the wall separating the system and the reservior is heat conducting and moveable. We can call this case the constant pressure ensemble.
a) [5 pts] Define the appropriate partition function Z(T, p, N) of the system in this new constant pressure ensemble.
b) [5 pts] If you defined Z properly in part (a), then the Gibbs free energy should be given by
G(T, p, N) = -k_BT ln Z(T, p, N)
To demonstrate this, show that using G defined from Z as above, the average volume of the system is correctly given by,
<V> = (∂G/∂p)_T,N
c) [5 pts] Derive a relation, in this constant pressure ensemble, between the isothermal compressibility κ_T and fluctuations in the volume V of the system. Show from this relation that the relative fluctuation in V vanishes in the thermodynamic limit.
d) [5 pts] Consider an ideal gas of indistinguishable, non-relativistic, non-interacting, point particles of mass m. Explicitly compute the partition function Z(T,p,N) of this gas. Use Z to compute G(T,p,N), and then from G compute the specific heat at constant pressure, C_p. Show that you get the correct answer for the ideal gas.