Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 37C (Statistical Physics)

Table of contents
- a 37C
  - Solution a
- b 37C
  - i b
    - Solution i
  - ii b
    - Solution ii

a

Solution

The microcanonical ensemble fixes energy, volume and particle number; the canonical ensemble fixes temperature, volume and particle number; and the grand canonical ensemble fixes temperature, volume and chemical potential while allowing particle exchange. For macroscopic systems with short-range interactions, away from phase coexistence and other nonconcave regimes, relative fluctuations vanish in the thermodynamic limit and the ensembles give equivalent local thermodynamics.

b

i

Solution

The center-of-mass and relative terms in the Hamiltonian are independent, so the phase-space integral factorizes as stated. Gaussian integration gives

Z_{trans} = \frac{V}{( 2 π ℏ ) ^{3}} (\frac{2 π M}{β})^{3/2} .

In the large-volume limit the relative coordinate may be integrated over all space. Since

4 π \int_{0}^{\infty} r^{2} e^{- β α r} d r = \frac{8 π}{( β α ) ^{3}},

we obtain

Z_{int} = \frac{1}{( 2 π ℏ ) ^{3}} (\frac{2 π μ}{β})^{3/2} \frac{8 π}{( β α ) ^{3}} .

The radial probability is proportional to

r^{2} e^{- β α r}

, hence

⟨ r ⟩ = \frac{3}{β α} = \frac{3 k _{B} T}{α} .

ii

Solution

For an ideal gas of indistinguishable classical mesons,

Z_{N} = \frac{Z _{1}^{N}}{N !} .

Apart from temperature-independent factors,

Z_{trans} \propto T^{3/2}

and

Z_{int} \propto T^{9/2}

, so

Z_{1} \propto T^{6}

. Therefore

U = - \frac{\partial}{\partial β} lo g Z_{N} = 6 N k_{B} T, C_{V} = 6 N k_{B} .

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