36A (Statistical Physics)

Table of contents
- a 36A
  - Solution a
- b 36A
  - Solution b
- c 36A
  - Solution c
- d 36A
  - i d
    - Solution i
  - ii d
    - Solution ii
  - iii d
    - Solution iii
- e 36A
  - Solution e

a

Solution

A grand canonical ensemble describes a system that exchanges both energy and particles with a reservoir, at fixed

T, V, μ

. If microstate

n

has occupation

N_{n}

and one-particle energy

E_{n}

, then

Z = {N_{n}} \sum exp [- β n \sum N_{n} (E_{n} - μ)], β = (k_{B} T)^{- 1},

(1)

with the allowed occupations chosen for the particle statistics.

b

 0  0

Solution

 0  0

The grand potential is

Φ = E - TS - μ ⟨ N ⟩ = - k_{B} T lo g Z .

(1)

c

 0  0

Solution

 0  0

Extensivity makes

Φ (T, V, μ)

proportional to

V

. Comparing its differential

d Φ = - S d T - p d V - ⟨ N ⟩ d μ

with Euler scaling gives

Φ = - p V .

(1)

d

 0  0

i

 0  0

Solution

 0  0

The one-particle translational partition function is

q_{1} = \frac{V}{h ^{3}} \int_{R^{3}} e^{- β p^{2} / (2 m)} d^{3} p = \frac{V}{λ _{T}^{3}}, λ_{T} = \frac{h}{2 πm k _{B} T} .

(1)

For a classical ideal gas,

Z = N = 0 \sum \infty \frac{( e ^{β μ} q _{1} ) ^{N}}{N !} = exp (\frac{V e ^{β μ}}{λ _{T}^{3}}),

(2)

and hence

Φ = - \frac{k _{B} T V e ^{β μ}}{λ _{T}^{3}} .

(3)

ii

 0  0

Solution

 0  0

Differentiating

lo g Z

with respect to

β μ

gives

⟨ N ⟩ = \frac{V e ^{β μ}}{λ _{T}^{3}}, (Δ N)^{2} = \frac{\partial ^{2} l o g Z}{\partial ( β μ ) ^{2}} = ⟨ N ⟩ .

(1)

Thus

\frac{Δ N}{⟨ N ⟩} = \frac{1}{⟨ N ⟩},

(2)

which vanishes in the thermodynamic limit.

iii

 0  0

Solution

 0  0

Combining

Φ = - p V

with part (i) and the expression for

⟨ N ⟩

gives the ideal-gas equation of state

p V = ⟨ N ⟩ k_{B} T .

(1)

e

 0  0

Solution

 0  0

For relativistic one-particle energy

ϵ (p) = p^{2} c^{2} + m^{2} c^{4}

, replace

q_{1}

q_{1}^{rel} = \frac{V}{h ^{3}} \int_{R^{3}} e^{- β ϵ (p)} d^{3} p .

(1)

Classical particle counting still gives

Z = exp (e^{β μ} q_{1}^{rel}), ⟨ N ⟩ = e^{β μ} q_{1}^{rel}, (Δ N)^{2} = ⟨ N ⟩ .

(2)

Since

q_{1}^{rel}

is proportional to

V

Φ = - k_{B} T ⟨ N ⟩ = - p V

, and therefore

p V = ⟨ N ⟩ k_{B} T, \frac{Δ N}{⟨ N ⟩} = \frac{1}{⟨ N ⟩} .

(3)

The classical ideal-gas equation of state is independent of the dispersion relation.

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 36A (Statistical Physics)

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