35A (Statistical Physics)

In a box of volume

V

, the electromagnetic mode density in a box, including the two transverse polarizations, is

d N = 2 \frac{V}{( 2 π ) ^{3}} d^{3} k = \frac{V}{π ^{2}} k^{2} d k .

(1)

Each photon has energy

ℏ ω = ℏ c k

, and the given mean occupation number is

(e^{ℏ c k / T} - 1)^{- 1}

. Omitting the temperature-independent zero-point energy, the total energy is therefore

E = \frac{V ℏ c}{π ^{2}} \int_{0}^{\infty} \frac{k ^{3}}{e ^{ℏ c k / T} - 1} d k .

(2)

With the change of variables

x = ℏ c k / T

E = \frac{V T ^{4}}{π ^{2} ℏ ^{3} c ^{3}} \int_{0}^{\infty} \frac{x ^{3}}{e ^{x} - 1} d x .

(3)

ii

 0  0

Solution

 0  0

Expanding

(e^{x} - 1)^{- 1} = \sum_{n \geq 1} e^{- n x}

and using the Gamma integral gives

\int_{0}^{\infty} \frac{x ^{3}}{e ^{x} - 1} d x = \sum_{n = 1}^{\infty} \int_{0}^{\infty} x^{3} e^{- n x} d x = 6 \sum_{n = 1}^{\infty} \frac{1}{n ^{4}} = \frac{π ^{4}}{15} .

(1)

Thus the energy of blackbody radiation is

E (T, V) = \frac{π ^{2}}{15 ℏ ^{3} c ^{3}} V T^{4} .

(2)

iii

 0  0

Solution

 0  0

The first law of thermodynamics is

d E = T d S - P d V,

(1)

so, regarding

S

as a function of

(T, V)

(\frac{\partial S}{\partial T})_{V} = \frac{1}{T} (\frac{\partial E}{\partial T})_{V}, (\frac{\partial S}{\partial V})_{T} = \frac{1}{T} [(\frac{\partial E}{\partial V})_{T} + P] .

(2)

Equality of mixed partial derivatives gives the thermodynamic derivation of blackbody radiation pressure

(\frac{\partial E}{\partial V})_{T} = T (\frac{\partial P}{\partial T})_{V} - P .

(3)

Writing

a = π^{2} / (15 ℏ^{3} c^{3})

, we have

E = aV T^{4}

, and hence

T \frac{d P}{d T} - P = a T^{4} .

(4)

Equivalently,

d (P / T) / d T = a T^{2}

, so

P = \frac{a}{3} T^{4} + CT .

(5)

The condition

P = o (T)

T \to 0

forces

C = 0

. Therefore the radiation pressure is

P = \frac{π ^{2} T ^{4}}{45 ℏ ^{3} c ^{3}} = \frac{E}{3 V} .

(6)

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 35A (Statistical Physics)

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