Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 35D (Statistical Physics)

Assume $A>0$ and $\alpha >0$, so the excitation energy increases with momentum and vanishes in the ground state. With one internal state, the number of states in a spherical momentum shell is $V4\pi k^{2}dk/(2\pi {)}^3$. Since $k=(ϵ/A{)}^{1/\alpha }/\hslash$,An internal degeneracy multiplies this density of states by its degeneracy factor.

The Bose-Einstein distribution is $\bar{n}(ϵ)=[e^{(ϵ-\mu )/(k_{B}T)}-1{]}^{-1}$. For a finite nonnegative occupation of the zero-energy state, its denominator must be positive, requiring $\mu <0$. At Bose-Einstein condensation the thermodynamic-limit chemical potential approaches $0$ from below, and the ground state must be counted separately; therefore “$\mu <0$” is a finite-system condition, while the condensed limit has $\mu =0$.

In a box whose shape is fixed, each momentum quantum number scales as $V^{-1/3}$, so each energy level scales as $V^{-\alpha /3}$. Differentiating the grand potential $\Omega =k_{B}T{\sum }_j\log (1-e^{-({ϵ}_j-\mu )/(k_{B}T)})$ at fixed $T,\mu$ givesThe zero-energy condensate contributes neither pressure nor kinetic energy.

The number outside the ground state isAt $\mu \uparrow 0$, its low-energy integrand is proportional to ${ϵ}^{3/\alpha -2}$. The integral is finite precisely when $3/\alpha >1$, soFor these exponents, the maximum excited population isThis follows by expanding the Bose denominator in positive exponentials and integrating each term. It tends to zero as $T\downarrow 0$, forcing excess particles into the ground state at fixed nonzero total number. At $\alpha =3$ the divergence is logarithmic; for $\alpha >3$ it is stronger, so there is no positive-temperature thermodynamic-limit condensation of this homogeneous gas.

Photons have $\alpha =1$ and two polarizations, but in ordinary blackbody equilibrium their number is not conserved and $\mu =0$ at every temperature. Cooling reduces their number rather than forcing an excess into a ground state, so an ordinary equilibrium photon gas does not condense. Systems imposing an effective photon-number constraint and an appropriate confined spectrum are different from the unconstrained photon gas considered here.