Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 35D (Statistical Physics)

The microcanonical ensemble describes an isolated system at fixed energy, volume and particle number. The canonical ensemble describes fixed particle number and volume with heat exchange at a prescribed temperature. The grand canonical ensemble additionally allows particles to exchange with a reservoir, prescribing temperature, volume and chemical potential. Their respective equilibrium probabilities are uniform on the accessible energy shell, proportional to $e^{-\beta E_n}$, and proportional to $e^{-\beta (E_n-\mu N_n)}$. For stable short-range systems in the thermodynamic limit, concentration of relative energy and particle-number fluctuations and concavity of entropy make their bulk predictions equivalent; phase coexistence or nonadditive interactions require care.

For the microcanonical case restrict the allowed states to the fixed-energy shell of size $\Omega$, imposing $p_n=0$ outside it. Maximize the Gibbs entropy subject to $\sum p_n=1$. A Lagrange multiplier gives $-k_B(\log p_n+1)+\lambda =0$ on the shell, soFor the canonical case impose both normalization and $\sum p_n{E}_n=U$. Stationarity of $-k_B\sum p_n\log p_n-\lambda (\sum p_n-1)-k_B\beta (\sum p_n{E}_n-U)$ gives $p_n\propto e^{-\beta E_n}$. ThusThe last identification follows from $\partial S/\partial U=1/T$. Entropy is strictly concave in the probabilities, so these stationary distributions are maxima.

For the independent three-level particles, put $z=ϵ/(k_{B}T)$ and $Z_1=1+2\cosh z$. The full partition function is $Z_1^N$ because the particles occupy labeled lattice sites. Differentiation givesThe heat capacity also equals $N\mathrm{Var}(E_1)/(k_B{T}^2)$, confirming positivity. As $T\to +\infty$, the three occupations become equal:As $T\downarrow 0$, all particles occupy their unique lowest state:

A negative temperature is possible because the total energy is bounded above. An equilibrated population inversion has more particles at $+ϵ$ than at $-ϵ$. For example probabilities $(p_{-},p_0,p_{+})=(1,2,4)/7$ correspond to $T=-ϵ/(k_B\log 2)$ and mean energy $3Nϵ/7$. This is a macroscopic occupation pattern; an isolated individual microstate is not itself assigned a temperature. At the positive zero-temperature limit the ground state is unique, so the entropy tends to zero: the system obeys the third law of thermodynamics. The negative-temperature branch likewise approaches the unique highest-energy state as $T\to 0^{-}$; it does not introduce residual ground-state entropy.