Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 34C (Statistical Physics)

Maximize the Gibbs entropy subject to normalization and the two mean constraints. Introduce multipliers $\eta ,A,B$ and differentiateStationarity gives ${\rho }_i\propto \exp [-(A{E}_i+B{N}_i)/k]$. Strict concavity of entropy gives the constrained maximum, on its support. Differentiating the maximum value with respect to the constrained means, or differentiating the entropy along the maximizing distribution and using ${\sum }_{i}d{\rho }_i=0$, gives $dS=AdE+BdN$. The thermodynamic identification is $A=1/T$, $B=-\mu /T$. Therefore

For independent weakly coupled systems, insert the product probabilities into the Gibbs entropy. Since each marginal sums to one and the logarithm of a product is a sum,At fixed total energy and particle number, $\delta E_2=-\delta E_1$ and $\delta N_2=-\delta N_1$. Part (i) givesIndependent exchange variations force both coefficients to vanish. Thus

The grand canonical partition function isThe probabilities in part (i) and their logarithms giveAt fixed volume and fixed $\mu$, differentiate the partition sum:Combining with the preceding identity yieldsThe variable held fixed is essential; this is the grand canonical derivative.

For occupation numbers zero or one, the energy and particle number are additive over modes. The grand canonical partition function factorizes:Dividing the product of Boltzmann weights by this product givesThe marginal mean occupation is thereforeThis is the Fermi-Dirac distribution of a noninteracting Fermi gas: the exclusion principle allows only zero or one fermion in each single-particle state, including its spin label.