Past exam of the thermal and statistical physics course of the University of Cambridge / 2021 / Section B / 5

A phase transition is a qualitative change of thermodynamic phase as control parameters such as temperature, pressure or chemical potential are varied. A genuine equilibrium singularity is a nonanalyticity of the appropriate thermodynamic potential in the thermodynamic limit; finite systems generally show rounded crossovers instead.

An order parameter is a macroscopic quantity that distinguishes the phases. It often vanishes in a symmetric disordered phase and becomes nonzero after spontaneous symmetry breaking, although density-driven transitions need not have that simple symmetry interpretation. It can be a scalar, vector or tensor. At a first-order phase transition an appropriate first derivative of the free energy is discontinuous, and the order parameter commonly jumps; at a continuous phase transition the order parameter can approach zero continuously while a susceptibility or a higher derivative becomes singular.

Three examples are:

These examples also illustrate why an order parameter must reflect the physical distinction between phases rather than merely be any quantity that changes with temperature.

For $N$ indistinguishable classical particles the canonical partition function is an integral over phase space,The factor $1/N!$ removes overcounting under particle permutations, and $h^{3N}$ makes the classical phase-space measure dimensionless. The pair potential depends only on positions, so all momentum integrations separate. The three-dimensional Gaussian integral for each particle givesThe grand canonical partition function is $\Xi ={\sum }_{N=0}^{\infty }{e}^{\beta \mu N}{Z}_N$, where chemical potential $\mu$ controls the mean number of particles. ThereforeFor $N=0$, the empty products and configurational integral equal one. The factor $1/2$ in the interaction prevents double counting of unordered particle pairs.

Write $M=\mathcal{N}=V/b$ for the number of cells; this is fixed even though the particle number $N$ fluctuates. Approximate each position integral by the cell sum $\int d^3{r}_i\simeq b{\sum }_{{\alpha }_i=1}^M$, evaluating the slowly varying intercell pair potential at cell centres. With the assumed single-occupancy restriction, the fixed-$N$ contribution becomesAn occupation configuration $\{n_{\alpha }\}$ with $N={\sum }_{\alpha }{n}_{\alpha }$ selects an unordered set of $N$ occupied cells. There are exactly $N!$ ways to assign the temporary particle labels to that set. They all have the same interaction energy, cancelling the prefactor $1/N!$.

Summing over all particle numbers is therefore equivalent to summing each cell occupation independently over $0,1$:The activity per cell, $e^{\beta \mu }b/{\lambda }^3$, is dimensionless. This is a lattice gas, with each cell either empty or occupied.

The exclusion of multiple occupation is an additional physical modeling assumption, explicitly imposed by the question. Making a cell small does not by itself forbid two point particles from entering it; a repulsive core or a dilute-occupancy approximation would be needed to justify that restriction starting from a general continuum pair potential. In the remaining parts it is part of the defined cell model.

Let $q$ be the coordination number of a lattice, the number of nearest neighbours of each cell away from boundaries. Define an effective chemical potential incorporating the cell's translational entropy:Use $⟨\alpha {\alpha }^{\prime }⟩$ for an unordered nearest-neighbour bond, counted once. The occupation Hamiltonian including the chemical-potential term isUnder $n_{\alpha }=(1+{\sigma }_{\alpha })/2$, one bond contributesIn a regular lattice with periodic boundary conditions, the number of bonds is $Mq/2$, and each spin occurs in $q$ linear bond terms. ThereforeThe pair sum equals one half of the ordered sum used in the paper. Consequently the effective field isThus $\Xi ={\sum }_{\{\sigma \}}{e}^{-\beta {\mathcal{H}}_{\mathrm{eff}}}$ is the Ising model partition sum, including the displayed spin-independent constant if its absolute normalization is required. Dropping that constant changes $\Xi$ by a multiplicative factor but leaves spin probabilities and the phase-transition argument unchanged. For simple cubic cells in three dimensions, $q=6$ and the first term in $h$ is $6J$. Open boundaries replace $q$ by the local coordination near the surface; the stated uniform field is the bulk or periodic-lattice result.

For attractive nearest-neighbour interactions, $J>0$, the transformed model is a ferromagnetic Ising model. In three dimensions and at $T<T_c$, its infinite-volume spontaneous magnetization has two branches. As the effective field crosses zero, the equilibrium mean spin changes discontinuously from a negative value to a positive value. Since $(\partial h/\partial \mu {)}_T=1/2$, varying the physical chemical potential crosses this same first-order phase transition.

The particle-hole-symmetric coexistence line isThe mean Ising spin is a rescaled occupation density. Writing $n=⟨N⟩/V$ for the physical number density,The negative-spin phase is a dilute gas and the positive-spin phase is a dense, liquid-like phase. If the spontaneous spin magnitude at coexistence is $m_0(T)>0$, their number densities areThus the physical order parameter is a density or occupation imbalance, not an actual magnetic moment of these particles. In a closed system with an intermediate mean density, the same transition appears as phase coexistence between the two densities. At $T=T_c$ the density contrast vanishes continuously; above $T_c$, sweeping the chemical potential gives no coexistence jump. A true discontinuity requires the thermodynamic limit; a finite sum over finitely many cells remains analytic.