Past exam of the thermal and statistical physics course of the University of Cambridge / 2017 / 2

Structure through pair correlations. For a homogeneous isotropic monatomic liquid of number density $n=N/V$, the radial distribution function $g(r)$ is defined byIt compares the conditional density around a tagged particle with the bulk density. A repulsive core makes close neighbours unlikely; a first peak identifies a preferred coordination shell, and damped peaks at larger radii describe short-range order. Far away, $g(r)\to 1$ in the bulk limit. Unlike a crystal, an ordinary liquid has no indefinitely repeated positional order. An ideal uncorrelated gas has $g(r)=1$ away from the tagged particle. The shell factor $4\pi r^2$ must not be confused with $g(r)$ itself.

The same pair structure enters the static structure factor measured through scattering. In the bulk isotropic convention,Thus real-space coordination and reciprocal-space scattering peaks describe the same correlations.

Energy from the structure. If interactions are sums of a central pair potential $v(r)$, the equipartition theorem gives the classical kinetic energy $3N{k}_{B}T/2$. Count the interaction of a tagged particle with its neighbours and divide by two to avoid counting each pair twice. The pair-correlation expression for internal energy isThis makes structure thermodynamically useful: short-distance repulsion and the attractive region contribute with different weights. Additional internal molecular modes or many-body interactions require extra terms. The formula is a classical bulk pair-potential result, not a universal energy formula for every liquid.

Pressure and the dilute expansion. The mechanical virial theorem relates the pair forces to pressure. The virial pressure from pair correlations isAt low density, $g(r)\simeq e^{-\beta v(r)}$, giving the virial expansionThe second virial coefficient has units of volume per particle; higher coefficients encode additional cluster correlations. A hard-sphere exclusion diameter $d$ gives $B_2=2\pi d^3/3>0$. Attractions contribute negative terms and can make $B_2$ negative. Although the pair-correlation formulas can describe a dense liquid when its actual $g(r)$ is known, truncating the virial series after a few terms is normally a dilute-fluid approximation. It should not be silently extrapolated through condensation or into a dense liquid.

Excited-state saturation. Consider first a uniform three-dimensional noninteracting gas of identical spinless bosons in a large box. Set the one-particle ground energy to zero. The Bose-Einstein distribution is $\bar{n}(ϵ)=[z^{-1}{e}^{\beta ϵ}-1{]}^{-1}$, with quantum gas fugacity $z=e^{\beta \mu }\le 1$. For dispersion $ϵ=p^2/(2m)$, the excited density of states isUsing the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi {\hslash }^2/(m{k}_{B}T)}$, integration givesThe polylogarithm is ${\mathrm{Li}}_s(z)={\sum }_{\ell \ge 1}{z}^{\ell }/{\ell }^s$. Its finite value at $z=1$ implies a maximum excited population $V\zeta (3/2)/{\lambda }_T^3$, with $\zeta (3/2)\simeq 2.612$. Once this is less than $N$, the excess particles occupy the ground state macroscopically. This is Bose-Einstein condensation, rather than occupation of a state by merely one or a few atoms.

Transition and condensate fraction. Setting the saturation number equal to $N$ gives the homogeneous ideal Bose gas condensation temperatureBelow this temperature, $\mu$ approaches zero from below in the thermodynamic limit, and particle conservation givesAbove $T_c$ there is no macroscopic condensate and $\mu <0$ adjusts to keep $N$ fixed. At finite $N$, the chemical potential remains slightly below the ground energy and the transition is rounded.

Energy and the role of geometry. The excited-state internal energy isThe zero-energy condensate contributes no term to this energy in the ideal model. Below $T_c$,Condensation depends on low-energy state counting: for a uniform quadratic-dispersion gas, a finite-temperature bulk condensate requires spatial dimension greater than two. A trap changes the density of states and therefore the temperature law; a three-dimensional harmonic trap gives a cubic, rather than $3/2$, condensate-fraction exponent. Finally, an ideal-gas condensate should not automatically be called a superfluid: its free quadratic spectrum has zero critical speed by the Landau criterion. Interactions matter for the robust collective-flow properties of real dilute condensates.

Quanta of collective motion. An elementary excitation is a low-energy departure from the many-body ground state that can often be treated as a quasiparticle. Near an ordered equilibrium, small collective motions form normal modes. Quantization turns a harmonic mode of frequency ${\omega }_{\mathbf{k}}$ into bosonic excitations with energy $\hslash {\omega }_{\mathbf{k}}$. Their number is not an independently conserved thermodynamic variable in equilibrium, so their chemical potential is zero andHere $\nu$ labels branches, and $U_0$ contains the ground-state contribution, including phonon zero-point energy where appropriate. The approximation is useful while interactions and damping of the excitations are weak enough that the modes remain identifiable.

Phonons. A phonon is a quantum of an elastic lattice-vibration mode. Long-wavelength acoustic phonon branches have $\omega \simeq c_{\nu }k$, with one longitudinal and two transverse polarizations in a three-dimensional monatomic solid. A multi-atom primitive cell can also support optical phonon branches, generally with nonzero frequency at $k=0$. Linear acoustic dispersion gives $D(ϵ)\propto {ϵ}^2$, so the energy integral has $U-U_0\propto T^4$ and the heat capacity obeys the Debye $T^3$ law. In the Debye model,At temperatures high compared with all phonon energies but below structural changes, the classical harmonic limit gives approximately $C_V=3N{k}_B$. Optical modes are frozen out at temperatures well below their gaps.

Spin waves. A spin wave is a spatially varying collective precession of ordered spins; its quantum is a magnon. In a simple isotropic ferromagnet, a long-wavelength gapless branch has energy ${ϵ}_{\mathbf{k}}\simeq D_s{k}^2$, where the spin stiffness $D_s$ has units $\mathrm{J}{\mathrm{m}}^2$. State counting gives $D(ϵ)\propto {ϵ}^{1/2}$. For one isotropic branch in three dimensions,This exponent is specific to a gapless quadratic ferromagnetic branch. Long-wavelength antiferromagnetic spin waves can be linear and instead give a $T^3$ heat-capacity law in three dimensions. Magnetic anisotropy or an applied field can open a gap, causing activated behaviour at still lower temperatures. Near a magnetic ordering transition, a dilute noninteracting spin-wave description is no longer adequate.

These examples illustrate the low-temperature energy of gapless bosonic excitations: for $ϵ\propto k^s$ in $d$ dimensions, $D(ϵ)\propto {ϵ}^{d/s-1}$, hence $U-U_0\propto T^{1+d/s}$ and $C_V\propto T^{d/s}$. The exponent records both the spatial dimension and the dispersion, making calorimetry a probe of elementary excitations.