Thermodynamics of the Bose Gas and Superfluidity

The condensation of an ideal Bose gas fixes the fugacity below $T_c$ at $z=1$. That single fact determines the energy, the pressure, and the heat capacity in the condensed phase, because every thermodynamic function reduces to a Bose function evaluated at unit fugacity. The ideal-gas results are exact and show the transition as a cusp in the heat capacity. Real superfluid helium condenses with the same statistics but differs quantitatively: the interactions between atoms reshape the excitation spectrum, and it is that spectrum, through the Landau criterion, that makes the flow frictionless.

Energy and pressure

The energy integrates $\epsilon g(\epsilon )⟨n⟩$ over the ${\epsilon }^{1/2}$ density of states, one power of $\epsilon$ higher than the particle count, giving the next Bose function:

The grand potential is $\Phi =-PV=-k_{B}T(V/{\lambda }^3)g_{5/2}(z)$, so the pressure is

The relation $PV=\frac{2}{3}U$ is the nonrelativistic result, unchanged by statistics; the statistics enter only through $g_{5/2}(z)$. The condensate — the particles in the zero-energy ground state — contributes nothing to either sum: it carries no energy, no momentum, and no pressure.

The condensed phase and the isotherm plateau

Below $T_c$ the fugacity is pinned at $z=1$, so $g_{5/2}(z)\to \zeta (5/2)=1.341$ and both $U$ and $P$ lose all dependence on the volume:¹

The pressure of the condensed gas is a function of temperature alone. Compressing the gas at fixed $T$ below $T_c$ pushes more particles into the condensate without raising the pressure, exactly as compressing a vapor along a coexistence line converts it to liquid at fixed pressure. On a $P$–$V$ isotherm the condensed region is a horizontal plateau, and the locus of transition points across isotherms is the curve $P_c\propto V_c^{-5/3}$.

The heat-capacity cusp

The heat capacity below $T_c$ follows from $U\propto V{T}^{5/2}$ at fixed $V,N$:

which rises as $T^{3/2}$ and reaches $\frac{15}{4}\zeta (5/2)/\zeta (3/2)=1.925$ at $T_c$. Above $T_c$ the fugacity falls below $1$ and

As $T\to T_c^{+}$ the fugacity approaches $1$, where $g_{1/2}(z)$ diverges, so the second term vanishes and $C_V/N{k}_B\to 1.925$ from above — the same value as from below. The heat capacity is therefore continuous at $T_c$, but its slope is not: the derivative $\mathbf{d}{C}_V/\mathbf{d}T$ jumps, producing a cusp. Far above the transition the gas becomes classical and $C_V\to \frac{3}{2}N{k}_B$.

Superfluid ${}^4$He shows a related but sharper feature. Its heat capacity diverges logarithmically at the lambda point $T_{\lambda }=2.17\mathrm{K}$, the shape that named the transition, rather than forming the finite ideal-gas cusp. The difference is the interactions: liquid helium is dense, its atoms strongly coupled, and the ideal-gas model captures only that a transition occurs near the right temperature. The ideal $T_c$ for helium's density is $3.1\mathrm{K}$, close enough to identify the mechanism, wrong in detail.²

The Landau criterion for superfluidity

Frictionless flow is a statement about excitations. Consider superfluid moving at velocity $\vec{v}$ through a stationary tube. Dissipation requires the fluid to create an excitation, which draws momentum and energy from the flow. Transforming to the fluid rest frame, an excitation of momentum $\vec{p}$ and energy $\epsilon (p)$ can be created only if it lowers the total energy, which happens only when the flow speed exceeds

Below $v_c$ no excitation can form and the flow persists without loss.

For a free particle $\epsilon =p^2/2m$, the ratio $\epsilon /p=p/2m\to 0$ as $p\to 0$, so $v_c=0$: an ideal Bose gas is not, strictly, a superfluid — any flow can decay by exciting a slow long-wavelength particle. Superfluidity is a property of the interacting system, whose spectrum is linear at small $p$ and has a positive minimum slope.

The phonon-roton spectrum and two sounds

The measured excitation spectrum of superfluid ${}^4$He has two regions:

• Phonons. At small momentum the excitations are quantized sound waves, $\epsilon =cp$ with $c\approx 240m/s$. The slope $\epsilon /p=c$ is finite, the first ingredient of a positive critical velocity.
• Rotons. At larger momentum the spectrum dips to a local minimum, parametrized by Landau as $\epsilon (p)=\Delta +(p-p_0{)}^2/2m^{\ast }$, with a gap $\Delta /k_B\approx 8.6\mathrm{K}$ at momentum $p_0/\hslash \approx 19{\mathrm{nm}}^{-1}$. The roton minimum, not the phonon slope, sets the critical velocity: $v_c\approx \Delta /p_0\approx 60m/s$.

The tangent from the origin touches the spectrum at the roton minimum, so the Landau critical velocity is the roton gap divided by the roton momentum. Measured critical velocities in wide channels are lower, limited by vortex formation rather than roton creation, but the roton value is recovered for ions moving fast through the bulk.³

The two-fluid model that describes helium II — a normal component of density ${\rho }_n$ and a superfluid component ${\rho }_s$ — supports two distinct sound modes:

• First sound. An ordinary pressure and density wave, in which the normal and superfluid components oscillate together. Its speed is the usual $\sqrt{(\partial P/\partial \rho {)}_S}$.
• Second sound. A temperature and entropy wave, in which the two components oscillate in antiphase at nearly constant total density, carrying heat as a wave rather than by diffusion. It exists only because the superfluid carries no entropy, and it has no counterpart in an ordinary fluid.

Summary

• The ideal Bose gas has $U=\frac{3}{2}{k}_{B}T(V/{\lambda }^3)g_{5/2}(z)$ and $P=(k_{B}T/{\lambda }^3)g_{5/2}(z)$ with $PV=\frac{2}{3}U$; the condensate contributes nothing to either.
• Below $T_c$ the fugacity is $1$, so $P=\zeta (5/2)k_{B}T/{\lambda }^3\propto T^{5/2}$ depends on temperature alone and the $P$–$V$ isotherm is flat.
• The heat capacity rises as $T^{3/2}$ to a peak $C_V/N{k}_B=1.925$ at $T_c$ and is continuous with a discontinuous slope — a cusp; real ${}^4$He diverges logarithmically because interactions matter.
• Superfluidity requires an excitation spectrum with positive minimum slope: the Landau critical velocity is $v_c={\min }_p\epsilon (p)/p$, set in helium by the roton minimum at about $60m/s$; the two-fluid model carries first sound (density) and second sound (temperature).