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Angular-relaxation collision time $\tau$ distinguishes first sound ($\omega \tau \ll 1$) from zero sound ($\omega \tau \gg 1$). Collisions establish local thermodynamic equilibrium in the former.^[ref]

For the figure's stipulated law $\omega \tau =(T/T_{\times }{)}^{-2}$, cooling from $T=5T_{\times }$ to $T=T_{\times }/5$ changes $\omega \tau$ from $0.04$ to $25$, without changing the driving angular frequency. The crossover temperature is defined by $\omega \tau (T_{\times })=1$. Calling zero sound a high-frequency mode therefore means high relative to the collision rate $1/\tau$, while its excitation energy must still remain within the low-energy domain of the Fermi liquid.

Consider a neutral, isotropic, unpolarized three-dimensional spin-one-half Fermi liquid, with only $F_0^s>0$ nonzero. A longitudinal density wave with angular frequency $\omega$ and wavevector magnitude $q$ has $s=\omega /(q{v}_F)=c_0/v_F$. The collisionless angular equation for the energy displacement $\nu (u)$ of the Fermi surface isAveraging givesThe bracket decreases from infinity to zero on $1<s<\infty$: each $F_0^s>0$ gives one undamped root above the small-$q$ particle-hole continuum.^[ref]

For an original example, $F_0^s=2$ gives $s=1.16474145$. Since $F_1^s=0$ here, the Principle of Galilean relativity gives $m^{\ast }=m$ and taking the hydrodynamic limit before $T\to 0$ gives $c_1/v_F=\sqrt{(1+F_0^s)/3}=1$. Stipulating $v_F=50\mathrm{m}{\mathrm{s}}^{-1}$ gives $c_0\simeq 58.24\mathrm{m}{\mathrm{s}}^{-1}$ and $c_1=50\mathrm{m}{\mathrm{s}}^{-1}$. These are model choices, not helium-3 measurements. At large $F_0^s$, expanding the logarithm gives $s\sim \sqrt{F_0^s/3}$; for small positive $F_0^s$, the root approaches $1$ exponentially closely, so numerical resolution matters. A charged electron liquid needs its long-range interaction included; this neutral model is not its long-wavelength density mode.

Landau quasiparticles describe the long-lived low-energy excitations of an interacting normal Fermi liquid. Starting from a free Fermi gas, imagine switching on interactions without changing the phase or destroying the one-to-one labelling of low-energy states. The resulting excitation carries the quantum numbers of a dressed particle but has a renormalized energy and interactions with other excitations. Its microscopic wavefunction need not resemble a bare electron state. The occupied momenta define a Fermi surface, and the quasiparticle energy measured from the chemical potential vanishes on that surface.

The Landau quasiparticle energy functional expresses the change near equilibrium in terms of occupation changes:Differentiation gives the energy of one quasiparticle in the distorted distribution. This is a phenomenological expansion around the true interacting ground state, not the assertion that its total energy equals that of independent particles. For an isotropic system, spin-symmetric and spin-antisymmetric components of $f$ are expanded in Legendre polynomials. Multiplication by the appropriate Fermi-surface density of states defines dimensionless Landau parameters $F_l^s,F_l^a$. The original formulation relates a change in the distribution to the energy shift.^[ref]

Pauli blocking makes the low-energy excitations sharp. The phase space for a quasiparticle to scatter while creating another particle–hole pair shrinks as the Fermi surface is approached, giving the regular three-dimensional low-energy Fermi-liquid quasiparticle lifetimeThus at zero temperature its width is small compared with its excitation energy. The single-particle spectral function has a pole with quasiparticle residue $Z$, generally less than one, and an incoherent remainder. In a conventional Fermi liquid the occupation distribution has a discontinuity of height $Z$ at the Fermi surface, rather than a unit free-particle step.

For an isotropic three-dimensional system, define $N^{\ast }(0)$ to include both spins. The quasiparticle density of states determines the low-temperature electronic heat capacity, $C_V/V=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$. In the usual Landau-parameter normalization, the spin response is modified by $1/(1+F_0^a)$ and the compressibility by $1/(1+F_0^s)$. In a Galilean-invariant continuum, $m^{\ast }/m=1+F_1^s/3$; that particular relation must not be imposed unchanged on a crystal band. A denominator approaching zero signals a possible instability of the assumed normal state.

The same interaction functional and a kinetic equation describe collective zero sound when the excitation frequency exceeds the collision rate. Ordinary first sound is the collision-dominated hydrodynamic limit. The theory organizes thermodynamics, response and transport through a small set of measured effective parameters while explaining why interactions coexist with long-lived fermionic excitations. It is not applicable without qualification in phases where the Fermi-liquid quasiparticle pole disappears or a gap opens over the relevant Fermi surface.