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Changing one Landau quasiparticle occupation changes the energies of others. To second order around equilibrium,Here $\mathcal{V}$ is volume, occupations are per momentum and spin state, and $f$ has units of energy times volume.^[ref]

For a toy two-state change at fixed volume, take $\delta n_a=\delta n_b=\eta$, $f_{aa}=f_{bb}=0$, $f_{ab}=f_{ba}=g\mathcal{V}$, and unperturbed energies ${\xi }_a={\xi }_b=e$. The expression gives $2e\eta +g{\eta }^2$. The factor $1/2$ counts the interaction of the same pair once; it does not halve each excitation's energy. This finite-state illustration only explains the bookkeeping of the Landau quasiparticle energy functional.

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.

The quasiparticle residue is the coherent pole weight of the single-particle Green's function. Its coherent spectral term $Z\delta (E-{\xi }_p)$ contributes $Z\Theta (-{\xi }_p)$ to the zero-temperature momentum occupation. A smooth remainder leaves a jump $Z$ at the Fermi surface; Landau quasiparticle occupations have a unit step.^[ref]

For the local illustrative distribution in the figure, put $x=(p-p_F)/p_F$ and$\Theta$ is the Heaviside step function. Its one-sided limits are $0.825$ and $0.175$, giving $Z=0.65$. The smooth term is stipulated; this local curve does not specify a normalized distribution over all momenta. In particular, $Z$ is neither the fraction of atoms above the Fermi surface nor $m/m^{\ast }$.