= Solution
<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:
$$
\delta(E-\mu N)=\sum_{\mathbf p\sigma}\xi^*_{\mathbf p}\delta n_{\mathbf p\sigma}
+\frac1{2V}\sum_{\mathbf p\sigma,\mathbf p'\sigma'}
f_{\sigma\sigma'}(\mathbf p,\mathbf p')\delta n_{\mathbf p\sigma}\delta n_{\mathbf p'\sigma'}.
$$
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.<On the Theory of the Fermi Liquid by Lev Davidovich Landau>[Page 70, quasiparticle interaction and distribution variation]{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 lifetime>
$$
\tau^{-1}\propto(\epsilon-\mu)^2+(\pi k_BT)^2.
$$
Thus 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 $N^*(0)$ to include both spins. The quasiparticle <density of states> determines the low-temperature electronic <heat capacity>, $C_V/V=(\pi^2/3)k_B^2N^*(0)T$. 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, $m^*/m=1+F_1^s/3$; 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. \b[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>.
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