= Solution
A normal <Fermi liquid> is an interacting fermion phase continuously connected to a <Fermi gas> without broken symmetry. Its <Landau quasiparticles> are dressed particle/hole excitations near the <Fermi surface>, with the same conserved quantum numbers, finite <quasiparticle residue> and a decay width small compared with their excitation energy. Their reference dispersion already includes zero-temperature interaction renormalization; “bare quasiparticle” here does not mean a free atom.<The Theory of a Fermi Liquid by Lev Davidovich Landau>[Low-energy quasiparticles and effective dispersion]{ref}
The rate assumes $k_BT,|\xi|\ll E_F$, $\xi=\epsilon-\mu$, regular scattering amplitudes and <density of states>, equilibrium <Fermi-Dirac distributions>, and two-body collisions conserving energy and momentum. <Pauli exclusion principle> restricts the available phase space. It describes a regular three-dimensional normal liquid, not a one-dimensional interacting electron liquid. On the surface, $\tau^{-1}\propto T^2$; at $T=0$ and $\xi>0$, it is proportional to $\xi^2$. Hence excitations become increasingly sharp. The quoted asymmetric occupation factor is for particle out-scattering: a hole decay or full spectral width also needs the complementary in-scattering process; it does not imply exponentially stable deep holes.
Symmetry $f_{ab}=f_{ba}$ cancels the quadratic functional's factor one-half, giving the <thermal Landau quasiparticle energy shift>
$$
\boxed{\widetilde\epsilon_a=\epsilon_a+\frac1V\sum_b f_{ab}\delta n_b+O(\delta n^2).}
$$
Here $a=(\mathbf k,\lambda)$, and the derivative is with respect to an occupation, not particle density. Replacing the homogeneous momentum sum by an energy integral defines a smooth angularly averaged kernel $H_a(\epsilon)$. With $\delta n=n_F(\epsilon;\mu,T)-\Theta(\mu_0-\epsilon)$,
$$
\delta\epsilon_a=H_a(\mu_0)\delta\mu+\frac{\pi^2}{6}(k_BT)^2H_a'(\mu_0)+O(T^4).
$$
Fixed density gives $\delta\mu=-(\pi^2/6)(k_BT)^2D'(\mu_0)/D(\mu_0)+O(T^4)$; fixed chemical potential instead has $\delta\mu=0$. Thus \b[$\widetilde\epsilon_a=\epsilon_a+O(T^2)$] in either case, provided the kernels are smooth and no instability intervenes. This is the <Sommerfeld expansion>; singular spectra require separate treatment.<Statistical Physics by David Tong>[§3.6.4, fixed-density Sommerfeld expansion]{ref}
The leading entropy is therefore that of quasiparticles with their zero-temperature <effective mass>:
$$
\boxed{C_V/T\longrightarrow\gamma=\frac{\pi^2k_B^2}{3}D^*(\mu_0).}
$$
$D^*$ here is the total, spin-summed <density of states>. The plotted low-temperature slopes require the enhanced masses $5.4m$ at molar volume $26.2\,\mathrm{cm^3\,mol^{-1}}$ and $2.7m$ at $36.8\,\mathrm{cm^3\,mol^{-1}}$, rather than the bare-mass predictions. At fixed molar particle number, $\gamma_{\rm mol}\propto m^*V_{\rm mol}^{2/3}$. Agreement is restricted to the low-temperature limit: the measured curves depart markedly from a fixed-mass ideal-gas extrapolation at higher temperature. This supports long-lived quasiparticles with smooth low-energy parameters, but heat capacity alone does not measure their lifetime or prove adiabatic continuity.
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