Ideal Fermi Gas

One-Sentence Description

The ideal Fermi gas is a free many-body model whose antisymmetric state space restricts every complete one-particle mode to occupation zero or one, producing a filled Fermi sea, degeneracy pressure, and a narrow thermally active shell even though the Hamiltonian contains no interparticle force.

This dossier fixes the model data, exactness claim, limits, observables, and validation targets. The Ideal Fermi Gas teaching article owns the full state-counting and thermodynamic derivations. Finite-Size Effects owns the cross-system diagnosis of mode spacing, shell parity, boundaries, and resolution. The Fermi Momentum and Fermi Energy, Fermi Surface, and Sommerfeld Expansion pages own their respective deeper treatments.

Baseline Model Record

Unless a variant is named explicitly, this dossier uses the following model.

Field	Baseline choice
particles	identical fermions with $g$ degenerate internal states
space	a cubic box of side $L$ and volume $V=L^3$
boundaries	periodic in all three directions
one-particle Hamiltonian	$h=\mathbf p^2/(2m)$
energy zero	$\epsilon_{\mathbf0}=0$
many-body space	fermionic Fock space, or its fixed-$N$ sector
conserved charge	total particle number $N$
equilibrium controls	$T,V,\mu$ or $T,V,N$
component convention	common spectrum and common chemical potential
thermodynamic limit	$N,V\to\infty$ at fixed total density $n=N/V$
interactions	absent

The internal label is part of a complete one-particle mode. Two fermions may occupy the same spatial orbital when their internal states differ; no two may occupy the same spatial-and-internal mode. A polarized gas, a trapped gas, a band gas, and an interacting Fermi liquid require additional data and are not silently included in this baseline.

Degrees of Freedom and Hilbert Space

The one-particle Hilbert space is

$$\mathcal h = L^2(\mathbb T_L^3) \otimes \mathbb C^g,$$

where $\mathbb T_L^3$ is the periodic cube. The fermionic Fock space is

$$\mathcal F_F(\mathcal h) = \bigoplus_{N=0}^{\infty} \bigwedge^N\mathcal h.$$

At fixed particle number, the physical space is

$$\mathcal H_N = \bigwedge^N\mathcal h.$$

Momentum-mode operators satisfy

$$\begin{aligned} \{c_{\mathbf k a},c_{\mathbf q b}^{\dagger}\} &= \delta_{\mathbf k\mathbf q}\delta_{ab}, \\ \{c_{\mathbf k a},c_{\mathbf q b}\} &= 0, \\ \{c_{\mathbf k a}^{\dagger},c_{\mathbf q b}^{\dagger}\} &= 0. \end{aligned}$$

Each complete-mode number operator

$$n_{\mathbf k a} = c_{\mathbf k a}^{\dagger}c_{\mathbf k a}$$

obeys

$$n_{\mathbf k a}^2 = n_{\mathbf k a}, \qquad n_{\mathbf k a} \in \{0,1\}.$$

Fermionic antisymmetry is already built into this occupation-number construction. Adding particle labels and antisymmetrizing again would double-count the same physical states. See Fermionic Fock Space for the underlying construction.

Hamiltonian

Periodic wave vectors and free-particle energies are

$$\mathbf k = \frac{2\pi}{L}\mathbf n, \qquad \mathbf n \in \mathbb Z^3,$$

and

$$\epsilon_{\mathbf k} = \frac{\hbar^2k^2}{2m}.$$

The isolated Hamiltonian and number operator are

$$\begin{aligned} H &= \sum_{\mathbf k,a} \epsilon_{\mathbf k} n_{\mathbf k a}, \\ N &= \sum_{\mathbf k,a} n_{\mathbf k a}. \end{aligned}$$

Grand-canonical equilibrium uses the grand Hamiltonian

$$K = H-\mu N = \sum_{\mathbf k,a} (\epsilon_{\mathbf k}-\mu) n_{\mathbf k a}.$$

An occupation configuration $\lvert\{n_{\mathbf k a}\}\rangle$ is an exact eigenstate:

$$\begin{aligned} H\lvert\{n\}\rangle &= \left( \sum_{\mathbf k,a} \epsilon_{\mathbf k}n_{\mathbf k a} \right) \lvert\{n\}\rangle, \\ N\lvert\{n\}\rangle &= \left( \sum_{\mathbf k,a} n_{\mathbf k a} \right) \lvert\{n\}\rangle. \end{aligned}$$

The absence of an interaction term does not set the many-body ground energy to zero. At fixed $N$, antisymmetry forces occupation of successively higher one-particle levels.

Energy-zero covariance

Changing the one-particle energy convention by a constant $\Delta$ gives

$$\epsilon_{\mathbf k}' = \epsilon_{\mathbf k}+\Delta, \qquad H' = H+\Delta N.$$

The same physical grand-canonical state requires

$$\mu' = \mu+\Delta,$$

so that

$$H'-\mu'N = H-\mu N.$$

Thus occupations depend on $\epsilon_{\mathbf k}-\mu$, not on either energy separately. At fixed $N$, the total energy and Helmholtz free energy shift by $N\Delta$, while entropy, pressure for volume-independent $\Delta$, and occupation probabilities remain unchanged.

Symmetries and Conserved Quantities

The baseline Hamiltonian has:

• global $U(1)$ particle-number symmetry;
• continuous translations on the torus and conserved total momentum;
• the cubic point-group symmetry at finite $L$, approaching rotational symmetry in the continuum thermodynamic limit;
• parity and time-reversal symmetry when the internal states transform conventionally and no external field is present;
• global $U(g)$ internal symmetry because the $g$ components have identical dispersions and no component-dependent fields;
• conservation of every mode occupation $n_{\mathbf k a}$.

The last property is much stronger than ordinary energy and particle-number conservation. It makes the model free and mode-factorized, but it also prevents collision-driven redistribution among occupations. An ideal gas can be assigned an equilibrium ensemble; its isolated Hamiltonian does not by itself generate thermalization.

In the infinite continuum the isolated model is Galilean invariant. A finite periodic box permits only boosts compatible with its momentum grid, so finite-volume statements should use the declared boundary conditions rather than an informal continuum symmetry.

Natural Scales and Control Parameters

For a balanced gas with total density $n$ and internal degeneracy $g$, define

$$k_{\mathrm F} = \left( \frac{6\pi^2n}{g} \right)^{1/3}.$$

The corresponding energy, temperature, and velocity are

$$\begin{aligned} E_{\mathrm F} &= \frac{\hbar^2k_{\mathrm F}^2}{2m}, \\ T_{\mathrm F} &= \frac{E_{\mathrm F}}{k_{\mathrm B}}, \\ v_{\mathrm F} &= \frac{\hbar k_{\mathrm F}}{m}. \end{aligned}$$

Useful dimensionless controls are

$$\theta = \frac{T}{T_{\mathrm F}}, \qquad \eta = \beta\mu, \qquad z = e^{\beta\mu}.$$

The thermal wavelength is

$$\lambda_T = \sqrt{ \frac{2\pi\hbar^2} {m k_{\mathrm B}T} }.$$

For this three-dimensional baseline,

$$\frac{n\lambda_T^3}{g} = \frac{4}{3\sqrt\pi} \theta^{-3/2}.$$

Consequently, $\theta\ll1$ and $n\lambda_T^3/g\gg1$ describe the same degenerate regime for the stated dispersion. This identity is not universal across dimensions or dispersions.

Finite-size control requires at least

$$k_{\mathrm F}L \gg 1.$$

Near the Fermi energy, the mean one-particle level spacing estimated from the smooth density of states is

$$\delta_{\mathrm F} \simeq \frac{1}{V\rho(E_{\mathrm F})} = \frac{2E_{\mathrm F}}{3N}.$$

Actual cubic-box levels occur in degenerate arithmetic shells, so $\delta_{\mathrm F}$ is a smoothing scale rather than a promise of uniformly spaced levels.

Exact Solution Status

For finitely many retained modes, the grand partition function factorizes exactly:

$$\Xi = \prod_{\mathbf k,a} \left[ 1+e^{-\beta(\epsilon_{\mathbf k}-\mu)} \right].$$

The mean occupation is

$$\overline n_{\mathbf k a} = \frac{1} {e^{\beta(\epsilon_{\mathbf k}-\mu)}+1}.$$

The exactness claim is quantity-dependent.

Quantity	Status	Caveat
finite-volume energies and eigenstates	exact occupation-number solution	requires the declared one-particle spectrum and boundary conditions
fixed-$N$ ground state	exact filling of the $N$ lowest complete modes	an open shell can be degenerate
grand partition function	exact product over modes	the infinite product must be regularized through the physical volume and spectrum
fixed-$N$ partition function	exactly the coefficient of $z^N$ in the grand product	it does not factor into independent fixed-$N$ mode ensembles
equilibrium occupations	exact Fermi–Dirac values	ensemble and chemical potentials must be specified
grand-canonical correlators	Gaussian and reducible by Wick contractions	a general canonical thermal state is not grand-canonical Gaussian
free dynamics	exact independent mode phases	there is no intrinsic collision rate or thermalization
continuum thermodynamics	exact leading thermodynamic-limit result	shell corrections and order of limits remain separate
Sommerfeld series	controlled asymptotic expansion	it is not an exact finite-temperature identity
real material or cold-atom response	not fixed by the ideal model alone	bands, trapping, interactions, preparation, and probes matter

In the Heisenberg picture,

$$c_{\mathbf k a}(t) = e^{-i\epsilon_{\mathbf k}t/\hbar} c_{\mathbf k a}(0).$$

The occupations are constants of motion. Exact free evolution therefore says nothing about how a generic nonequilibrium distribution would approach a Fermi–Dirac form.

Thermodynamic Fingerprints

The model-factorization identities are

$$\begin{aligned} \ln\Xi &= \sum_{\mathbf k,a} \ln\left[ 1+e^{-\beta(\epsilon_{\mathbf k}-\mu)} \right], \\ N &= \sum_{\mathbf k,a} \overline n_{\mathbf k a}, \\ U &= \sum_{\mathbf k,a} \epsilon_{\mathbf k} \overline n_{\mathbf k a}. \end{aligned}$$

For one mode,

$$\operatorname{Var}(n_{\mathbf k a}) = \overline n_{\mathbf k a} \left( 1-\overline n_{\mathbf k a} \right).$$

The density of one-particle states per volume, including $g$, is

$$\rho(\epsilon) = \frac{g}{4\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt\epsilon.$$

The continuum number and energy densities are therefore

$$\begin{aligned} n &= \int_0^\infty d\epsilon\, \rho(\epsilon)f(\epsilon), \\ u &= \int_0^\infty d\epsilon\, \epsilon\rho(\epsilon)f(\epsilon), \end{aligned}$$

where

$$f(\epsilon) = \frac{1} {e^{\beta(\epsilon-\mu)}+1}.$$

At zero temperature and fixed density,

$$f(\epsilon) \longrightarrow \Theta(E_{\mathrm F}-\epsilon),$$

and the defining fingerprints are

$$\begin{aligned} n &= \frac{gk_{\mathrm F}^3}{6\pi^2}, \\ \frac{U_0}{N} &= \frac35 E_{\mathrm F}, \\ P_0 &= \frac25 nE_{\mathrm F}, \\ \kappa_T(0) &= \frac{3}{2nE_{\mathrm F}}. \end{aligned}$$

Here

$$\kappa_T = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_T.$$

For a quadratic continuum dispersion in the thermodynamic limit,

$$P = \frac23\frac{U}{V}$$

holds at every temperature. It is not a universal identity for a lattice band or a different dispersion.

At fixed density and $\theta\ll1$, the leading Sommerfeld fingerprints are

$$\begin{aligned} \frac{\mu}{E_{\mathrm F}} &= 1- \frac{\pi^2}{12}\theta^2 +O(\theta^4), \\ \frac{U}{NE_{\mathrm F}} &= \frac35 \left[ 1+ \frac{5\pi^2}{12}\theta^2 +O(\theta^4) \right], \\ \frac{C_V}{Nk_{\mathrm B}} &= \frac{\pi^2}{2}\theta +O(\theta^3). \end{aligned}$$

Only a shell of energy width $O(k_{\mathrm B}T)$ around $E_{\mathrm F}$ is thermally active. The occupied sea remains essential for the ground energy and pressure, but most particles do not contribute the classical heat capacity.

In the dilute classical regime,

$$x \equiv \frac{n\lambda_T^3}{g} \ll 1,$$

and the pressure begins as

$$\frac{P}{nk_{\mathrm B}T} = 1+ \frac{x}{2^{5/2}} +O(x^2).$$

The positive exchange correction is a statistical effect. It should not be described as a microscopic repulsive potential.

Important Limits

Limit	What survives	What requires care
finite $N$, $T\to0$	exact shell filling	open-shell degeneracy and chemical-potential intervals
thermodynamic limit, $T=0$	sharp Fermi sea and smooth density formulas	the limit replaces arithmetic shells by a continuum
$T/T_{\mathrm F}\ll1$	Sommerfeld organization and active shell	fixed-$N$ and fixed-$\mu$ coefficients differ
$n\lambda_T^3/g\ll1$	Maxwell–Boltzmann leading behavior	the first exchange correction remains positive
$g=1$	fully polarized or spinless gas	the same total density gives a larger $k_{\mathrm F}$
lower dimension	mode factorization remains exact	state-counting powers and density of states change
lattice dispersion	independent Bloch modes remain exact	no universal spherical sea or $P=2u/3$ identity
relativistic dispersion	Pauli filling remains	energy-zero, equation of state, and scaling change

The order of limits matters. Taking $T\to0$ at finite $N$ resolves individual shells. Taking the thermodynamic limit first produces a smooth Fermi surface. Neither description is wrong, but they answer different questions.

Finite Volume and Ensemble Audit

Order the complete one-particle energies, including internal multiplicity, as

$$\epsilon_{(1)} \leq \epsilon_{(2)} \leq \cdots.$$

The fixed-$N$ ground energy is

$$E_0(N) = \sum_{j=1}^{N} \epsilon_{(j)}.$$

Its removal and addition thresholds are

$$\begin{aligned} \mu_-(N) &= E_0(N)-E_0(N-1) = \epsilon_{(N)}, \\ \mu_+(N) &= E_0(N+1)-E_0(N) = \epsilon_{(N+1)}. \end{aligned}$$

At strict zero temperature, a grand-canonical chemical potential in

$$\mu_-(N) < \mu < \mu_+(N)$$

selects particle number $N$. If the highest shell is partially filled, the two thresholds can coincide and the canonical ground state may be degenerate. A single grand-canonical step function then does not encode an arbitrary symmetry-preserving partial filling without an additional limiting prescription.

At finite temperature the canonical partition function is obtained by coefficient extraction:

$$Z_N(\beta) = [z^N] \prod_{\alpha} \left( 1+ze^{-\beta\epsilon_\alpha} \right).$$

The grand-canonical product is often computationally simpler, but finite-system number fluctuations are then physical features of that ensemble rather than numerical error. Ensemble equivalence applies to suitable intensive observables in the thermodynamic limit, not to every finite-size fluctuation.

A continuum low-temperature calculation additionally needs a window

$$\delta_{\mathrm F} \ll k_{\mathrm B}T \ll E_{\mathrm F}.$$

The left inequality smooths individual levels; the right keeps the gas degenerate. For a very small system no broad window may exist.

Typical Observables

The natural observables include:

• mode occupations $n_{\mathbf k a}$ and the momentum distribution;
• total density, energy, pressure, entropy, and heat capacity;
• compressibility and spin or component susceptibilities;
• the one-body density matrix and equal-time Green function;
• density correlations, exchange holes, and the static structure factor;
• particle–hole response around the Fermi surface.

The Fourier density operator is

$$\rho_{\mathbf q} = \sum_{\mathbf k,a} c_{\mathbf k+\mathbf q,a}^{\dagger} c_{\mathbf k a}.$$

For equal component densities, same-component coincidence is forbidden,

$$g_{aa}^{(2)}(0) = 0,$$

whereas the unresolved total-density coincidence is

$$g^{(2)}(0) = 1- \frac1g.$$

This exchange hole is an observable consequence of antisymmetry, not evidence for a two-body repulsive term in $H$. Dynamic response, screening, and collective modes require the appropriate response page and, when interactions are added, a new model.

Minimal Worked Example

Consider two spatial orbitals with energies $0$ and $\Delta$, each carrying internal labels $\uparrow$ and $\downarrow$. At fixed $N=2$, there are

$$\binom42 = 6$$

allowed Slater determinants.

Occupation pattern	Energy	Degeneracy
both internal states in the lower orbital	$0$	$1$
one fermion in each spatial orbital	$\Delta$	$4$
both internal states in the upper orbital	$2\Delta$	$1$

With

$$x = e^{-\beta\Delta},$$

the canonical partition function is

$$Z_2 = 1+4x+x^2.$$

It is also the coefficient of $z^2$ in

$$\Xi = (1+z)^2 (1+zx)^2.$$

The mean energy is

$$U = \Delta \frac{4x+2x^2} {1+4x+x^2}.$$

Therefore

$$\begin{aligned} T\to0 &: U\to0, \\ T\to\infty &: U\to\Delta. \end{aligned}$$

The lower spatial orbital contains two particles in the ground state, but the occupied complete modes differ by their internal labels. The example demonstrates Pauli counting and coefficient extraction; it has neither a thermodynamic Fermi surface nor a phase transition.

Numerical Benchmark

The canonical numerical contract is MB-B007: Ideal Fermi Gas.

It uses $g=2$ and units

$$\frac{\hbar^2}{2m} = 1,$$

with density

$$n = \frac{1}{3\pi^2}.$$

Hence, numerically in these units,

$$k_{\mathrm F} = 1, \qquad E_{\mathrm F} = 1.$$

The number and energy densities are evaluated from

$$\begin{aligned} n &= \frac{1}{\pi^2} \int_0^\infty k^2 f(k^2)\,dk, \\ u &= \frac{1}{\pi^2} \int_0^\infty k^4 f(k^2)\,dk. \end{aligned}$$

At zero temperature, the exact targets are

$$\mu = 1, \qquad \frac{U}{N} = 0.6, \qquad \frac{P}{nE_{\mathrm F}} = 0.4.$$

As $\theta=T/T_{\mathrm F}\to0$, the scaled corrections must satisfy

$$\begin{aligned} \frac{1-\mu/E_{\mathrm F}} {\theta^2} &\longrightarrow \frac{\pi^2}{12}, \\ \frac{U/(NE_{\mathrm F})-3/5} {\theta^2} &\longrightarrow \frac{\pi^2}{4}, \\ \frac{C_V} {Nk_{\mathrm B}\theta} &\longrightarrow \frac{\pi^2}{2}. \end{aligned}$$

A robust implementation should:

1. solve the finite-temperature number equation for $\mu(T)$ rather than holding $\mu=E_{\mathrm F}$;
2. evaluate the occupation with a branch-stable logistic form so large positive exponents do not overflow;
3. bracket the unique number root using monotonicity;
4. report root residual, quadrature truncation, and asymptotic truncation separately;
5. compare a decreasing sequence of $\theta$ values instead of treating a low-order series as exact;
6. obtain $C_V$ through analytic derivatives or a convergence-tested fit rather than a noisy two-point difference.

Root uniqueness follows from

$$\begin{aligned} \left( \frac{\partial n}{\partial\mu} \right)_T &= \beta \int_0^\infty d\epsilon\, \rho(\epsilon) \\ &\qquad {} \times f(\epsilon) \left[1-f(\epsilon)\right] > 0. \end{aligned}$$

The notebook name ideal_fermi_gas_sommerfeld.ipynb is currently a planned artifact, not a committed executable. Its release status and promotion requirements belong to Reproducible Notebooks.

Variants and Handoffs

Polarized and multicomponent gases

If component densities differ, each component has its own Fermi scale:

$$k_{{\mathrm F},a} = \left( 6\pi^2 n_a \right)^{1/3}.$$

Using one total-density formula with a degeneracy factor $g$ assumes equal spectra, equal chemical potentials, and balanced populations. A Zeeman field or separately conserved populations changes that record.

Harmonic traps

A trap replaces translation invariance by a discrete one-particle spectrum. Mode factorization remains exact, but the density of states, shell closures, and thermodynamic limit differ. A local-density treatment uses

$$\mu(\mathbf r) = \mu_0-V_{\mathrm{trap}}(\mathbf r).$$

Use Quantum Gases in Traps for the canonical trapped treatment.

Lattices and bands

For noninteracting fermions in a band,

$$H = \sum_{\mathbf k,a} \epsilon_{\mathrm{band}}(\mathbf k) n_{\mathbf k a}.$$

The model is still quadratic, but the Brillouin zone, filling, band degeneracies, van Hove singularities, and possible filled-band gaps replace the universal spherical picture. The continuum relation $P=2u/3$ generally fails.

Relativistic dispersion

Relativistic fermions use, for example,

$$\epsilon_{\mathbf k} = \sqrt{ (\hbar ck)^2+m^2c^4 }.$$

Subtracting the rest energy is an allowed convention only if $\mu$ is shifted consistently. The ultrarelativistic equation of state and astrophysical scaling are not supplied by the nonrelativistic baseline.

Interactions and quasiparticles

Adding

$$\begin{aligned} H_{\mathrm{int}} &= \frac12 \int d^3x\,d^3y\, \psi^{\dagger}(\mathbf x) \psi^{\dagger}(\mathbf y) \\ &\qquad {} \times V(\mathbf x-\mathbf y) \psi(\mathbf y) \psi(\mathbf x). \end{aligned}$$

destroys independent conservation of the free momentum occupations. Same-component contact scattering is suppressed by antisymmetry, but opposite components, finite-range forces, and higher partial waves can interact. Hartree–Fock Approximation, Random Phase Approximation, and pairing theories then describe different controlled questions.

An effective-mass quasiparticle gas may reproduce low-energy thermodynamics of a Fermi liquid. It is an effective interacting description, not evidence that the microscopic particles are ideal.

Canonical Boundaries

This dossier owns:

• the convention-complete baseline record;
• the energy-zero, symmetry, and ensemble audit;
• the quantity-specific exactness statement;
• the comparison of finite-shell and thermodynamic claims;
• the model-variant and MB-B007 benchmark handoffs.

It does not rederive:

• the one-mode occupation law, owned by Fermi–Dirac Statistics;
• the uniform three-dimensional equation of state, owned by Ideal Fermi Gas;
• the low-temperature physical regime and applications, owned by Degenerate Fermi Gas;
• dimension-specific Fermi scales, owned by Fermi Momentum and Fermi Energy;
• general surface geometry and low-energy kinematics, owned by Fermi Surface;
• the asymptotic method, owned by Sommerfeld Expansion;
• interacting, material-specific, and experimental interpretations, owned by their respective volumes.

Common Mistakes

• Writing “ideal Fermi gas” without specifying spectrum, dimension, internal degeneracy, ensemble, and boundaries.
• Applying Pauli exclusion to a spatial orbital while ignoring a distinct internal label.
• Counting the degeneracy factor $g$ twice, or omitting it entirely.
• Using total density in a per-component Fermi-momentum formula.
• Treating $H-\mu N$ as the isolated Hamiltonian.
• Shifting one-particle energies without shifting the chemical potential.
• Setting the free many-body ground energy to zero because interactions vanish.
• Equating $\mu(T)$ with $E_{\mathrm F}$ at every temperature.
• Calling degeneracy pressure a microscopic repulsive force.
• Treating a finite open shell as a unique spherical Fermi sea.
• Applying continuum Sommerfeld coefficients near a band edge or unresolved discrete spectrum.
• Assuming exact free evolution supplies collisions, transport relaxation, or equilibration.
• Using $P=2u/3$ for a lattice band or relativistic dispersion.
• Calling an interacting electron, neutron, or cold-atom system ideal without stating a controlled approximation.

Exercises

Exercise 1: Shift the energy zero

Let every one-particle energy shift by the same constant $\Delta$. Show how the canonical partition function, Helmholtz free energy, chemical potential, and grand partition function transform.

Solution

Every fixed-$N$ many-body energy gains $N\Delta$, so

$$Z_N' = e^{-\beta N\Delta} Z_N.$$

Therefore

$$F_N' = -k_{\mathrm B}T\ln Z_N' = F_N+N\Delta.$$

The chemical potential shifts by $\Delta$. Setting $\mu'=\mu+\Delta$ leaves every exponent invariant:

$$\epsilon_\alpha'-\mu' = \epsilon_\alpha-\mu.$$

Hence

$$\Xi'(T,V,\mu') = \Xi(T,V,\mu).$$

Occupations and entropy are unchanged. A volume-independent $\Delta$ also leaves pressure unchanged, while the reported internal energy shifts by $N\Delta$.

Exercise 2: Recover the zero-temperature fingerprints

Starting from the three-dimensional density of states, derive $N$, $U_0/N$, and $P_0$ at fixed density.

Solution

At $T=0$, integrate through $E_{\mathrm F}$:

$$n = \int_0^{E_{\mathrm F}} d\epsilon\, \rho(\epsilon).$$

Since $\rho(\epsilon)=C\epsilon^{1/2}$,

$$n = \frac{2C}{3} E_{\mathrm F}^{3/2}.$$

Likewise,

$$u_0 = \int_0^{E_{\mathrm F}} d\epsilon\, \epsilon\rho(\epsilon) = \frac{2C}{5} E_{\mathrm F}^{5/2}.$$

Dividing gives

$$\frac{U_0}{N} = \frac{u_0}{n} = \frac35E_{\mathrm F}.$$

For a quadratic continuum gas, $P=2u/3$, so

$$P_0 = \frac25nE_{\mathrm F}.$$

Substituting the explicit constant $C$ recovers $n=gk_{\mathrm F}^3/(6\pi^2)$.

Exercise 3: Number fluctuations in the active shell

Show that the grand-canonical number variance obeys $\operatorname{Var}(N)=k_{\mathrm B}T(\partial N/\partial\mu)_T$. Estimate its leading low-temperature value for the three-dimensional gas.

Solution

Independent modes give

$$\operatorname{Var}(N) = \sum_\alpha f_\alpha(1-f_\alpha).$$

Because

$$\frac{\partial f_\alpha}{\partial\mu} = \beta f_\alpha(1-f_\alpha),$$

one obtains

$$\operatorname{Var}(N) = k_{\mathrm B}T \left( \frac{\partial N}{\partial\mu} \right)_T.$$

At low temperature, the derivative is the total density of states at the Fermi energy:

$$\begin{aligned} \left( \frac{\partial N}{\partial\mu} \right)_T &= V\rho(E_{\mathrm F}) +O(T^2), \\ &= \frac{3N}{2E_{\mathrm F}} +O(T^2). \end{aligned}$$

Therefore

$$\operatorname{Var}(N) = \frac32N\theta +O(N\theta^3).$$

Only the thermal shell contributes. Relative root-mean-square fluctuations scale as $\sqrt{3\theta/(2N)}$ and vanish in the thermodynamic limit at fixed $\theta$.

Exercise 4: Diagnose an open shell

Use the ordered finite-volume spectrum to show when a zero-temperature chemical-potential interval selects exactly $N$ particles. What changes when $\epsilon_{(N)}=\epsilon_{(N+1)}$?

Solution

The $N$-particle grand energy is

$$\Phi_N = E_0(N)-\mu N.$$

Requiring $\Phi_N<\Phi_{N-1}$ gives

$$\mu > E_0(N)-E_0(N-1) = \epsilon_{(N)}.$$

Requiring $\Phi_N<\Phi_{N+1}$ gives

$$\mu < E_0(N+1)-E_0(N) = \epsilon_{(N+1)}.$$

Thus a nonempty interval exists when

$$\epsilon_{(N)} < \epsilon_{(N+1)}.$$

If these energies are equal, the shell is only partially filled and several particle numbers or configurations meet at the same grand energy. The fixed-$N$ canonical problem remains well defined, but a strict zero-temperature grand-canonical step needs an additional prescription to represent partial filling.

Exercise 5: Derive the leading exchange correction

Let $x=n\lambda_T^3/g\ll1$. Using the small-fugacity expansions, derive the first correction to the classical pressure and interpret its sign.

Solution

For fermions,

$$\begin{aligned} x &= z- \frac{z^2}{2^{3/2}} +O(z^3), \\ \frac{P\lambda_T^3} {gk_{\mathrm B}T} &= z- \frac{z^2}{2^{5/2}} +O(z^3). \end{aligned}$$

Inverting the first relation gives

$$z = x+ \frac{x^2}{2^{3/2}} +O(x^3).$$

Substitution into the pressure yields

$$\frac{P\lambda_T^3} {gk_{\mathrm B}T} = x+ \frac{x^2}{2^{5/2}} +O(x^3).$$

Dividing by $x$ gives

$$\frac{P}{nk_{\mathrm B}T} = 1+ \frac{x}{2^{5/2}} +O(x^2).$$

The positive sign reflects reduced same-state occupancy from exchange. It is a statistical correction, not a repulsive pair potential.

References

1. R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
2. K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987).
3. M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
4. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
5. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston (1976).
6. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980).
7. S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases”, Reviews of Modern Physics 80, 1215–1274 (2008).
8. I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases”, Reviews of Modern Physics 80, 885–964 (2008).
9. B. DeMarco and D. S. Jin, “Onset of Fermi degeneracy in a trapped atomic gas”, Science 285, 1703–1706 (1999).
10. D. Pines and P. Nozières, The Theory of Quantum Liquids, Volume I: Normal Fermi Liquids, CRC Press (2018 reprint).

Cross-Links

• Model Encyclopedia Overview for the shared dossier and exactness conventions.
• Ideal Fermi Gas for the full continuum thermodynamic derivation.
• Degenerate Fermi Gas for the low-temperature regime, Pauli blocking, and applications.
• Fermi Momentum and Fermi Energy for dimension- and degeneracy-aware formulas.
• Fermi Surface for surface geometry and low-energy kinematics.
• Sommerfeld Expansion for the controlled asymptotic method.
• Fermi–Dirac Distribution for quick formula lookup.
• Ideal Fermi Gas Model Card for the short Reference locator.
• MB-B007 Benchmark for the stable numerical contract.
• Reproducible Notebooks for the planned artifact and promotion gates.