From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 11
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This is where the kinetic derivation connects with modern quantum fluids. Up to now we have treated quantum statistics as a modification of the equilibrium distribution. In this chapter we will explore what happens in the strongly quantum regimes, where the gas is no longer a small correction to the classical picture.
There are two fundamentally different limits:
- Degenerate Fermi gases
- dominated by Pauli exclusion,
- finite pressure exists even at zero temperature,
- transport is controlled by excitations near the Fermi surface.
- Degenerate Bose gases
- dominated by macroscopic occupation of quantum states,
- Bose enhancement becomes extreme,
- a condensate component appears and a single-fluid kinetic description becomes incomplete.
The interesting point is that the Chapman–Enskog framework still tells us where these effects enter.
Chapter 11 — Degenerate Quantum Fluids: Fermi and Bose Limits of Quantum Navier–Stokes
11.1 The Quantum Degeneracy Parameter
The crossover between classical and quantum behavior is controlled by
\[\boxed{n\lambda_T^3}\]where
\[\lambda_T= \frac{h}{\sqrt{2\pi mk_BT}}\]is the thermal wavelength.
Three regimes exist:
\[n\lambda_T^3\ll1\]classical gas,
\[n\lambda_T^3\sim1\]quantum gas,
\[n\lambda_T^3\gg1\]strongly degenerate gas.
The Chapman–Enskog derivation remains formally valid as long as local equilibrium exists, but the thermodynamics and collision physics change dramatically.
Part I — Degenerate Fermi Fluids
11.2 The Zero-Temperature Fermi Gas
For fermions,
\[f_F = \frac{1} {z^{-1}e^{mc^2/(2k_BT)}+1}.\]At very low temperature,
\[T\rightarrow0\]the distribution approaches a step function:
\[\boxed{f_F(p) = \Theta(p_F-p)}\]where
\[p_F = \hbar(3\pi^2n)^{1/3}\]is the Fermi momentum.
The particles fill all states up to the Fermi surface.
11.3 Fermi Pressure
The pressure becomes
\[p_F = \frac{2}{5} nE_F\]where
\[\boxed{E_F= \frac{p_F^2}{2m}}\]is the Fermi energy.
This is a remarkable result:
\[\boxed{p_F\neq0 \quad\text{even when}\quad T=0.}\]The pressure is not thermal.
It is a purely quantum mechanical consequence of the Pauli exclusion principle.
11.4 Quantum Euler Equation for a Fermi Fluid
The momentum equation becomes
\[\rho \frac{D u_i}{Dt} = -\partial_i p_F .\]Using
\[p_F = \frac{2}{5}nE_F\]we obtain
\[\boxed{\rho \frac{D\mathbf{u}}{Dt} = -\nabla \left( \frac{2}{5}nE_F \right).}\]This is the macroscopic manifestation of degeneracy pressure.
It appears in:
- electron gases in metals,
- white dwarf stars,
- neutron matter,
- ultracold Fermi atoms.
11.5 Fermi Surface and Transport
At low temperature, almost all states are blocked.
A collision can only occur if particles are excited near the Fermi surface.
The available phase space scales as
\[\sim \left(\frac{k_BT}{E_F}\right)^2.\]Therefore,
\[\boxed{\tau_F^{-1} \propto T^2.}\]This is the famous Fermi-liquid result.
Consequently,
\[\boxed{\mu_F \propto \tau_F \propto T^{-2}.}\]The viscosity increases strongly as the temperature decreases.
11.6 The Fermi Heat Flux
Unlike viscosity, heat transport requires excitations.
At zero temperature:
\[\boxed{\kappa_F\rightarrow0.}\]There are no thermal excitations available.
At low temperature:
\[C_V \propto T\]and
\[\kappa_F \sim C_V v_F^2\tau_F.\]Therefore,
\[\boxed{\kappa_F \propto T^{-1}.}\]This is a hallmark of degenerate Fermi liquids.
Part II — Degenerate Bose Fluids
11.7 Bose–Einstein Condensation
For bosons,
\[f_B = \frac{1}{ z^{-1}e^{mc^2/(2k_BT)}-1 }.\]The fugacity satisfies
\[0<z\le1.\]At the critical temperature:
\[z\rightarrow1.\]The excited states cannot accommodate all particles.
The excess particles accumulate in the ground state:
\[\boxed{N_0/N = 1- \left( \frac{T}{T_c} \right)^{3/2}.}\]This is Bose–Einstein condensation.
11.8 Why Single-Fluid BGK Breaks Down
The kinetic distribution describes thermal particles.
But the condensate is a coherent quantum state.
The system naturally separates into:
Condensate
\[n_0,\mathbf{u}_0\]Thermal cloud
\[n_T,\mathbf{u}_T\]Therefore the hydrodynamics becomes:
\[\boxed{\text{two-fluid hydrodynamics}.}\]This is the foundation of Landau’s theory of superfluidity.
11.9 Superfluid Equations
The condensate velocity is not determined by collisions.
Instead,
\[\boxed{\mathbf{u}_s = \frac{\hbar}{m}\nabla\phi}\]where \(\phi\) is the condensate phase.
The superfluid component obeys
\[\boxed{m \frac{\partial\mathbf{u}_s}{\partial t} + \nabla \left( \frac{1}{2}mu_s^2 +\mu \right) =0.}\]This is no longer a dissipative Navier–Stokes equation.
It is closer to an inviscid Euler equation.
11.10 Thermal Bose Component
The normal component still obeys kinetic theory.
Its transport coefficients involve Bose integrals:
\[G_\nu^{(-)}(z).\]Near condensation:
\[z\rightarrow1^-\]these integrals become strongly enhanced.
The collision term contains
\[1+f_B\]which increases scattering into occupied states.
This leads to:
- enhanced scattering,
- unusual viscosity,
- anomalous thermal transport.
11.11 Quantum Navier–Stokes Structure
For a degenerate quantum gas we can write schematically:
\[\boxed{\rho \frac{D\mathbf{u}}{Dt} = -\nabla p_q + \nabla\cdot\Pi_q}\]with
\[\Pi_q = -\mu_q \left[ \nabla\mathbf{u} + (\nabla\mathbf{u})^T -\frac{2}{3}I\nabla\cdot\mathbf{u} \right].\]The new physics is entirely in:
\[p_q\] \[\mu_q\] \[\kappa_q.\]11.12 Comparison of Extreme Limits
| Classical | Degenerate Fermi | Condensed Bose | |
|---|---|---|---|
| Pressure source | thermal | Pauli exclusion | interactions/thermal cloud |
| \(T=0\) pressure | zero | finite | depends on interactions |
| Collisions | normal | Pauli blocked | Bose enhanced |
| Viscosity | finite | grows at low \(T\) | strongly modified |
| Heat conduction | classical | vanishes as \(T\to0\) | two-fluid transport |
| Hydrodynamics | Navier–Stokes | Fermi liquid | superfluid hydrodynamics |
11.13 The Deeper Connection
The Chapman–Enskog expansion reveals something subtle:
The Navier–Stokes equations are not fundamentally “classical.”
Their structure comes from:
- conservation laws,
- symmetry,
- near-equilibrium expansion.
Those ingredients also exist in quantum gases.
What changes is the equation of state and transport physics.
The same mathematical skeleton can describe:
- air,
- electron fluids,
- neutron matter,
- ultracold atomic gases,
provided the appropriate kinetic equilibrium and collision physics are inserted.
Where this leads next
A particularly interesting path would be:
Chapter 12 — From Quantum BGK to Quantum Hydrodynamics: the connection with the Gross–Pitaevskii equation, Wigner functions, and quantum pressure terms.
That chapter would address a subtle point: the Navier–Stokes equations derived here are quantum statistical equations, but they do not yet contain the dispersive “quantum pressure” term familiar from Bose–Einstein condensates:
\[-\frac{\hbar^2}{2m} \nabla \left( \frac{\nabla^2\sqrt n}{\sqrt n} \right).\]Understanding why that term is absent from BGK-based quantum hydrodynamics—and how it emerges from a different kinetic starting point—is one of the most interesting bridges between kinetic theory and modern quantum fluids.