From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 7
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We have now completed the classical derivation up to the Navier–Stokes closure. This gives us a very useful reference point, because the quantum derivation will not be a completely new calculation—it will be a deformation of the classical calculation.
The central question we will answer is:
If the particles obey Bose–Einstein or Fermi–Dirac statistics instead of Maxwell–Boltzmann statistics, how does the Chapman–Enskog derivation change?
The short answer is:
- The kinematic structure of the derivation is almost unchanged.
- The equilibrium distribution changes.
- The velocity moments change.
- The equation of state and transport coefficients become functions of quantum statistics.
Let’s begin.
Chapter 7 — Quantum Kinetic Theory: From BGK to Quantum Hydrodynamics
7.1 Why Classical Kinetic Theory Must Be Modified
The classical Boltzmann distribution assumes that particles are distinguishable and that occupation numbers of phase-space states are small.
This requires the gas to be dilute in the quantum sense:
\[n\lambda_T^3\ll1\]where
\[\lambda_T = \frac{h}{\sqrt{2\pi mk_BT}}\]is the thermal de Broglie wavelength.
When
\[n\lambda_T^3\sim1\]the wave nature of particles becomes important.
The classical Maxwell–Boltzmann distribution
\[f^{eq}_{MB} = n \left( \frac{m}{2\pi k_BT} \right)^{3/2} e^{-\frac{mc^2}{2k_BT}}\]is no longer correct.
The occupation of quantum states must be included.
7.2 Quantum Statistics: Bose and Fermi Symmetry
Identical particles are not distinguishable.
The many-particle wavefunction must satisfy:
Bosons
Particles with integer spin satisfy
\[\Psi(\ldots,x_i,\ldots,x_j,\ldots) = -\Psi(\ldots,x_j,\ldots,x_i,\ldots).\]They may occupy the same quantum state.
Examples:
- photons,
- helium-4 atoms,
- many composite particles.
Fermions
Particles with half-integer spin satisfy
\[\Psi(\ldots,x_i,\ldots,x_j,\ldots) = -\Psi(\ldots,x_j,\ldots,x_i,\ldots).\]They obey the Pauli exclusion principle.
Examples:
- electrons,
- neutrons,
- helium-3 atoms.
This symmetry modifies the equilibrium occupation number.
7.3 The Quantum Equilibrium Distribution
The equilibrium distribution is obtained by maximizing the quantum entropy, not the classical Boltzmann entropy.
The result is
\[\boxed{f^{eq}_q = \frac{g}{h^3} \frac{1} {\exp \left[ \frac{\frac{1}{2}m c^2-\mu}{k_BT} \right] +\eta}.}\]Here:
\[\eta= \begin{cases} +1,&\text{fermions},\[4pt] -1,&\text{bosons}, \end{cases}\]and
- \(g\) is the internal degeneracy (spin states),
- \(\mu\) is the chemical potential.
The variable
\[c=v-u\]is still the peculiar velocity.
This is important:
\[\boxed{\text{The Galilean structure is unchanged.}}\]The equilibrium is still isotropic in the local rest frame.
7.4 Classical Limit
The Maxwellian must emerge when quantum effects disappear.
Define the fugacity
\[z=e^{\mu/(k_BT)}.\]Then
\[f^{eq}_q = \frac{g}{h^3} \frac{1} {z^{-1}e^{\frac{mc^2}{2k_BT}}+\eta}.\]If
\[z\ll1\]then
\[z^{-1}e^{mc^2/(2k_BT)}\gg1.\]Therefore,
\[\frac{1}{z^{-1}e^x+\eta} \approx ze^{-x}.\]Hence
\[f^{eq}_q \rightarrow \frac{gz}{h^3} e^{-\frac{mc^2}{2k_BT}}.\]After normalization, this becomes the Maxwellian.
Therefore,
\[\boxed{\text{Maxwell–Boltzmann theory is the dilute limit of quantum kinetic theory.}}\]7.5 The Quantum BGK Equation
The classical BGK equation was
\[\partial_tf+v_i\partial_if = -\frac{1}{\tau}(f-f^{eq}).\]The natural quantum analogue is
\[\boxed{\partial_tf+v_i\partial_if = -\frac{1}{\tau} (f-f^{eq}_q).}\]At first sight, this looks identical.
However, there is an important subtlety.
For the classical BGK model,
\[f^{eq}\]is the Maxwellian.
For the quantum model,
\[f^{eq}_q\]must be the Bose–Einstein or Fermi–Dirac distribution.
Also, conservation requires
\[\int \begin{pmatrix} 1\ v_i\ \frac{1}{2}v^2 \end{pmatrix} (f-f_q^{eq}) \,d^3v\]The relaxation must preserve:
- particle number,
- momentum,
- energy.
Exactly the same collision invariants survive.
7.6 Quantum Macroscopic Variables
The definitions remain unchanged:
Number density:
\[\boxed{n = \int f\,d^3v.}\]Velocity:
\[\boxed{nu_i = \int v_i f\,d^3v.}\]Internal energy:
\[\boxed{ne = \frac{1}{2}m \int c^2f\,d^3v.}\]Stress tensor:
\[\boxed{P_{ij} = m \int c_ic_jf\,d^3v.}\]Heat flux:
\[\boxed{q_i = \frac{1}{2}m \int c^2c_if\,d^3v.}\]The reason is simple:
These are definitions of moments, not consequences of Maxwell–Boltzmann statistics.
7.7 Quantum Euler Equations
Because the collision invariants are unchanged, taking moments of the quantum BGK equation gives exactly the same conservation laws:
Mass:
\[\partial_t\rho+\partial_i(\rho u_i)=0.\]Momentum:
\[\partial_t(\rho u_j) + \partial_i(\rho u_i u_j+P_{ij}) =0.\]Energy:
\[\partial_tE+ \partial_i(Eu_i+P_{ij}u_j+q_i)=0.\]The difference appears only when we impose equilibrium.
For classical gases:
\[P_{ij}^{eq}=nk_BT\delta_{ij}.\]For quantum gases:
\[P_{ij}^{eq}=p_q\delta_{ij}\]where
\[\boxed{p_q\neq nk_BT.}\]The equation of state is modified by quantum statistics.
7.8 Quantum Thermodynamic Integrals
All equilibrium moments can be expressed using the quantum integral
\[\boxed{G_\nu^{(\eta)}(z) = \frac{1}{\Gamma(\nu)} \int_0^\infty \frac{x^{\nu-1}} {z^{-1}e^x+\eta} dx.}\]where:
- \(\eta=+1\): Fermi–Dirac,
- \(\eta=-1\): Bose–Einstein.
The important moments become:
Number density:
\[\boxed{n = \frac{g}{\lambda_T^3} G_{3/2}^{(\eta)}(z).}\]Pressure:
\[\boxed{p_q = \frac{gk_BT}{\lambda_T^3} G_{5/2}^{(\eta)}(z).}\]Energy density:
\[\boxed{\rho e = \frac{3}{2}p_q.}\]Therefore,
\[\boxed{p_q = n k_BT \frac{ G_{5/2}^{(\eta)}(z) }{ G_{3/2}^{(\eta)}(z) }.}\]This ratio is the first major quantum correction.
For the classical limit,
\[G_\nu(z)\rightarrow z\]so
\[p_q\rightarrow nk_BT.\]7.9 The Chapman–Enskog Expansion Again
Now comes the important comparison.
We again write
\[\boxed{f = f_q^{(0)} + \varepsilon f_q^{(1)} +\cdots.}\]The zeroth-order equation gives
\[\boxed{f_q^{(0)} = f_q^{eq}.}\]The first-order equation gives
\[\boxed{f_q^{(1)} = -\tau (\partial_{t_1}+v_i\partial_i) f_q^{eq}.}\]This equation is formally identical to the classical one.
The difference is entirely contained inside
\[f_q^{eq}.\]7.10 What Will Change in the Next Steps?
When we repeat the Chapter 5 calculation, we will again differentiate
\[\ln f_q^{eq}.\]The result will again separate into:
Shear mode
\[\left( c_ic_j-\frac{1}{3}c^2\delta_{ij} \right) \partial_j u_i\]producing viscosity.
Heat mode
\[c_i(\text{function of }c^2)\partial_iT\]producing heat conduction.
The tensor structure survives.
The only changes are:
Classical:
\[\int c^{2n}f_{MB}\,d^3c\]becomes
Quantum:
\[\int c^{2n}f_{FD/BE}\,d^3c.\]Those integrals introduce the functions
\[G_{1/2},G_{3/2},G_{5/2},G_{7/2},\ldots\]into the transport coefficients.
Chapter 7 Summary
We have established the quantum counterpart of the kinetic foundation:
| Classical | Quantum |
|---|---|
| Maxwellian | Bose/Fermi distribution |
| Boltzmann entropy | Quantum entropy |
| \(p=nk_BT\) | \(p=p(n,T,z)\) |
| Gaussian moments | Quantum statistical integrals |
| Same conservation laws | Same conservation laws |
| Same Chapman–Enskog structure | Same Chapman–Enskog structure |
The next chapter will repeat the Chapter 5 calculation for
\[f_q^{eq} = \frac{g}{h^3} \frac{1} {z^{-1}e^{mc^2/(2k_BT)}+\eta}\]and derive the explicit quantum \(f_q^{(1)}\). The algebra is more subtle because derivatives of the fugacity and chemical potential appear, but the same cancellations and tensor decomposition will occur. After that, we will calculate the Fermi and Bose corrections to viscosity and thermal conductivity.