From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 8
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We now proceed to repeat the Chapter 5 calculation for \(f_q^{eq}\) and derive the explicit quantum \(f_q^{(1)}\).
Chapter 8 — Explicit Quantum Chapman–Enskog Correction \(f_q^{(1)}\)
8.1 The Quantum BGK Expansion
We start from the quantum BGK equation:
\[\partial_t f+v_i\partial_i f = -\frac{1}{\tau}(f-f_q^{eq}).\]The Chapman–Enskog expansion is
\[f = f_q^{(0)} + \varepsilon f_q^{(1)} +\cdots .\]As before, the zeroth-order equation gives
\[\boxed{f_q^{(0)}=f_q^{eq}.}\]The first-order equation is
\[\boxed{f_q^{(1)} = -\tau (\partial_{t_1}+v_i\partial_i) f_q^{eq}.}\]The entire problem is therefore evaluating the streaming derivative of the quantum equilibrium distribution.
8.2 Quantum Local Equilibrium Distribution
We write
\[\boxed{f_q^{eq} = \frac{g}{h^3} \frac{1}{ z^{-1} e^{\beta_E mc^2/2} +\eta }}\]where
\[\beta_E=\frac{1}{k_BT}\]and
\[\eta= \begin{cases} +1,&\text{Fermi gas},\\ -1,&\text{Bose gas}. \end{cases}\]It is useful to define
\[X = z^{-1}e^{\beta_E mc^2/2}.\]Then
\[f_q^{eq} = \frac{g}{h^3} \frac{1}{X+\eta}.\]8.3 The Quantum Generalization of the Logarithmic Derivative
For the Maxwellian we used
\[Df^{eq} = f^{eq}D\ln f^{eq}.\]For quantum statistics this is slightly modified.
Differentiate:
\[D f_q^{eq} = -\frac{g}{h^3} \frac{DX}{(X+\eta)^2}.\]But
\[f_q^{eq} = \frac{g}{h^3} \frac{1}{X+\eta}.\]Therefore,
\[D f_q^{eq} = -f_q^{eq} \frac{DX}{X+\eta}.\]Now,
\[\frac{X}{X+\eta} = \frac{h^3}{g}f_q^{eq}X\]is inconvenient, so we introduce the quantum occupation factor
\[\boxed{\Phi_q = 1-\eta\frac{h^3}{g}f_q^{eq}.}\]For fermions:
\[\Phi_F=1-\frac{h^3}{g}f_q^{eq}\]and for bosons:
\[\Phi_B=1+\frac{h^3}{g}f_q^{eq}.\]Then the derivative becomes
\[\boxed{Df_q^{eq} = -f_q^{eq}\Phi_q,D\ln X.}\]This is one of the key differences from the classical theory.
8.4 Physical Meaning of the Quantum Factor
The factor
\[\boxed{\Phi_q = 1-\eta f}\](up to normalization conventions) is the Bose enhancement / Pauli blocking factor.
For collisions:
- Fermions: \(1-f\) suppresses transitions into occupied states.
- Bosons: \(1+f\) enhances transitions into occupied states.
This factor is the kinetic origin of quantum statistical effects.
In the dilute limit,
\[f_q^{eq}\ll1\]and therefore
\[\Phi_q\rightarrow1.\]We recover the classical calculation.
8.5 Differentiating the Quantum Exponent
We have
\[X=z^{-1}e^{mc^2/(2k_BT)}.\]Taking the logarithm:
\[\ln X = -\ln z + \frac{mc^2}{2k_BT}.\]Therefore,
\[D\ln X = -D\ln z + D \left( \frac{mc^2}{2k_BT} \right).\]The second term is identical to the classical calculation.
The new term is
\[D\ln z.\]8.6 The Role of Fugacity
The fugacity is
\[z=e^{\mu/(k_BT)}.\]Therefore,
\[\ln z = \frac{\mu}{k_BT}.\]Hence
\[D\ln z = D \left( \frac{\mu}{k_BT} \right).\]Unlike the classical case, chemical potential becomes an independent thermodynamic variable.
This is where quantum hydrodynamics differs fundamentally from the dilute gas.
8.7 Thermodynamic Derivative Identities
The hydrodynamic variables are now
\[n,\quad T,\quad u_i.\]The chemical potential is not independent; it is determined by
\[n=n(T,z).\]For an ideal quantum gas,
\[n= \frac{g}{\lambda_T^3} G_{3/2}^{(\eta)}(z).\]Taking the logarithm,
\[\ln n = \ln g -\frac{3}{2}\ln\lambda_T^2 + \ln G_{3/2}^{(\eta)}(z).\]Since
\[\lambda_T\propto T^{-1/2}\]we obtain
\[d\ln n = \frac{3}{2}d\ln T + d\ln G_{3/2}^{(\eta)}.\]The derivative identity is
\[\boxed{z\frac{d}{dz}G_\nu^{(\eta)}(z) = G_{\nu-1}^{(\eta)}(z).}\]Therefore,
\[d\ln G_{3/2} = \frac{G_{1/2}}{G_{3/2}} d\ln z.\]Hence
\[\boxed{d\ln z = \frac{G_{3/2}}{G_{1/2}} \left( d\ln n -\frac{3}{2}d\ln T \right).}\]This replaces the simple classical relation between density, temperature, and the Maxwellian normalization.
8.8 Streaming Derivative
Now we repeat the classical calculation.
The operator is
\[D = \frac{D_h}{Dt} + c_i\partial_i.\]The quantity
\[D\ln X\]contains:
- density gradients,
- temperature gradients,
- velocity gradients.
After substituting the Euler equations for a quantum ideal gas,
\[\frac{D_hn}{Dt} = -n\nabla\cdot u\] \[\frac{D_hu_i}{Dt} = -\frac{1}{mn}\partial_ip_q\]and
\[\frac{D_hT}{Dt} = -\frac{2}{3}T\nabla\cdot u\]the same cancellation mechanism occurs.
The result can be organized as
\[\boxed{\begin{aligned} D f_q^{eq} =& -f_q^{eq}\Phi_q \Bigg[ 2A_q \left( c_ic_j-\frac{1}{3}c^2\delta_{ij} \right) \partial_j u_i \ & + B_q(c^2) c_i\partial_iT \Bigg]. \end{aligned}}\]The coefficients \(A_q\) and \(B_q\) depend on the quantum thermodynamics.
8.9 Quantum First-Order Distribution
Therefore,
\[\boxed{\begin{aligned} f_q^{(1)} =& \tau f_q^{eq}\Phi_q \Bigg[ 2A_q \left( c_ic_j-\frac{1}{3}c^2\delta_{ij} \right) \partial_j u_i \ & + B_q(c^2)c_i\partial_iT \Bigg]. \end{aligned}}\]This is the quantum analogue of the classical result.
The important structural statement is:
\[\boxed{\text{Quantum statistics modify coefficients, not tensor structure.}}\]The shear mode is still:
\[c_ic_j-\frac{1}{3}c^2\delta_{ij}\]and the heat mode is still a vector proportional to
\[c_i\times\text{function}(c^2).\]8.10 Classical Limit Check
Let
\[z\ll1.\]Then
\[G_\nu^{(\eta)}(z)\rightarrow z\]and
\[\Phi_q\rightarrow1.\]Therefore,
\[A_q\rightarrow\beta_E\]and
\[B_q(c^2) \rightarrow \frac{1}{T} \left( \beta_E c^2-\frac{5}{2} \right).\]Thus,
\[f_q^{(1)} \rightarrow f^{(1)}_{MB}.\]The classical Chapman–Enskog correction is recovered exactly.
8.11 What Remains to Be Done
We now have the quantum nonequilibrium distribution.
The next chapter will perform the final step:
\[P_{ij} = m\int c_ic_j f_q^{(1)}\,d^3c\]and
\[q_i = \frac{1}{2}m\int c^2c_i f_q^{(1)}\,d^3c.\]The tensor reductions are identical to Chapter 6, but the scalar moments become
\[G_{5/2}^{(\eta)}, \quad G_{7/2}^{(\eta)}, \quad G_{9/2}^{(\eta)}\]and the transport coefficients become:
\[\boxed{\mu_F,\mu_B}\]and
\[\boxed{\kappa_F,\kappa_B}\]with explicit dependence on fugacity.
The next chapter will therefore be the quantum analogue of Chapter 6: deriving the Fermi and Bose corrections to viscosity and thermal conductivity, and showing explicitly how the classical BGK result emerges as the dilute limit.