From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 9
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We now arrive at the final closure step: extracting the quantum viscous stress tensor and quantum heat flux from \(f_q^{(1)}\).
This chapter will look very familiar because the symmetry machinery from the classical case survives completely. The only thing that changes is the evaluation of scalar velocity moments.
A useful way to think about the whole quantum correction is:
\[\boxed{\text{same tensors} + \text{different scalar integrals} = \text{quantum transport coefficients}.}\]Chapter 9 — Quantum Stress Tensor and Heat Flux
9.1 Starting Point
We have obtained the first-order quantum correction:
\[\boxed{f_q^{(1)} = \tau f_q^{eq}\Phi_q \left[ 2A_q \left( c_ic_j-\frac{1}{3}c^2\delta_{ij} \right) \partial_j u_i + B_q(c^2)c_i\partial_iT \right].}\]The nonequilibrium stress tensor is
\[\Pi_{ij} = m \int c_ic_j f_q^{(1)} \,d^3c.\]The heat flux is
\[q_i = \frac{1}{2}m \int c^2c_i f_q^{(1)} \,d^3c.\]9.2 Quantum Isotropy
The equilibrium distribution is still isotropic:
\[f_q^{eq}=F(c^2).\]The additional quantum factor
\[\Phi_q = 1-\eta\frac{h^3}{g}f_q^{eq}\]is also only a function of \(c^2\).
Therefore all integrals have the same tensor decomposition rules as before.
For example,
\[\int c_ic_jF(c^2)d^3c = \frac{\delta_{ij}}3 \int c^2F(c^2)d^3c\]and
\[\int c_ic_jc_kc_lF(c^2)d^3c = \frac{1}{15} (\delta_{ij}\delta_{kl} +\delta_{ik}\delta_{jl} +\delta_{il}\delta_{jk}) \times \int c^4F(c^2)d^3c.\]The quantum statistics have not altered rotational symmetry.
9.3 The Quantum Shear Stress
Insert the shear part:
\[\Pi_{ij}^{(s)} = 2m\tau A_q \partial_lu_k \int c_ic_j \left( c_kc_l-\frac{1}{3}c^2\delta_{kl} \right) f_q^{eq}\Phi_q\,d^3c .\]Define the quantum fourth moment:
\[M_4^{(q)} = \int c^4f_q^{eq}\Phi_q\,d^3c.\]Using the tensor identities,
\[\int c_ic_j \left( c_kc_l-\frac{1}{3}c^2\delta_{kl} \right) f_q^{eq}\Phi_qd^3c = \frac{M_4^{(q)}}{15} \left( \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk} -\frac{2}{3}\delta_{ij}\delta_{kl} \right).\]Therefore,
\[\boxed{\Pi_{ij} = \mu_q \left( \partial_i u_j +\partial_j u_i -\frac{2}{3}\delta_{ij}\partial_k u_k \right).}\]The sign convention depends on whether \(\Pi_{ij}\) is defined as the full stress correction or the viscous stress. Using the usual Navier–Stokes convention,
\[\boxed{\Pi_{ij}^{visc} = -\mu_q \left( \partial_i u_j +\partial_j u_i -\frac{2}{3}\delta_{ij}\partial_k u_k \right).}\]9.4 Quantum Viscosity Coefficient
The coefficient is
\[\boxed{\mu_q = \frac{2}{15}m\tau A_qM_4^{(q)}.}\]This is the quantum replacement of the classical BGK result
\[\mu=p\tau.\]The entire problem is now reduced to evaluating
\[M_4^{(q)}.\]9.5 Evaluating Quantum Moments
The generic quantum moment is
\[I_s = \int c^{2s} f_q^{eq}\Phi_q\,d^3c .\]The velocity integral can be transformed into an energy integral.
Define
\[x= \frac{mc^2}{2k_BT}.\]Then
\[c^2 = \frac{2k_BT}{m}x.\]The volume element becomes
\[d^3c = 4\pi \left( \frac{2k_BT}{m} \right)^{3/2} \frac{1}{2} x^{1/2}dx .\]Thus,
\[I_s \propto \int_0^\infty \frac{x^{s+1/2}} {z^{-1}e^x+\eta} \Phi_q,dx.\]The factor
\[\Phi_q = 1-\eta f_q^{eq}\]has a useful identity:
\[\boxed{f_q^{eq}\Phi_q = -z\frac{\partial f_q^{eq}}{\partial z}.}\]This allows the moment to be converted into derivatives of Fermi/Bose integrals.
Using integration identities, one obtains
\[\boxed{M_4^{(q)} = 15n \left(\frac{k_BT}{m}\right)^2 \frac{ G_{5/2}^{(\eta)}(z) }{ G_{3/2}^{(\eta)}(z) }.}\]Therefore,
\[\boxed{\mu_q = p_q\tau}\]for the single-relaxation-time quantum BGK model.
This is a beautiful result:
\[\boxed{\text{BGK viscosity keeps the form }p\tau}\]but the pressure is now quantum.
9.6 Fermi and Bose Limits
Classical dilute gas
For
\[z\ll1\] \[G_\nu(z)\rightarrow z\]so
\[p_q\rightarrow nk_BT\]and therefore
\[\boxed{\mu_q\rightarrow nk_BT\tau.}\]The classical BGK result returns.
Degenerate Fermi gas
For
\[z\gg1\]the Fermi integrals approach the Sommerfeld expansion:
\[G_\nu^{(+)}(z) \sim \frac{(\ln z)^\nu}{\Gamma(\nu+1)} \left[ 1+O(T^2) \right].\]The viscosity becomes controlled by the Fermi energy scale rather than \(k_BT\).
Bose gas
For bosons,
\[\eta=-1.\]As
\[z\rightarrow1^-\]the moments become sensitive to Bose condensation.
The viscosity behavior depends strongly on whether the gas is above or below the condensation temperature.
The simple normal-gas BGK model does not describe the condensate component; a two-fluid kinetic theory is required below \(T_c\).
9.7 Quantum Heat Flux
Now consider
\[q_i = \frac{1}{2}m \int c^2c_i f_q^{(1)} d^3c .\]The shear contribution vanishes because it contains an odd number of \(c_i\).
Only the temperature-gradient term survives:
\[q_i = -\kappa_q\partial_iT.\]The coefficient becomes
\[\boxed{\kappa_q = \frac{m\tau}{2} \int c^4 B_q(c^2) f_q^{eq}\Phi_q d^3c .}\]After evaluating the quantum moments,
\[\boxed{\kappa_q = \frac{5}{2} \frac{k_B}{m} p_q\tau ,\mathcal R_\eta(z)}\]where
\[\boxed{\mathcal R_\eta(z)}\]is a dimensionless quantum correction factor constructed from ratios of quantum integrals.
For the classical limit,
\[\mathcal R_\eta(z)\rightarrow1\]giving
\[\boxed{\kappa = \frac{5}{2} \frac{k_B}{m}\mu .}\]9.8 Quantum Prandtl Number
The BGK Prandtl number becomes
\[Pr_q = \frac{\mu_q c_p}{\kappa_q}.\]Unlike the classical BGK model, the heat capacity and thermal conductivity are both modified by quantum statistics.
Therefore,
\[\boxed{Pr_q \neq1}\]in general.
The value depends on:
- fugacity \(z\),
- Fermi versus Bose statistics,
- temperature.
This is one of the experimentally relevant signatures of quantum kinetic transport.
9.9 Final Quantum Navier–Stokes Equations
We can now write the macroscopic equations.
Continuity
\[\boxed{\partial_t\rho+\nabla\cdot(\rho\mathbf{u})=0}\]Momentum
\[\boxed{\rho \frac{D\mathbf{u}}{Dt} = -\nabla p_q + \nabla\cdot\boldsymbol{\Pi}}\]with
\[\boxed{\Pi_{ij} = -\mu_q \left( \partial_i u_j +\partial_j u_i -\frac{2}{3}\delta_{ij}\nabla\cdot\mathbf{u} \right).}\]Energy
\[\boxed{\partial_tE + \nabla\cdot (E\mathbf{u}+p_q\mathbf{u}) = \nabla\cdot(\boldsymbol{\Pi}\cdot\mathbf{u}) -\nabla\cdot(\kappa_q\nabla T).}\]This is the kinetic-theory quantum Navier–Stokes system.
9.10 The Main Lesson
The Chapman–Enskog derivation reveals a deep separation:
Universal kinetic structure
\[\boxed{\text{conservation laws} + \text{tensor symmetry}}\]does not care whether the gas is classical, fermionic, or bosonic.
Statistical physics
\[\boxed{\text{equation of state} + \text{transport coefficients}}\]contains all the quantum information.
The next chapter would be a very interesting one: a side-by-side derivation table comparing classical BGK, Fermi BGK, and Bose BGK term by term, including the equilibrium distribution, moments, viscosity, thermal conductivity, and the classical limit. This will make the similarities and differences completely transparent.