From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 13
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This is the point where the two threads we have developed—quantum kinetic theory and wave-mechanical quantum hydrodynamics—are brought together.
A small warning before we start: a fully rigorous derivation of a dissipative quantum Navier–Stokes equation with both Bose/Fermi statistics and Bohm quantum pressure requires a specific quantum kinetic model (for example, a Wigner–Boltzmann equation with a chosen collision operator). There is no single universally accepted “quantum Navier–Stokes equation.” What we derive here is the structure obtained from a semiclassical Wigner–Chapman–Enskog expansion and show where each term originates.
Chapter 13 — Quantum Navier–Stokes–Wigner Equations: Combining Statistics, Dissipation, and Quantum Pressure
13.1 The Goal
We want a hydrodynamic equation containing three types of physics:
1. Thermodynamic pressure
from quantum statistics:
\[p_q(n,T)\]with Bose/Fermi corrections.
2. Dissipation
from kinetic collisions:
\[\Pi_{ij}^{visc}, \qquad q_i .\]3. Quantum coherence
from wave mechanics:
\[Q = -\frac{\hbar^2}{2m} \frac{\nabla^2\sqrt n}{\sqrt n}.\]The final momentum equation should therefore have the schematic form:
\[\boxed{\rho\frac{D u_i}{Dt} = -\partial_i p_q + \partial_j\Pi_{ij} = n\partial_i Q .}\]Each term has a different microscopic origin.
13.2 The Wigner–Boltzmann Equation
The starting point is the Wigner function:
\[W(\mathbf{x},\mathbf{p},t).\]The evolution equation is:
\[\boxed{\partial_tW + \frac{\mathbf{p}}{m}\cdot\nabla_xW = \nabla_xV\cdot\nabla_pW + \mathcal{M}_\hbar[W] = C[W].}\]Here:
- (C[W]) is the collision operator,
- \(\mathcal{M}_\hbar\) is the quantum correction.
The Moyal operator is
\[\mathcal{M}_\hbar[W] = \sum_{l=1}^{\infty} \frac{(-1)^l}{(2l+1)!} \left( \frac{\hbar}{2} \right)^{2l} \nabla_x^{2l+1}V \cdot \nabla_p^{2l+1}W .\]The first correction is:
\[\boxed{\mathcal{M}_\hbar \approx -\frac{\hbar^2}{24} \nabla_x^3V \cdot \nabla_p^3W .}\]The classical Boltzmann equation is recovered as:
\[\hbar\rightarrow0.\]13.3 Hydrodynamic Moments of the Wigner Function
The definitions are analogous to kinetic theory.
Density:
\[\boxed{n= \int W\,d^3p}\]Velocity:
\[\boxed{nu_i = \int \frac{p_i}{m}W\,d^3p}\]Stress tensor:
\[\boxed{P_{ij} = \int \frac{(p_i-mu_i)(p_j-mu_j)}{m} W\,d^3p}\]Heat flux:
\[\boxed{q_i = \frac{1}{2} \int c^2c_iW\,d^3p}\]13.4 Zeroth Moment: Continuity Equation
Integrating the Wigner equation over momentum:
\[\int d^3p,(\text{Wigner equation})\]gives:
\[\boxed{\partial_t n + \nabla\cdot(n\mathbf{u}) = 0 .}\]The quantum corrections vanish because they are total derivatives in momentum space.
This is exactly the same conservation law obtained from BGK theory.
13.5 First Moment: Momentum Equation
Multiply by \(p_i/m\):
\[\int \frac{p_i}{m} (\text{Wigner equation}) d^3p .\]The result is:
\[\boxed{mn \frac{Du_i}{Dt} = -\partial_jP_{ij} = n\partial_iV + F_i^{Q}.}\]The pressure tensor now has two parts:
\[\boxed{P_{ij} = P_{ij}^{kin} + P_{ij}^{Q}.}\]13.6 Separation of the Pressure Tensor
The kinetic part is:
\[P_{ij}^{kin} = p_q\delta_{ij} + \Pi_{ij}.\]Here:
\[p_q\]is the Bose/Fermi equation of state.
The viscous correction is:
\[\boxed{\Pi_{ij} = -\mu_q \left( \partial_i u_j+ \partial_j u_i -\frac{2}{3}\delta_{ij}\nabla\cdot u \right).}\]This is exactly the Chapman–Enskog result.
13.7 Quantum Pressure Contribution
The coherent part is:
\[\boxed{P_{ij}^{Q} = \frac{\hbar^2}{4m} \left( \frac{\partial_i n\partial_j n}{n} - \partial_i\partial_j n \right).}\]Taking the divergence:
\[\partial_jP_{ij}^{Q} = n\partial_iQ.\]Therefore:
\[\boxed{F_i^{Q} = -n\partial_iQ .}\]The momentum equation becomes:
\[\boxed{mn \frac{D u_i}{Dt} = -\partial_i p_q + \partial_j\Pi_{ij} = n\partial_iQ .}\]This is the quantum Navier–Stokes–Wigner equation.
13.8 Energy Equation
The total energy density is:
\[E = \frac{1}{2}mn u^2 + \epsilon_q + \epsilon_Q .\]The contributions are:
Classical kinetic energy:
\[\frac{1}{2}mn u^2\]Quantum statistical internal energy:
\[\epsilon_q = \epsilon(n,T)\]Quantum coherence energy:
\[\boxed{\epsilon_Q = \frac{\hbar^2}{2m} |\nabla\sqrt n|^2 .}\]The energy equation becomes:
\[\boxed{\begin{aligned} \partial_tE + \nabla\cdot \left[ (E+p_q)\mathbf{u} \right] =& \nabla\cdot(\Pi\cdot\mathbf{u}) \ & -\nabla\cdot q \ & +\text{quantum flux}. \end{aligned}}\]13.9 Final Quantum Navier–Stokes–Wigner System
We have:
Continuity
\[\boxed{\partial_t n+ \nabla\cdot(n\mathbf{u})=0}\]Momentum
\[\boxed{mn \frac{D\mathbf{u}}{Dt} = -\nabla p_q + \nabla\cdot\Pi = n\nabla Q}\]where
\[\boxed{Q = -\frac{\hbar^2}{2m} \frac{\nabla^2\sqrt n}{\sqrt n}}\]and
\[\boxed{\Pi_{ij} = -\mu_q \left( \partial_i u_j+\partial_j u_i -\frac{2}{3}\delta_{ij}\nabla\cdot u \right).}\]Heat equation
\[\boxed{q=-\kappa_q\nabla T .}\]13.10 Classical and Quantum Limits
Classical limit
If:
\[\hbar\rightarrow0\]then:
\[Q\rightarrow0.\]The equation becomes:
\[\boxed{\text{quantum BGK Navier–Stokes} \rightarrow \text{classical Navier–Stokes}.}\]Collisionless quantum limit
If:
\[\mu_q,\kappa_q\rightarrow0\]then:
\[\boxed{\text{quantum Navier–Stokes} \rightarrow \text{quantum Euler/Gross–Pitaevskii hydrodynamics}.}\]13.11 Physical Interpretation of the Three Forces
The final momentum equation:
\[mn\frac{D\mathbf{u}}{Dt} = -\nabla p_q + \nabla\cdot\Pi = n\nabla Q\]contains:
Statistical pressure
\[-\nabla p_q\]“particles resist compression.”
Viscous stress
\[\nabla\cdot\Pi\]“collisions dissipate gradients.”
Quantum pressure
\[-n\nabla Q\]“wavefunctions resist localization.”
These are three fundamentally different mechanisms.
13.12 The Unified View
We can now summarize the whole journey:
\[\boxed{\text{Boltzmann}}\] \[\downarrow\] \[\boxed{\text{BGK}}\] \[\downarrow\] \[\boxed{\text{Chapman–Enskog}}\] \[\downarrow\] \[\boxed{\text{Navier–Stokes}}\]then extend:
\[+\] \[\boxed{\text{Fermi/Bose statistics}}\] \[\downarrow\] \[\boxed{\text{quantum kinetic hydrodynamics}}\]and finally:
\[+\] \[\boxed{\text{Wigner coherence}}\] \[\downarrow\] \[\boxed{\text{quantum Navier–Stokes–Wigner equations}.}\]The next chapter could now explore an especially fascinating application:
Chapter 14 — Ultracold Atomic Gases: Applying Quantum Navier–Stokes Theory to Unitary Fermi Gases and Bose–Einstein Condensates.
This would connect the formal derivations to real systems where these equations are actually tested: viscosity minima, superfluidity, collective modes, and experimental measurements of quantum transport.