From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 12
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This chapter takes us into a subtle but very important distinction in quantum fluid theory.
So far we have derived quantum Navier–Stokes equations from quantum kinetic theory. These equations contain quantum statistics through:
- Bose–Einstein or Fermi–Dirac equilibrium,
- quantum equation of state,
- quantum transport coefficients.
However, they do not automatically contain the dispersive quantum correction
\[-\frac{\hbar^2}{2m} \nabla \left( \frac{\nabla^2\sqrt n}{\sqrt n} \right)\]usually called the quantum pressure or Bohm potential term.
Why?
Because the Chapman–Enskog expansion we performed starts from a phase-space distribution and describes particles through local statistical equilibrium. The quantum pressure originates from a different aspect of quantum mechanics: the coherence of the wavefunction itself.
This chapter builds the bridge between these two descriptions.
Chapter 12 — From Quantum Kinetic Theory to Quantum Hydrodynamics
12.1 Two Meanings of “Quantum Hydrodynamics”
The phrase quantum hydrodynamics is used for two related but distinct theories.
A) Quantum statistical hydrodynamics
Derived from:
\[\boxed{\text{quantum Boltzmann/BGK equation}}\]through Chapman–Enskog expansion.
It gives:
\[\boxed{\text{Navier–Stokes equations with quantum statistics}}\]The quantum information enters through:
\[p(n,T)\] \[\mu(n,T)\] \[\kappa(n,T).\]B) Wave-mechanical quantum hydrodynamics
Derived from:
\[\boxed{\text{Schrödinger equation}}\]or
\[\boxed{\text{Gross–Pitaevskii equation}}\]using the Madelung transformation.
It gives:
\[\boxed{\text{Euler equations with quantum pressure}.}\]The quantum information enters through:
\[\hbar.\]These are not competing theories.
They describe different physics.
12.2 Starting from the Schrödinger Equation
Consider a single particle:
\[i\hbar \frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2\psi + V\psi .\]Write the wavefunction as
\[\boxed{\psi(\mathbf{r},t) = \sqrt{n(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}.}\]This is the Madelung transformation.
The density is:
\[|\psi|^2=n.\]The velocity is identified as:
\[\boxed{\mathbf{u} = \frac{\nabla S}{m}.}\]12.3 The Continuity Equation
Substituting into the imaginary part of Schrödinger’s equation gives:
\[\boxed{\frac{\partial n}{\partial t} + \nabla\cdot(n\mathbf{u}) = 0}\]This is identical to the kinetic-theory continuity equation.
So both approaches agree here.
12.4 The Momentum Equation
The real part gives:
\[\frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V + Q = 0\]where
\[\boxed{Q = -\frac{\hbar^2}{2m} \frac{\nabla^2\sqrt n}{\sqrt n}}\]is the quantum potential.
Taking a gradient:
\[m \frac{D\mathbf{u}}{Dt} = -\nabla(V+Q).\]Therefore,
\[\boxed{m n \frac{D\mathbf{u}}{Dt} = -n\nabla V = n\nabla Q .}\]The last term is the quantum pressure contribution.
12.5 The Quantum Pressure Tensor
The quantum force can be written as a stress divergence:
\[-n\nabla Q = -\partial_jP_{ij}^{Q}.\]The quantum stress tensor is:
\[\boxed{P_{ij}^{Q} = \frac{\hbar^2}{4m} \left[ \frac{\partial_i n\partial_j n}{n} - \partial_i\partial_j n \right].}\]This is a completely different object from the kinetic pressure tensor:
\[P_{ij} = m\int c_ic_j f\,d^3c.\]The first comes from wave coherence.
The second comes from particle motion.
12.6 Why BGK Does Not Produce Quantum Pressure
The BGK equation evolves:
\[f(\mathbf{x},\mathbf{v},t).\]The microscopic variables are:
\[\mathbf{x},\mathbf{v}.\]The phase of the wavefunction,
\[S(\mathbf{x},t)\]is not present.
Therefore the information required to create:
\[\nabla^2\sqrt n\]is absent.
The quantum BGK equation knows about:
- occupation numbers,
- exchange statistics,
- Pauli blocking,
- Bose enhancement.
It does not know about:
- phase coherence,
- interference,
- wavefunction curvature.
Therefore:
\[\boxed{\text{Quantum statistics} \neq \text{wave-mechanical quantum effects}.}\]12.7 The Wigner Function: The Bridge
To include both effects, we introduce the Wigner function:
\[W(\mathbf{x},\mathbf{p},t).\]It is a quantum phase-space distribution.
Unlike the classical distribution:
\[f(x,p)\]it can contain negative regions because it retains phase information.
The Wigner equation is:
\[\frac{\partial W}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla_xW = \mathcal{Q}[W]\]where
\[\mathcal{Q}\]is the quantum Moyal operator.
Expanding the Moyal term:
\[\mathcal{Q} = -\nabla V\cdot\nabla_pW + \frac{\hbar^2}{24} \nabla^3V\cdot\nabla_p^3W +\cdots\]The classical limit is:
\[\hbar\rightarrow0.\]12.8 Wigner Hydrodynamics
Taking moments of the Wigner equation gives:
Continuity
\[\boxed{\partial_tn+\nabla\cdot(nu)=0}\]Momentum
\[\boxed{mn \frac{Du_i}{Dt} = -\partial_jP_{ij} = n\partial_iV + F_i^{Q}.}\]The quantum force is generated by the higher-order Wigner moments.
In the pure condensate limit:
\[F_i^{Q} = -n\partial_iQ.\]12.9 Gross–Pitaevskii Equation
For a weakly interacting Bose condensate:
\[\boxed{i\hbar\partial_t\psi = \left( -\frac{\hbar^2\nabla^2}{2m} + V + g|\psi|^2 \right)\psi .}\]The interaction parameter is:
\[g= \frac{4\pi\hbar^2a_s}{m}\]where \(a_s\) is the scattering length.
Applying the Madelung transformation gives:
Continuity:
\[\partial_tn+\nabla\cdot(nu)=0.\]Momentum:
\[\boxed{m \frac{Du}{Dt} = -\nabla \left( gn + V + Q \right).}\]The terms are:
| Term | Origin |
|---|---|
| \(gn\) | particle interactions |
| \(V\) | external potential |
| \(Q\) | quantum coherence |
12.10 Comparing the Two Quantum Hydrodynamics
| Quantum BGK | Gross–Pitaevskii | |
|---|---|---|
| Starting point | kinetic equation | wave equation |
| Variable | \(f(x,v)\) | \(\psi(x)\) |
| Statistics | BE/FD | usually Bose condensate |
| Pauli blocking | yes | no |
| Bose enhancement | yes | no |
| Quantum pressure | no | yes |
| Viscosity | yes | negligible |
| Dissipation | yes | no |
12.11 The Complete Picture
A realistic quantum fluid may require all three ingredients:
1. Quantum statistics
\[f_{FD/BE}\]controls equilibrium.
2. Collisions
\[C[f]\]controls viscosity and heat transport.
3. Coherence
\[\hbar\nabla^2\sqrt n\]controls quantum pressure.
A unified theory attempts to combine them:
\[\boxed{\text{Wigner} + \text{quantum collision operator} + \text{Chapman–Enskog expansion}}\]This is an active research area.
12.12 Final Conceptual Map
We can now distinguish three levels:
\[\boxed{\text{Boltzmann}}\]classical particles:
\[f_{MB}\]↓
\[\boxed{\text{Quantum BGK}}\]quantum statistics:
\[f_{FD/BE}\]↓
\[\boxed{\text{Wigner/Gross–Pitaevskii}}\]quantum coherence:
\[\hbar\text{-dependent pressure}.\]The next natural chapter is a very rich one:
Chapter 13 — Deriving the quantum Navier–Stokes–Wigner equations: combining quantum statistics, viscosity, and the Bohm quantum pressure term.
That chapter will attempt to put the two worlds together: a dissipative quantum fluid equation containing both
\[-\nabla p_q\] \[\nabla\cdot\Pi_q\]and
\[-\frac{\hbar^2}{2m} \nabla \left( \frac{\nabla^2\sqrt n}{\sqrt n} \right).\]This is the point where kinetic theory, quantum mechanics, and fluid dynamics truly meet.