Part 3, Chapter 1
Moment Equations
Deriving fluid equations from kinetic theory through velocity moments
1.1 From Kinetic to Fluid Description
The kinetic description via the Boltzmann equation contains complete information through the distribution function f(r, v, t). By taking velocity moments, we derive fluid equations for macroscopic quantities like density, velocity, and pressure.
$$\int (\text{Boltzmann equation}) \times \{1, \mathbf{v}, \tfrac{1}{2}m v^2\} \, d^3v$$
1.2 Macroscopic Quantities
Number Density (Zeroth Moment)
$$n(\mathbf{r}, t) = \int f(\mathbf{r}, \mathbf{v}, t) \, d^3v$$
Mean Velocity (First Moment)
$$\mathbf{u}(\mathbf{r}, t) = \frac{1}{n} \int \mathbf{v} \, f \, d^3v$$
Pressure Tensor (Second Moment)
$$P_{ij} = m \int (v_i - u_i)(v_j - u_j) f \, d^3v$$
Heat Flux (Third Moment)
$$\mathbf{q} = \frac{m}{2} \int |\mathbf{v}-\mathbf{u}|^2 (\mathbf{v}-\mathbf{u}) f \, d^3v$$
1.3 The Fluid Equations
Continuity Equation
$$\frac{\partial n}{\partial t} + \nabla \cdot (n\mathbf{u}) = 0$$
Mass conservation
Momentum Equation
$$mn\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = qn(\mathbf{E} + \mathbf{u} \times \mathbf{B}) - \nabla \cdot \mathbf{P} + \mathbf{R}$$
Force balance with Lorentz force, pressure, and friction
Energy Equation
$$\frac{3}{2}\frac{dp}{dt} + \frac{5}{2}p\nabla \cdot \mathbf{u} + \nabla \cdot \mathbf{q} + \boldsymbol{\pi}:\nabla\mathbf{u} = Q$$
Thermal energy evolution
Closure Problem: Each moment equation involves the next higher moment. The energy equation needs heat flux q, which needs the fourth moment, etc. We must close with approximations.
1.4 Closure Approximations
Cold Plasma (T → 0)
$$p = 0, \quad \mathbf{q} = 0$$
Isothermal
$$p = nk_BT, \quad T = \text{const}$$
Adiabatic
$$\mathbf{q} = 0, \quad p \propto \rho^\gamma$$
Braginskii
$$\mathbf{q} = -\kappa \nabla T$$
Key Takeaways
- ✓ Fluid equations are velocity moments of the kinetic equation
- ✓ Zeroth → continuity, First → momentum, Second → energy
- ✓ Each moment couples to the next (closure problem)
- ✓ Common closures: cold, isothermal, adiabatic, Braginskii