Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 2, Chapter 4

Fokker-Planck Equation

Collision operators and diffusion

4.1 Introduction

The Fokker-Planck equation describes the time evolution of a probability distribution function under the influence of drag forces and random forces. In plasma physics, it provides a kinetic description of Coulomb collisions, which are the dominant mechanism for thermalization and transport in weakly coupled plasmas.

Unlike the collisionless Vlasov equation, the Fokker-Planck equation includes collision operators that account for the cumulative effect of many small-angle scattering events. This approach is valid when the plasma parameter $g = n \lambda_D^3 \gg 1$, meaning there are many particles in a Debye sphere.

Key Concept:

The Fokker-Planck equation bridges the gap between microscopic particle dynamics and macroscopic transport coefficients, providing expressions for diffusion, friction, and heating rates in plasmas.

4.2 Derivation from Binary Collisions

Small-Angle Scattering

Consider a test particle of species $a$ with velocity $\mathbf{v}$ colliding with field particles of species $b$. In a plasma, most collisions are at small angles due to the long-range nature of the Coulomb force. The cumulative effect of many small deflections dominates over rare large-angle scattering.

For a single collision with impact parameter $b$, the scattering angle $\theta$ in the center-of-mass frame is:

$\theta \approx \frac{2 Z_a Z_b e^2}{4\pi\epsilon_0 \mu v_{rel}^2 b}$

where $\mu = m_a m_b/(m_a + m_b)$ is the reduced mass, $v_{rel}$ is the relative velocity, and $Z_a, Z_b$ are the charge numbers.

The Rosenbluth Potentials

The change in the distribution function due to collisions can be expressed using the Rosenbluth potentials$h(\mathbf{v})$ and $g(\mathbf{v})$:

$h(\mathbf{v}) = \int f_b(\mathbf{v}') \, |\mathbf{v} - \mathbf{v}'| \, d^3v'$
$g(\mathbf{v}) = \int \frac{f_b(\mathbf{v}')}{|\mathbf{v} - \mathbf{v}'|} \, d^3v'$

These potentials encode the velocity-space structure of the collision operator and allow efficient computation of collision rates.

4.3 The Fokker-Planck Collision Operator

General Form

The Fokker-Planck equation for the distribution function $f_a(\mathbf{v}, t)$ of species $a$ is:

$\frac{\partial f_a}{\partial t} + \mathbf{v} \cdot \nabla f_a + \frac{q_a}{m_a}(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \frac{\partial f_a}{\partial \mathbf{v}} = \left(\frac{\partial f_a}{\partial t}\right)_{\text{coll}}$

The collision operator on the right-hand side can be written as:

$\left(\frac{\partial f_a}{\partial t}\right)_{\text{coll}} = -\frac{\partial}{\partial \mathbf{v}} \cdot \left( \langle\Delta\mathbf{v}\rangle f_a \right) + \frac{1}{2} \frac{\partial^2}{\partial v_i \partial v_j} \left( \langle\Delta v_i \Delta v_j\rangle f_a \right)$

This has the form of a continuity equation in velocity space with a drift term and a diffusion term.

Friction and Diffusion Coefficients

The dynamical friction coefficient is:

$\langle\Delta\mathbf{v}\rangle = -\nu_s \mathbf{v}$

where $\nu_s$ is the collision frequency for momentum loss (slowing-down frequency).

The velocity diffusion coefficient tensor is:

$\langle\Delta v_i \Delta v_j\rangle = D_{ij}^{ab}$

In terms of Rosenbluth potentials:

$\left(\frac{\partial f_a}{\partial t}\right)_{\text{coll}} = \Gamma_{ab} \frac{\partial}{\partial \mathbf{v}} \cdot \left[ \frac{\partial g}{\partial \mathbf{v}} f_a + \frac{m_a}{m_b} \frac{\partial h}{\partial \mathbf{v}} \frac{\partial f_a}{\partial \mathbf{v}} \right]$

where the collision coefficient is:

$\Gamma_{ab} = \frac{4\pi Z_a^2 Z_b^2 e^4 \ln\Lambda}{m_a^2}$

The Coulomb Logarithm:

The factor $\ln\Lambda$ arises from integrating over impact parameters from the Debye length$\lambda_D$ (maximum) down to the classical distance of closest approach (minimum). Typically$\ln\Lambda \approx 10-20$ in laboratory and astrophysical plasmas.

4.4 Conservation Properties

The Fokker-Planck collision operator must satisfy fundamental conservation laws:

Particle Conservation

$\int \left(\frac{\partial f_a}{\partial t}\right)_{\text{coll}} d^3v = 0$

Collisions redistribute particles in velocity space but do not create or destroy them.

Momentum Conservation

$\sum_a m_a \int \mathbf{v} \left(\frac{\partial f_a}{\partial t}\right)_{\text{coll}} d^3v = 0$

Total momentum is conserved in elastic collisions (summed over all species).

Energy Conservation

$\sum_a \frac{m_a}{2} \int v^2 \left(\frac{\partial f_a}{\partial t}\right)_{\text{coll}} d^3v = 0$

Total kinetic energy is conserved (elastic collisions).

H-Theorem

The Fokker-Planck equation satisfies Boltzmann's H-theorem, ensuring that entropy increases and the system evolves toward a Maxwellian distribution:

$\frac{dS}{dt} = -k_B \sum_a \int f_a \ln f_a \left(\frac{\partial f_a}{\partial t}\right)_{\text{coll}} d^3v \geq 0$

4.5 Collision Frequencies and Time Scales

Electron-Ion Collisions

For a Maxwellian plasma, the characteristic collision frequencies are:

Electron-ion momentum exchange:

$\nu_{ei} = \frac{4\sqrt{2\pi} n_e Z^2 e^4 \ln\Lambda}{3 m_e^{1/2} (k_B T_e)^{3/2}} \approx 2.9 \times 10^{-6} \frac{n_e Z \ln\Lambda}{T_e^{3/2}} \, \text{s}^{-1}$

where $n_e$ is in m⁻³ and $T_e$ is in eV.

Ion-Ion Collisions

$\nu_{ii} = \frac{4\sqrt{\pi} n_i Z^4 e^4 \ln\Lambda}{3 m_i^{1/2} (k_B T_i)^{3/2}}$

Temperature Equilibration

When electrons and ions have different temperatures, they equilibrate on a time scale:

$\tau_{eq} \sim \frac{m_i}{m_e} \frac{1}{\nu_{ei}} \sim 43 \frac{A}{\nu_{ei}}$

where $A$ is the ion mass number. This is much longer than the electron-electron or ion-ion collision times due to the large mass ratio.

Example: Tokamak Core

For ITER-like parameters: $n_e = 10^{20}$ m⁻³, $T_e = 10$ keV, $\ln\Lambda = 17$:

  • $\nu_{ei} \approx 5 \times 10^4$ s⁻¹ (electron-ion collision frequency)
  • $\tau_{ei} \approx 20$ μs (mean collision time)
  • This is much longer than the plasma oscillation period $\omega_{pe}^{-1} \sim 0.1$ ps

4.6 Transport Coefficients

The Fokker-Planck equation provides the foundation for calculating classical transport coefficients in magnetized plasmas.

Electrical Resistivity

The parallel (to $\mathbf{B}$) electrical resistivity is:

$\eta_{\parallel} = \frac{m_e \nu_{ei}}{n_e e^2} = \frac{m_e}{n_e e^2} \frac{4\sqrt{2\pi} n_e Z e^4 \ln\Lambda}{3 m_e^{1/2} (k_B T_e)^{3/2}}$

Numerically:

$\eta_{\parallel} \approx 5.2 \times 10^{-5} \frac{Z \ln\Lambda}{T_e^{3/2}} \, \Omega \cdot \text{m}$

Thermal Conductivity

The parallel electron thermal conductivity is:

$\kappa_{\parallel e} \approx \frac{3.2 n_e k_B^2 T_e}{m_e \nu_{ei}}$

Ion thermal conductivity:

$\kappa_{\parallel i} \approx \frac{3.9 n_i k_B^2 T_i}{m_i \nu_{ii}}$

Viscosity

The ion viscosity coefficient is:

$\mu_i \approx \frac{0.96 n_i k_B T_i}{\nu_{ii}}$

4.7 Applications and Extensions

Runaway Electrons

In the presence of an electric field, electrons with velocities above the critical velocity can "run away" because the friction force decreases with velocity faster than the electric force:

$v_c = \sqrt{\frac{n e^3 \ln\Lambda}{4\pi\epsilon_0^2 m_e E}}$

This phenomenon is important in tokamak disruptions and lightning physics.

Plasma Heating

The Fokker-Planck equation describes how energetic particles (from neutral beam injection or fusion reactions) thermalize and transfer energy to the bulk plasma.

Neoclassical Transport

In toroidal confinement devices, particle orbits in non-uniform magnetic fields modify the collision operator, leading to enhanced "neoclassical" transport. The Fokker-Planck equation must be solved in the presence of trapped and passing particle populations.

Limitations

The Fokker-Planck approach assumes:

  • Small-angle scattering dominates (valid when $g \gg 1$)
  • Binary collisions (breaks down in strongly coupled plasmas)
  • No collective effects beyond Debye screening
  • Classical physics (quantum effects ignored)

For strongly coupled plasmas ($g \lesssim 1$) or degenerate plasmas, more sophisticated approaches like molecular dynamics or quantum kinetic theory are required.

Key Takeaway:

The Fokker-Planck equation provides a rigorous kinetic framework for collision dynamics in weakly coupled plasmas, enabling calculation of transport coefficients, thermalization rates, and the approach to thermal equilibrium. It bridges microscopic scattering physics with macroscopic plasma behavior.

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