Part I, Chapter 3

Collisions & Mean Free Path

Coulomb collisions and their role in plasma dynamics

3.1 Coulomb Collisions

In a plasma, particles interact via the Coulomb force. Unlike neutral gases where collisions are hard-sphere interactions, plasma collisions are long-range and cumulative.

The Coulomb potential between two charges is:

$$V(r) = \frac{q_1 q_2}{4\pi\epsilon_0 r}$$

Small-Angle Scattering Dominates

Consider a particle passing another with impact parameter b. The scattering angle θ is approximately:

$$\theta \approx \frac{q_1 q_2}{4\pi\epsilon_0 m v^2 b}$$

Large impact parameters b give small deflections θ. Since most encounters have large b,small-angle scattering dominates over 90° collisions.

3.2 Impact Parameter and Cross Section

Define b₉₀ as the impact parameter for 90° scattering:

$$b_{90} = \frac{q_1 q_2}{4\pi\epsilon_0 m v^2}$$

The cross section for 90° scattering is:

$$\sigma_{90} = \pi b_{90}^2$$

Debye Cutoff

For large impact parameters, Debye shielding cuts off the Coulomb interaction at b ≈ λ_D. The effective collision cross section involves the Coulomb logarithm:

$$\ln \Lambda = \ln\left(\frac{\lambda_D}{b_{90}}\right) \approx \ln\left(\frac{12\pi n_e \lambda_D^3}{Z}\right)$$

Typically ln Λ ≈ 10-20 for most plasmas. This logarithmic dependence means collision rates are relatively insensitive to plasma parameters.

3.3 Collision Frequency

The electron-ion collision frequency is:

$$\nu_{ei} = \frac{n_e e^4 \ln\Lambda}{4\pi\epsilon_0^2 m_e^2 v^3}$$

Averaging over a Maxwellian distribution with temperature T_e:

$$\boxed{\nu_{ei} = \frac{n_e e^4 \ln\Lambda}{6\sqrt{2\pi}\epsilon_0^2 m_e^{1/2} (k_B T_e)^{3/2}}}$$

In practical units:

$$\nu_{ei}[\text{s}^{-1}] \approx 2.9 \times 10^{-12} \frac{n_e[\text{m}^{-3}] \ln\Lambda}{T_e[\text{eV}]^{3/2}}$$

Physical Interpretation

• ν_ei ∝ n_e: Higher density → more collisions
• ν_ei ∝ T_e^−3/2: Hotter plasmas are more collisionless
• ln Λ weak dependence: Relatively universal across plasmas

3.4 Mean Free Path

The mean free path for collisions is:

$$\lambda_{mfp} = \frac{v_{th}}{\nu_{ei}} \approx \frac{(k_B T_e/m_e)^{1/2}}{\nu_{ei}}$$

Using v_th = √(k_BT_e/m_e):

$$\lambda_{mfp} \approx \frac{6\sqrt{2\pi}\epsilon_0^2 (k_B T_e)^2}{n_e e^4 \ln\Lambda m_e^{1/2}}$$

Comparison with System Size

The Knudsen number Kn = λ_mfp/L determines the plasma regime:

• Kn << 1 (collisional): Fluid description valid (MHD)
• Kn ∼ 1 (transitional): Kinetic effects important
• Kn >> 1 (collisionless): Kinetic theory required (Vlasov equation)

Examples

• Tokamak core: λ_mfp ∼ 10 m, L ∼ 2 m → Kn ∼ 5 (collisionless)
• Solar wind: λ_mfp ∼ 1 AU → highly collisionless
• Dense Z-pinch: λ_mfp << L → collisional

3.5 Collision Operators

In kinetic theory, collisions are represented by the collision term in the Boltzmann equation:

$$\frac{\partial f}{\partial t} + \vec{v} \cdot \nabla f + \frac{\vec{F}}{m} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}}$$

BGK Collision Operator

The simplest collision model is the Bhatnagar-Gross-Krook (BGK) operator:

$$\left(\frac{\partial f}{\partial t}\right)_{\text{BGK}} = -\nu(f - f_M)$$

where f_M is the local Maxwellian distribution. This relaxes f toward equilibrium at rate ν.

Fokker-Planck Operator

For Coulomb collisions, the full operator is:

$$C(f) = \frac{\partial}{\partial v_i}\left[A_i f + \frac{\partial}{\partial v_j}(D_{ij} f)\right]$$

where A_i is the dynamical friction coefficient and D_ij is the diffusion tensor (covered in Part II).

3.6 Collisional Transport

Collisions enable diffusion and transport. The classical resistivity is:

$$\eta = \frac{m_e \nu_{ei}}{n_e e^2} \propto T_e^{-3/2}$$

The thermal diffusivity is:

$$\chi = \frac{v_{th}^2}{\nu_{ei}} = \frac{\lambda_{mfp} v_{th}}{3}$$

These classical transport coefficients set lower bounds on confinement times. In magnetized plasmas, anomalous transport (from turbulence) often exceeds classical predictions by orders of magnitude.

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