A derivation of transverse electron diffusion coefficient from electron velocity distribution function under parallel electric and magnetic fields based on a two-dimensional random walk model*

A derivation of transverse diffusion coefficient D_⊥ of electrons in gas under parallel electric and magnetic fields (E‖B fields) applied in the z-direction is presented assuming a model of two-dimensional random walk in the xy-plane and a constant collision frequency ν. 

Because the electron velocity distribution function (EVDF) does not have the information on spatial position of the electrons and is identical irrespective of |B| under the action of conformal rotation around the V_z-axis by the Lorentz force in the E‖B fields, variance 〈λ²〉 of step λ of random walk is evaluated from the Larmor radius r_L = V_⊥/ω (V_⊥² = V_x² + V_y², ω = e|B|/m: angular gyrofrequency) as a factor representing microscopic electron motion in real space. Here, D_⊥ = 〈λ²〉ν/4 = 〈λ_x²〉ν/2 = 〈λ_y²〉ν/2. λ is the displacement of an electron gyrating in real space from its start of free flight beginning with an isotropic scattering. 

A well-known factor 1/[1 + (ω/ν) ²] representing suppression of the electron diffusion across magnetic field lines, which can be utilized for confinement or guide of charged particles, is derived naturally from an integral ∫2r_L²(1 − cosωt)exp(−νt)νdt in [0, ∞) for 〈λ²〉. The D_⊥ values estimated in the random walk model from the EVDF obtained by Monte Carlo simulations agree well with the values of (d/dt)〈x²〉/2 and (d/dt)〈y²〉/2 directly obtained from the real-space transverse diffusion of the electrons.