Neutral Particle Transport Through a Stochastically Inhomogeneous Medium with the Gaussian Statistics of the Extinction Function

Abstract

The stationary one‐speed transport equation for neutral particles in the slab geometry is considered. The medium between two planes, z=0 and z=L, is taken as absorbing and isotropically scattering. The extinction function σ(r) is defined as a Gaussian random function with a constant mean value σ¯=⟨σ(r)⟩, a constant variance η_σ ²=⟨[σ(r)−σ¯]²⟩, and a given autocorrelation function W _σ (r ₂−r ₁)=⟨[σ(r ₂)−σ¯][σ(r ₁)−σ¯]⟩. The albedo ω (0<ω<1) is taken as a constant. Considering a perpendicular influx of particles from the left and no influx from the right, we focus attention on the solution I(ζ, μ) of the transport equation obtained within the framework of the Pomraning‐Eddington approximation. Our boundary conditions read I(0, 1)=I _L and I(Z, −1)=0. (ζ=ζ(z) is the length of the projection of the optical path on the z‐axis, and θ is the angle between the general flight direction and the z‐axis, μ=cos θ.) Since the randomness of σ(r) is Gaussian, the optical thickness Z=ζ(L) is also Gaussian. For finite values of L, we show that the transmission and reflection coefficients are correlated random quantities. We calculate their first‐order and second‐order statistical moments.

Keywords:

• Radiative transfer
• inhomogenous media
• Gaussian statistics
• Pomraning–Eddington approximation

Acknowledgments

This work has been supported by the Grant Agency VEGA of the Slovak Academy of Sciences and of the Ministry of Education of the Slovak Republic under contract No. 1/0251/03.