Neutral particle transport through a stochastically inhomogeneous medium with the Gaussian statistics of the extinction function

The stationary one-speed transport equation for neutral particles in the slab
geometry is considered. The medium between two planes, z ¼ 0 and z ¼ L, is taken as
absorbing and isotropically scattering. The extinction function s(r) is defined as a
Gaussian random function with a constant mean value s¯ ¼ ks(r)l, a constant variance
hs 2 ¼ k[s(r)2s¯]2l, and a given autocorrelation function Ws (r22r1)¼k[s(r2)2s¯]
[s(r1)2s¯]l. The albedov (0 , v , 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(z, m) of the transport equation obtained within the framework of the
Pomraning-Eddington approximation. Our boundary conditions read I(0, 1) ¼ IL and I(Z,
21) ¼ 0. (z ¼ z(z) is the length of the projection of the optical path on the z-axis, and u
is the angle between the general flight direction and the z-axis, m ¼ cosu.) Since the
randomness of s(r) is Gaussian, the optical thickness Z ¼ z(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