Average transmission probability of a random stack

TL;DR

This paper introduces a novel method to directly calculate the average transmission probability through a random stack of slabs, providing analytical bounds and a conjecture on the accuracy of the upper bound approximation.

Contribution

It presents a recurrence relation for computing the average transmission probability and derives bounds, offering a new approach beyond the traditional logarithmic averaging.

Findings

Upper bound approximation is highly accurate

Derived analytical bounds for transmission probability

Conjecture on asymptotic exactness of the upper bound

Abstract

The transmission through a stack of identical slabs that are separated by gaps with random widths is usually treated by calculating the average of the logarithm of the transmission probability. We show how to calculate the average of the transmission probability itself with the aid of a recurrence relation and derive analytical upper and lower bounds. The upper bound, when used as an approximation for the transmission probability, is unreasonably good and we conjecture that it is asymptotically exact.