Diffusion approximation and short-path statistics at low to intermediate Knudsen numbers

In the field of first-return statistics in bounded domains, short paths may be defined as those paths for which the diffusion approximation is inappropriate. However, general integral constraints have been identified that make it possible to address such short-path statistics indirectly by application of the diffusion approximation to long paths in a simple associated first-passage problem. This approach is exact in the zero Knudsen limit (Blanco S. and Fournier R., Phys. Rev. Lett., 97 (2006) 230604). Its generalization to the low to intermediate Knudsen range is addressed here theoretically and the corresponding predictions are compared to both one-dimension analytical solutions and three-dimension numerical experiments. Direct quantitative relations to the solution of the Schwarzschild-Milne problem are identified.