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

¹ Université de Toulouse; Centre RAPSODEE, UMR CNRS 5302, Mines Albi; Campus Jarlard F-81013 Albi Cedex 09, France
² Université de Toulouse, UPS, INPT; LAPLACE (Laboratoire Plasma et Conversion d'Energie) 118 route de Narbonne, F-31062 Toulouse cedex 9, France
³ CNRS, LAPLACE - F-31062 Toulouse cedex 9, France
⁴ LMB, UMR CNRS 6623, Université de Franche-Comté - 16, route de Gray, 25030 Besançon Cedex, France

Received: 7 January 2015
Accepted: 15 April 2015

Abstract

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.