Cauchy's formulas for random walks in bounded domains

Abstract

Cauchy's formula was originally established for random straight paths crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average chord length through $B$ to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in $B$. Similar results are obtained also for the average number of collisions performed by the walker in $B$, and an extension to absorbing media is discussed.Comment: 12 pages, 6 figure

Cauchy’s theorem and generalization

“…Generally this theorem is established by performing the integrations on the point A and the inward direction [1,Sect. 14.7,2]. But a much more direct and simple neutronics 2 reasoning is also possible [1,Sect.…”

“…He is the author of almost 800 articles and seven books. 2 If you prefer, you can replace the neutrons with any neutral particles which do not interfere each other: photons, neutrinos, atoms of a very deluted gas... 3 We use the term "flux" with its meaning in neutronics: the flux F = nv is the density n (mean number of particles per unit volume) multiplied by the velocity v. 4 The cosmic microwave background is an example of such a flux.…”

“…An extension of this theorem, established in particular in [2], was noticed in a large audience newspaper [3] as a curious feature concerning not the neutrons but the photons crossing water or scattered in milk. Here we will consider neutrons but that does not change the reasoning.…”

Monte Carlo particle transport in random media: The effects of mixing statistics

“…where S leak is the surface area of the boundaries where leakage conditions are applied Blanco and Fournier (2003); Zoia et al (2012); Mazzolo et al (2014);de Mulatier et al (2014). This formula, which can be understood as a non-trivial generalization of the Cauchy's formula applying to the average chord lengths Blanco and Fournier (2003); Mazzolo et al (2014), holds true provided that particles enter the domain uniformly and isotropically, which is ensured here by the source that we have chosen and by symmetry considerations. Hence, the flux ϕ depends exclusively on the ratio of purely geometrical quantities, namely, ϕ = 4V/S leak , which for our benchmark yields ϕ = 20.…”

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

“…A very significant and recent step forward was also made in [22], where the property could be rigorously extended to the scattering of waves in resonant, chaotic or Anderson-localized structures. Major advances can also be expected from the numerous contributions of Mazzolo and co-workers that have closely considered the links between the physics and mathematics literatures, in particular with the introduction of this property in the field of integral geometry as a generalization of the Cauchy formula [23][24][25][26][27], and a reconciliation with the Feynman-Kac formalism [28][29][30][31]. Their researches, especially those addressing the full length distribution [32], led to the identification of the following second property [33]: for any function f of the trajectory length, with a defined limit f 0 in zero, ⟨f…”