A universal property of random trajectories in bounded domains

TL;DR

This paper reveals a universal law relating the average in-domain path length, total length, and domain geometry for random trajectories in bounded spaces, applicable across various physical and biological systems.

Contribution

It generalizes the classical invariance property to a broad class of stochastic and deterministic trajectories using integral geometry, independent of step-length statistics and other complexities.

Findings

The law applies to diverse dynamics including ballistic and diffusive processes.

Monte Carlo simulations confirm the universality across different geometries and dimensions.

Local measurements of in-domain length can infer total length without full trajectory tracking.

Abstract

The celebrated invariance property states that particles entering a bounded domain, with isotropic and uniform incidence, spend on average $\langle \ell \rangle=4V/S$ length inside, no matter how they scatter. We show that this remarkable property is merely the infinite-length limit of an even broader law: for any curves randomly placed and oriented in space -- stochastic or deterministic, generated by ballistic or diffusive dynamics, with possible stopping or branching, in two or more dimensions -- $ \displaystyle \frac{1}{\langle \ell \rangle}= \frac{1}{\langle L\rangle}+ \frac{1}{\langle \sigma \rangle} $, with $\langle\ell\rangle$ its mean in-domain path, $\langle L\rangle$ its mean total length, and $\langle\sigma\rangle$ the mean chord of the domain, a known geometric quantity related to the volume-to-surface ratio. Derived solely from the kinematic formula of integral geometry, the result is independent of step-length statistics, memory, absorption, and branching, making it equally relevant to photons in turbid tissue, active bacteria in micro-channels, cosmic rays in molecular clouds, or neutron chains in nuclear reactors. Monte-Carlo simulations spanning straight needles, Y-shapes, and isotropic random walks in 2D and 3D confirm the universality and demonstrate how a local measurement of $\langle \ell \rangle$ yields $\langle L\rangle$ without ever tracking the full trajectory.