A universal property of random trajectories in bounded domains

Abstract

The celebrated invariance property states that particles entering a bounded domain, with isotropic and uniform incidence, spend on average $⟨\ell ⟩=4V/S$ 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 -- $\frac{1}{⟨\ell ⟩}=\frac{1}{⟨L⟩}+\frac{1}{⟨\sigma ⟩}$, with $⟨\ell ⟩$ its mean in-domain path, $⟨L⟩$ its mean total length, and $⟨\sigma ⟩$ 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 $⟨\ell ⟩$ yields $⟨L⟩$ without ever tracking the full trajectory.

9 pages, 6 figures proof simplified, 2d case fully treated, Monte Carlo simulations added

References in corpus (5)

• Run-and-Tumble Dynamics of Self-Propelled Particles in Confinement
• Invariance property of wave scattering through disordered media
• Properties of branching exponential flights in bounded domains
• Cauchy's formulas for random walks in bounded domains
• Invariance property in inhomogeneous scattering media with refractive-index mismatch