Universal property of random trajectories in bounded domains.

Abstract

Can we deduce the total length of a random trajectory by observing only its local path segments within a confined domain? Surprisingly, the answer is yes-for curves randomly placed and oriented in space, whether stochastic or deterministic; generated by ballistic or diffusive dynamics; possibly interrupted by stopping or branching; and in two or more dimensions. More precisely, the mean total length 〈L〉 relates to the mean in-domain path length 〈ℓ〉 and the mean chord length of the domain 〈σ〉 via the following simple and universal relation:1/〈ℓ〉=1/〈L〉+1/〈σ〉.Here, 〈σ〉 is a purely geometric quantity, dependent only on the volume-to-surface ratio of the domain. 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 microchannels, cosmic rays in molecular clouds, or neutron chains in nuclear reactors. Monte Carlo simulations spanning straight needles, Y shapes, and isotropic random walks in two and and three dimensions confirm the universality and demonstrate how a local measurement of 〈ℓ〉 yields 〈L〉 without ever tracking the full trajectory.