A universal property of random trajectories in bounded domains

TL;DR

The paper shows that the classic invariance property for mean path length in a bounded domain is the infinite-length limit of a broader universal law: $\frac{1}{\langle \ell\rangle} = \frac{1}{\langle L\rangle} + \frac{1}{\langle \sigma\rangle}$, where $\langle \sigma\rangle$ is the mean chord length and $\langle L\rangle$ the mean total trajectory length. The approach is purely geometric, leveraging integral geometry (including Santalo's and Blaschke's kinematic formulas) and applies to curves of arbitrary stochastic or deterministic dynamics, including stopping and branching. The authors validate the universality with Monte Carlo simulations across straight needles, Y-shaped and open-triangle geometries, and isotropic 3D random walks, demonstrating that a local measurement of $\langle \ell\rangle$ yields $\langle L\rangle$ without tracking entire trajectories. This yields a practical, scale-agnostic diagnostic for transport phenomena in complex domains, with broad relevance from photons in turbid media to cosmic rays and neutrons in reactors.

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 σ\rangle} $, with $\langle\ell\rangle$ its mean in-domain path, $\langle L\rangle$ its mean total length, and $\langleσ\rangle$ 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.

Figures (6)

• Figure 1: An example of a random trajectory traversing a nonconvex domain $K_0$, defined as the union of 3 volumes, one of which contains a hole.
• Figure 2: Right: an experimental ant trajectory containing four loops, two of which lie inside the observation area. The path can be converted into a loop‑free curve (bottom‑right inset) by selecting, at each node, a branch that bypasses the crossing. In both representations, the number of in‑domain segments is 3. While the loop‑free construction makes the invariance property manifest, it is not practically available, so the segment count must be obtained differently. A graph‑theoretic analogy provides the required counting; note that the loop‑free decomposition is not unique.
• Figure 3: Example of 2D traces where two trajectories — one of which (in red) branches and terminates within the domain — give rise to three independent paths ($N$) in the observation region. In two dimensions, such traces can be interpreted as a graph. In this example, the graph consists of 13 vertices ($v$), indicated by black circles, including 5 4-branch vertices ($v_4$), and 15 edges ($e$), confirming the relation $N = v - e + v_4 = 3$, as expected.
• Figure 4: Comparison between the generalized IP, Eq. \ref{['eq_cauchy_nD_random_final']} (red dashed lines), and Monte Carlo simulations (black filled circles) for various configurations. Distances $\ell$ traveled within the detectors are in arbitrary units; relative differences are given in absolute values. Error bars represent three standard deviations of the mean. (a) Distance $\langle \ell \rangle$ traveled within a disk-shaped detector by 2D straight needles of random length, uniformly distributed. Simulations used $10^6$ needles in a square domain (half-side 50 a.u.) with a disk detector of radius 1 a.u. (b) Same comparison for 2D 'Y-shaped' random curves, using $10^6$ samples under identical geometric conditions. (c) Same as (b), but for 2D open triangles (marked by blue circles at one angle). Since loops are not permitted under the generalized IP, only open shapes are included. Again, $10^6$ samples were used. (d) Comparison of $\langle \ell \rangle$ versus capture probability $p_c$ for a 3D isotropic random walk with fixed jump size $a = 10$ a.u. Simulations used $10^7$ walks in a cubic domain (half-side 1000 a.u.) with a cubic detector of half-side 25 a.u.
• Figure 5: The mean path length $\langle \ell \rangle$ of the fly through the cube —including trajectories that start, end, or pass through the volume— is given by the generalized IP formula in 3 dimensions: $1 / \langle \ell \rangle = 1 / \langle L \rangle+ 1/{(4 V/S)}$, where $\langle L \rangle$ denotes the mean total length of the fly’s trajectory.