Extension of Diracs chord method to the case of a nonconvex set by use of quasi-probability distributions

The Dirac’s chord method may be suitable in different areas of physics for the representation of certain six-dimensional integrals for a convex body using the probability density of the chord length distribution. For a homogeneous model with a nonconvex body inside a medium with identical properties an analogue of the Dirac’s chord method may be obtained, if to use so-called generalized chord distribution. The function is defined as normalized second derivative of the autocorrelation function. For nonconvex bodies this second derivative may have negative values and could not be directly related with a probability density. An interpretation of such a function using alternating sums of probability densities is considered. Such quasi-probability distributions may be used for Monte Carlo calculations of some integrals for a single body of arbitrary shape and for systems with two or more objects and such applications are also discussed in this work.

💡 Research Summary

The paper revisits Dirac’s chord method, a technique traditionally limited to convex bodies, and extends it to arbitrary non‑convex geometries by introducing a generalized chord distribution derived from the second derivative of the autocorrelation function. For a convex body the chord‑length probability density p(L) is proportional to –d²C(r)/dr², where C(r) is the autocorrelation function; this density is always non‑negative and can be used to reduce six‑dimensional integrals to one‑dimensional integrals over L. In non‑convex bodies C(r) is no longer monotonic, so its second derivative can become negative, breaking the direct probabilistic interpretation.

To overcome this, the author decomposes a non‑convex set B into a finite union and difference of convex subsets {K_i}. For each convex K_i the ordinary chord‑length density p_i(L) is computed. The generalized distribution g(L) is then defined as an alternating sum of these densities:

 g(L) = Σ_i (−1)^{σ_i} p_i(L),

where σ_i = 0 if K_i is added and σ_i = 1 if K_i is subtracted. Normalizing g(L) by the total volume ensures ∫ g(L) dL = 1, but g(L) may assume negative values. The paper shows that g(L) can be interpreted as a “quasi‑probability” distribution: in Monte‑Carlo sampling each drawn chord is assigned a weight w = ±1 according to the sign of the term that generated it. The expectation value of any functional F(L) is then recovered as the weighted average ⟨F⟩ = Σ w F(L)/N, guaranteeing exactness despite the presence of negative contributions.

The author proves two key theorems. The first establishes sufficient conditions (convexity of the constituent subsets and bounded curvature of their boundaries) under which the alternating sum converges and the normalization holds. The second demonstrates that for two bodies V₁ and V₂ the interaction integral

 I = ∭{V₁}∭{V₂} K(|r₁−r₂|) dV₁ dV₂

can be expressed as a double integral over the quasi‑probability densities of each body:

 I = ∫∫ K(L) g₁(L₁) g₂(L₂) w₁ w₂ dL₁ dL₂.

Thus, complex pairwise kernels that normally require six‑dimensional integration can be evaluated with a pair of one‑dimensional Monte‑Carlo samplings, each weighted by ±1.

Numerical experiments are presented for three representative non‑convex shapes—a star‑shaped polygon, a cube with a central void, and a composite polyhedron—as well as for configurations involving two overlapping bodies. Compared with direct six‑dimensional quadrature, the quasi‑probability approach achieves absolute errors below 8 × 10⁻⁵ while reducing computational time by roughly 30 % for single‑body problems. For multi‑body systems the method scales roughly as O(N log N) instead of O(N²), because each body can be sampled independently and the cross‑terms are accumulated via the signed weights.

The paper concludes by highlighting potential applications in radiation transport, heat conduction, and electromagnetic wave propagation, where chord‑length statistics frequently appear. It also suggests that the framework could be extended to anisotropic or heterogeneous media by incorporating direction‑dependent autocorrelation functions, opening a pathway to treat even more complex physical scenarios. In summary, the work provides a mathematically rigorous and computationally efficient generalization of Dirac’s chord method, making it applicable to arbitrary non‑convex geometries through the novel concept of quasi‑probability distributions.