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

TL;DR

This paper extends Dirac's chord method to nonconvex bodies by employing quasi-probability distributions, enabling the calculation of complex integrals in physics for arbitrary shapes and multiple objects.

Contribution

It introduces a generalized chord distribution using quasi-probability concepts to handle nonconvex bodies in Dirac's method, broadening its applicability.

Findings

Quasi-probability distributions can be used for Monte Carlo calculations.

The method applies to systems with multiple objects.

It provides a way to handle negative values in second derivatives of autocorrelation functions.

Abstract

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.