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

### 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.

© 2011 American Institute of Physics

Received 21 January 2011
Accepted 19 April 2011
Published online 24 May 2011

Article outline:

I. INTRODUCTION
II. RAY METHOD
A. Ray length distribution
B. Method Monte Carlo with rays
III. HELPFUL ANALYTICAL EQUATIONS
IV. CHORD METHOD
A. Chord length distribution
B. Method Monte Carlo with chords
V. MULTI-BODY CASE
A. Some equations with two different bodies
B. Relation with methods for single body
C. Application to calculations with rays
D. Application to calculations with chords
E. Analytical expressions for two bodies
VI. CONCLUSION