Discrepancy between Monte-Carlo Results and Analytic Values for the Average Excluded Volume of Rectangular Prisms

TL;DR

This study compares Monte Carlo simulation results with analytical calculations for the average excluded volume of various 3D shapes, revealing an initial discrepancy for rectangular prisms that was later attributed to a coding error.

Contribution

The paper identifies and corrects a coding error in Monte Carlo simulations, aligning simulation results with analytical values for the excluded volume of rectangular prisms.

Findings

Initial discrepancy found between simulation and analytical results for rectangular prisms.

Correction of simulation code resolved the discrepancy, confirming analytical predictions.

Simulations and analytics agree for ellipsoids and capped cylinders.

Abstract

We perform Monte Carlo simulations to determine the average excluded volume <V_{ex}> of randomly oriented rectangular prisms, randomly oriented ellipsoids and randomly oriented capped cylinders in 3-D. There is agreement between the analytically obtained <V_{ex}> and the results of simulations for randomly oriented ellipsoids and randomly oriented capped cylinders. However, we find that the <V_{ex}> for randomly oriented prisms obtained from the simulations differs from the analytically obtained results. In particular, for cubes, the percentage difference is 3.92, far exceeding the bounds of statistical error in our simulation.{\bf Added in Revision 2: We recently found the cause of the discrepancy between the simulation result and the analytic value of the excluded volume to be the effect of an error in our simulation code. Upon rectification of the simulation code, the simulation yields $ 11.00 \pm 0.002 $ as the excluded volume of a pair of randomly oriented cubes of unit volume. The simulation also yields results as predicted by the analytic formula for all other cases of rectangular prisms that we study.}