Statistical mechanics guides the motions of cm scale objects

Abstract

Calculations and mechanistic explanations for the probabilistic movement of objects at the highly relevant cm to m length scales has been lacking and overlooked due to the complexity of current techniques. Predicting the final-configuration probability of flipping cars for example remains extremely challenging. In this paper we introduce new statistical methodologies to solve these challenging macroscopic problems. Boltzmann's principles in statistical mechanics have been well recognized for a century for their usefulness in explaining thermodynamic properties of matter in gas, liquid and solid phases. Studied systems usually involve a large number of particles (e.g. on the order of Avogadro's number) at the atomic and nanometer length scales. However, it is unusual for Boltzmann's principles to be applied to individual objects at centimeter to human-size length scales. In this manuscript, we show that the concept of statistical mechanics still holds for describing the probability of a tossed orthorhombic dice landing on a particular face. For regular dice, the one in six probability that the dice land on each face is well known and easily calculated due to the 6-fold symmetry. In the case of orthorhombic dice, this symmetry is broken and hence we need new tools to predict the probability of landing on each face. Instead of using classical mechanics to calculate the probabilities, which requires tedious computations over a large number of conditions, we propose a new method based on Boltzmann's principles which uses synthetic temperature term. Surprisingly, this approach requires only the dimensions of the thrown object for calculating potential energy as the input, with no other fitting parameters needed. The statistical predictions for landing fit well to experimental data of over fifty-thousand samplings of dice in 23 different dimensions. We believe that the ability to predict, in a simple and tractable manner, the outcomes of macroscopic movement of large scale probabilistic phenomena opens up a new line of approach for explaining many phenomena in the critical centimeter-to-human length scale.