Throw of a die: Dependance on initial conditions and chaos

[greypilgrim]

Hi,

We normally use a simple symmetry argument to show that the probability of each outcome of a throw of a fair, cube-shaped die is 1/6. However, is it possible to actually model the physics of the throw and show that the probabilities are 1/6?

Since this is classical physics, the outcome can in principle be predicted knowing the inital conditions of the throw. So I guess we'd have to show that very similar initial conditions lead to any of the six outcomes. A numerical simulation of the throw might get nasty, but maybe there's a simpler chaos-theoretic argument?

 

[Dr. Courtney]

In practice, one needs a model based not only on initial conditions, but accurate empirical descriptions of all the collisions and interactions as the die is rolling and bouncing along. To my knowledge, an accurate model for all this is unavailable.

 

[greypilgrim]

If already this is unavailable, is there a way to compute the probability distribution of an asymmetric die, consisting of a general rectangular cuboid with different sides a,b,c (or maybe let's first have a=b) with constant mass distribution, if the initial conditions are "random enough"?