Expansion

Expansion Records

The number of trajectories to reach an element of the subclass `R_(k+dk,o +do)` coming from the subclass `R_(k,o)` is `((dk),(do))`, and the total number of trajectories of `dk` steps is `2^(dk)`.
Each element of that subclass is such that:
`{(\stackrel{~}{u}_(k+dk,o +do),=(k+dk)*ln(2)-(o +do)*ln(3)),(,=\stackrel{~}{u}_(k,o)+dk*ln(2)-do*ln(3)):}`

So, an number of the subclass `R_(k+dk,o +do)` has a probability `(((dk),(do)))/2^(dk)` to have an expansion:
`s=1+(dk*ln(2)-do*ln(3))/(k*ln(2)-o*ln(3))`

If we consider that this event of probability `p` may occur for `n` around `1/p`:
`u~~dk*ln(2)-ln(((dk),(do)))`
`s=1+(dk*ln(2)-do*ln(3))/(dk*ln(2)-ln(((dk),(do))))`

We can build a model for maximum expansion by:
`s_max(dk)=max_(do)(1+(dk*ln(2)-do*ln(3))/(dk*ln(2)-ln(((dk),(do)))))`

The Figure bellow shows this model and real expansion records at `dk`, number of steps to reach the Maximum Height.