Are there well known probability distributions defined on a n-dimensional simplex besides the Dirichlet distribution where the variation of of each component doesn't vary as much when the mean of the component changes as in the Dirichlet distribution?
In the Dirichlet distribution, the variance changes too much when the mean changes between $(0, 1)$.
I'm thinking of something more like a multivariate normal distribution but defined on the $n$-dimensional simplex.