Well known probability distributions defined on a $n$-dimensional simplex besides the Dirichlet distribution?

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.

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As for the comment about changig variance: for a distribution which lives on $(0,1)$, when the mean gets close to one of the borders, the variance is bound to be small! Thats a fact of life. – kjetil b halvorsen Sep 15 '12 at 3:24

This is studied in compositional data analysis, there is a book by Aitchison. Define the simplex by $$S^n =\{(x_1, \dots,x_{n+1}) \in {\mathbb R}^{n+1} \colon x_1>0,\dots, x_{n+1}>0, \sum_{i=1}^{n+1} x_i=1\}.$$ Note that we use the index $n$ to indicate dimension! Define the geoemtric mean of an element of the simplex, $x$ as $\bar{x}$. Then we can define the logratio transformation (introduced by Aitchison) as $x=(x_1, \dots, x_{n+1}) \mapsto (\log(x_1/\bar{x}), \dots, \log(x_n/\bar{x})$. This transformation is onto ${\mathbb R}^n$, so have an inverse which I leave to you to calculate (There are also other versions of this transformation that can be used).
Now you can take a normal (or whatever) distribution defined on $\mathbb R^n$ and use this inverse transformation to define a distribution on the simplex.