Normally sampling from the standard simplex I want to be able to generate values from an $n$-dimensional multivariate Gaussian distribution truncated to $[0, 1]$ range with given means and a covariance matrix, such that they sum to one.
I think this is the same as sampling from the standard $n$-simplex according to the Gaussian distribution, but how would I go about doing this?
 A: These papers describe how to sample from a multivariate normal truncated on a (p - 1) simplex ([http://ieeexplore.ieee.org/abstract/document/6884588/], [dobigeon.perso.enseeiht.fr/papers/Dobigeon_TechReport_2007b.pdf]). Sampling is done via Gibbs sampling or HMC. In short, it uses ideas from the (conditional) multivariate Normal distribution. Assume that you want to sample a vector $\alpha_{(p\times 1)}$ which is contrained to a Multivariate Normal truncated on a $p-1$ simplex, i.e., $\alpha\sim N(\mu,\Sigma)I(\alpha\in\mathbb{S}^{p-1})$. You can sample the $j^{th}$ component ($\alpha_j$) conditional on $\alpha_{-j}$ (i.e., the remaining components of $\alpha$), and set the last component ($p^{th}$) to $1 - \sum_{j=1}^{p-1}\alpha_j$ with conditional mean $\mu_{j|-j}$ and conditional variance $\Sigma_{j|-j}$. The papers I mentioned describe how to calculate these. Note that there's only $p - 1$ pieces of information.
A: It sounds like you want the logit-normal distribution. This distribution shows up a lot in Compositional Data Analysis (CDA). CDA is often used in geology to measure the composition of minerals within soil or rock samples. The logit-normal takes a logit tranform of your random variable and this logit-transformed random variable is a normally distributed random variable. Formally, 
$$Y=log\left(\frac{X}{1-X}\right)$$
where $X$ is logit-normal and $Y$ is normal. Multivariate extensions exist and are the more commonly used forms of the density. 
If this is not what you want and you truly want a normal random variable that is restricted by a collection of constraints to always sum to 1 and have all entries be non-negative, you'll need to resort to other simulation techniques to get draws from the distribution. It is fairly complicated to perform these draws. John Geweke wrote a paper about doing this and Christian Robert also wrote a paper on sampling from this type of distribution. 
