Diffeomorphism of sorted vectors and a box

Question

Let $A = \left\{ a \in \mathbb R^n \;\middle|\; a_1 \geq \cdots \geq a_n \geq 0 \right\}$ be the set of sorted non-negative vectors in $\mathbb R^n$. I'm looking for a diffeomorphism $\varphi : A \to B$, where $B$ is a "box" in $\mathbb R^n$. By a box, I mean a set of the form $B = \prod_{i=1}^n I_i$, where $I_i$ is an interval or ray in $\mathbb R$. (Extra points if $B = \mathbb R^n$.)