This arose in the context of statistical physics and is related to the total energy distribution of $N$ Gibbs subsystems.
\begin{align} &\prod_{i=1}^N \left(\int_0^\infty d x_i e^{-x_i}\right)\delta\left(x_t - \sum_{i=1}^N x_i\right) \\ &=e^{-x_t}\frac{d}{dx_t}\prod_{i=1}^N \left(\int_0^\infty d x_i\right)\theta\left(x_t - \sum_{i=1}^N x_i\right) =e^{-x_t}\frac{d}{dx_t}\frac{x_t^N}{N!} \end{align}
$x_t$ is a positive real constant - the system energy, and the $x_i$ represent subsystem energies. (So I want to say that $x_t=\sum x_i$, as these represent a system energy and the sum of subsystem energies. The presence of the integration variables makes it awkward to say that, though). The delta and step functions are supposedly part of the integrand, but I'm having difficultly understanding the notation. Supposedly the final integral,
$$\prod_{i=1}^N \left(\int_0^\infty d x_i\right)\theta\left(x_t - \sum_{i=1}^N x_i\right) =\frac{x_t^N}{N!}$$
is a volume of a pyramid cut from an $N$-dimensional cube with edge length $x_t$. Can someone explain where this is coming from? Specifically, I don't understand how the machinery of the delta and step function is working here.
So $x_t$ is a constant, and so it isn't one of the variables $x_1,\ldots,x_n$? Not great notation. Can I call it $c$ (for constant) instead?
The integral is just the volume of the simplex defined by $x_i-\le 0$ and $\sum_1^n x_i\le c$. This is $c^n$ times the volume of the simplex defined by $x_i\ge0$ and $\sum_1^n x_i\le 1$. This volume is $1/n!$.
Of course, this is well-known, but comes recursively from the fact that if one has an $n$-simplex with a face $F$ of $(n-1)$-volume $v$ and height $h$ (distance of the other vertex from hyperplane containing $F$) then the $n$-simplex has volume $h^n v/n$.