How do you turn this statement into a Mathematica formula?

Question

I'm reading a book on Nucleosynthesis. It has this formula for the number density of a species:$$n_i=g_i\space e^{\mu_i/T}\int\frac{d^3p}{(2\pi)^3}e^{-E_1/T}$$I don't recognize this notation. How do you implement $\int{d^3p}$ in Mathematica?

Answer 1

$d^3p$ is the volume element of momentum space, I think. So, $p$ is a point in 3 dimensional space; therefore an infinitesimal volume element in this space will be sort of...thrice $d$'d, one time for each axis. In other words, $d^3p = dp_1\ dp_2\ dp_3$, where $p=(p_1, p_2, p_3)$.

Note that $\int d^3p$ is notationally equivalent to $\iiint dp_1\ dp_2\ dp_3$, and unless you're in a weird analysis setting, you can perform each separate integration in whatever order you choose, e.g. $\int dp_2 \left(\int dp_1 \left(\int dp_3 f(p_1,p_2,p_3)\right)\right)$. Luckily, Mathematica doesn't require you to explicitly nest integrals like this; you can account for all of the integrated-over variables in a single Integrate.

So, you could implement the integral part of that formula as something like

Integrate[Exp[-Ei[{p1, p2, p3}] / T] / (2 Pi)^3,
          {p1, -Infinity, Infinity},
          {p2, -Infinity, Infinity},
          {p3, -Infinity, Infinity}]

which will integrate over the full extent of all variables (newlines not important, just for layout here).