Sethna 3.2.1 - What does it mean to integrate over configuration space?

Question

I'm having trouble understanding the examples Sethna uses in this section to illustrate the microcanonical ensemble.

First he talks about the probability density $\rho(Q)$ that $N$ ideal gas particles will be in position space configuration $Q\in R^{3N}$ inside a box of volume $V$. He says that since $\int \rho dQ=1$ and integrating over the positions gives a factor of $V$ for each of the $N$ particles, $\rho (Q)=1/V^N$.

I am confused by a few things here. First, what does it mean to integrate over position space, and where does the factor of $V$ come from for each position? This is confusing to me because $Q$ is a position vector and the volume is continuous, so why can we give any answer besides there are an infinite number of spatial configurations? Second, what is $R^{3N}$?

Answer 1

A spatial microscopic configuration of a set of $N$ non-interacting point-like particles in 3D is given once we have all the positions, i.e. the set of all the position vectors $Q=\{{\bf r}_1, \dots , {\bf r}_N \}$.

If the particles are confined in a finite volume, each cartesian component of the position ${\bf r}_i$ is an element of a subset of the real numbers $R$, each position vector ${\bf r}_i$ is inside a finite volume $V$, and the $3N$ position $Q$ is an element of a finite volume subset of $R^{3N}$, the $3N$-fold cartesian product of $R$, of volume $V^{N}$.

Notice that the continuity of the values of the positions has nothing to do with the spatial dimensionality. When we say that the continuous values of the position ${\bf r}_i$ belong to a 3D space, we mean that we need $3$ real numbers to uniquely assign a position in the space.