If you have a gas with $n$ particles, can you model this as a random walk of a single particle in $3n$-space? If you have $n$ particles in a box that undergo diffusion, this is basically a random walk of $n$ particles. Can this exactly be modeled by a single random walk in $3n$ space? Does the variance of that single random walk correspond to the variance of the particles in the box? Or is there no one-to-one analogy?
If this is true, does someone have a good introduction to the mathematics of random walks in physics for a gas?
My specific question
If you have $n$ random walkers in $3$ dimensions, is it equivalent to say that you have one random walker in $3n$ dimensions?
 A: 
My specific question If you have n random walkers in 3 dimensions, is it equivalent to say that you have one random walker in 3n dimensions?

I would say no. Some properties of the random walk are dimension-dependent in a way that cannot be absorbed by changing the number of particles. For example, the first passage time scales with the dimensions as $P^{1st}\sim t^{-\frac{d}{2}}$ . Since the dependence on $d$ in this quantity cannot be absorbed with a constant, nor "balanced" changing the number of particles, I would say that you cannot proceed with the mapping that you propose.

If this is true, does someone have a good introduction to the mathematics of random walks in physics for a gas?

I don't know if they treat the case of the gas with random walks, but in "Krapivsky, P. L., Redner, S., & Ben-Naim, E. (2010). A kinetic view of statistical physics. Cambridge University Press." they show how to compute quantities such as the mentioned first passage time on random walks having into account the role of dimensionality.
