Phase space is the abstract space constructed from the spatial coordinates and momentum components of all of the atoms in a system.
6N dimensional for N particles in 3D. 2D space for a single particle in 1D.
The phase space of a physical system in statistical mechanics is the set of all possible positions and velocities of the particles in the system, and counting microstates in these systems corresponds to finding the volume of surfaces and regions in phase space.

Macrostate of a classical ideal gas:

• Count the microstates consistent with a given macrostate
• Specify the macrostate by giving three parameters
  • N – number of atoms
  • V – volume of the gas
  • U – internal energy (for an ideal gas, it's only the sum of kinetic energies)
  $$U=\sum _{i=1}^N\frac{p_i^2}{2m}$$

Microstates are given by $(x,p_x)$

The phase space of a model composed of $N$ particles with momenta and coordinates is the set of all possible values of the $\overrightarrow{p_i}$ and $\overrightarrow{x_i}$. In general for a system composed of N particles in $d$ spatial dimensions, the phase space is a $2\times d\times N$ dimensional space.

The configuration space is the product of the volume V of space available to each particle (note for 2d billiards, ‘volume V’ really means area).

• For example, the probability of the macrostate where all the particles are on the left half of the box is ${\omega }_q(V/2,N)=(V/2{)}^N$
  Monatomic ideal gas derivation