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More complicated systems demand even higher-dimensional phase spaces!
For example, here are 32 atoms moving around in a square box.
It takes 64 numbers to describe their positions, and 64 more to describe all their velocities. So, their phase space is 128-dimensional.
The physicists Boltzmann and Gibbs were the first to clearly understand this idea.
In 1871, Boltzmann introduced the word 'phase' in analogy to Lissajous figures! In 1902, Gibbs emphasized that the motion of many particles could be modeled as a single point in a high-dimensional space.