Space representing all possible states of the universe. A point in phase space completely describes the state at a given instant. In a classic Newtonian world with n particles, there would be 6n dimensions in its phase space; three spatial dimensions and three momentum dimensions for each particle, representing its position and momentum vector.

A Hamiltonian function describes the evolution of points in phase space over time. Not only points are interesting, but solids as well. For instance, a hypersphere in phase space with radius one would describe all possible universes described by measurements with an rms error of one. An interesting theorem by Joseph Liouville proves that for any given region in phase space, the volume will remain the same. However, this does not confine the region; it continually spreads out and grows more and more convoluted.

A little more practcally, a phase space is any imaginary space, plotting measurements of a few phemonena pertinent to a particular physical or mathematical system.

If you were showing the states of a pendulum, for example, you might choose a two-dimensional phase space so that you could plot the pendulum's horizontal position versus its speed at any one given point in time.

You might even throw time in to your phase space to get three dimensions (although time with one other dimension is usually called a time series rather than a phase space), but you would definitely *not* add a dimension to plot the pendulum's vertical position, since that doesn't give you any new information.

One well-known phase space is the Hertzsprung-Russell Diagram used by astronomers to plot the luminosity of a star (as evidenced by its absolute magnitude) against its temperature (as derived from the star's color).

If you pick the right phenomena to include as dimensions in your phase space, when you plot your observations, patterns will emerge to tell you something about your system.

This happened in the early 1960s when Edward Lorenz reduced complicated equations about convection to differential equations in three dimensions. When he plotted his observations in the phase space those dimensions represented, he got the famous strange attractor which bears his name.

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