In mathematics and physics, a **phase space**, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables i.e. the cotangent space of configuration space.

A plot of position and momentum variables as a function of time is sometimes called a **phase plot** or a **phase diagram**. Phase diagram, however, is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of pressure, temperature, and composition.

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system, or allowed combination of values of the system's parameters, a point is plotted in the multidimensional space. Often this succession of plotted points is analogous to the system's state evolving over time. In the end, the phase diagram represents all that the system can be, and its shape can easily elucidate qualities of the system that might not be obvious otherwise. A phase space may contain a great many dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle's *x*, *y* and *z* positions and momenta as well as any number of other properties.

In classical mechanics, any choice of generalized coordinates qi for the position (i.e. coordinates on configuration space) defines conjugate generalized momenta p_{i} which together define co-ordinates on phase space. More abstractly, in classical mechanics phase space is the cotangent space of configuration space, and in this interpretaton the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local Darboux coordinates for the standard symplectic structure on a cotangent space.

The motion of an ensemble of systems in this space is studied by classical statistical mechanics. The local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant. Within the context of a model system in classical mechanics, the phase space coordinates of the system at any given time are composed of all of the system's dynamical variables. Because of this, it is possible to calculate the state of the system at any given time in the future or the past, through integration of Hamilton's or Lagrange's equations of motion.

Read more about Phase Space: Quantum Mechanics, Thermodynamics and Statistical Mechanics, Phase Integral

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### Famous quotes containing the words space and/or phase:

“As photographs give people an imaginary possession of a past that is unreal, they also help people to take possession of *space* in which they are insecure.”

—Susan Sontag (b. 1933)

“I had let preadolescence creep up on me without paying much attention—and I seriously underestimated this insidious *phase* of child development. You hear about it, but you’re not a true believer until it jumps out at you in the shape of your own, until recently quite companionable child.”

—Susan Ferraro (20th century)