Phase Space

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 pi 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

Other articles related to "phase space, phase, space":

Phase Space - Phase Integral
... In classical statistical mechanics (continuous energies) the concept of phase space provides a classical analog to the partition function (sum over states) known as the phase integral ... quantum numbers for each degree of freedom) one may integrate over continuous phase space ... parts integration of the momentum component of all degrees of freedom (momentum space) and integration of the position component of all degrees of freedom (configuration space) ...
Graph Dynamical System - Formal Definition
... The phase space associated to a dynamical system with map F Kn → Kn is the finite directed graph with vertex set Kn and directed edges (x, F(x)) ... The structure of the phase space is governed by the properties of the graph Y, the vertex functions (fi)i, and the update scheme ... The research in this area seeks to infer phase space properties based on the structure of the system constituents ...
Adiabatic Invariant - Thermodynamics - Adiabatic Expansion of An Ideal Gas
... In a molecular interpretation, S is the logarithm of the phase space volume of all gas states with energy E(T) and volume V ... Since each gas molecule can be anywhere within the volume V, the volume in phase space occupied by the gas states with energy E is ... the N gas molecules are indistinguishable, the phase space volume is divided by, the number of permutations of N molecules ...
Mc Gehee Transformation
... The transformation blows up the single point in phase space where the collision occurs into a collision manifold, the phase space point is cut out and in its place a smooth manifold is pasted ... This allows the phase space singularity to be studied in detail ...
Sequential Dynamical System - Definition
... The phase space associated to a sequential dynamical system with map F Kn → Kn is the finite directed graph with vertex set Kn and directed edges (x, F(x)) ... The structure of the phase space is governed by the properties of the graph Y, the vertex functions (fi)i, and the update sequence w ... A large part of SDS research seeks to infer phase space properties based on the structure of the system constituents ...

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