Space of all possible states that a system can take
For other uses, see Phase space (disambiguation).
Diagram showing the periodic orbit of a mass-spring system in simple harmonic motion. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)
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In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented, with each possible state 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. It is the direct product of direct space and reciprocal space.[clarification needed] The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.[1]
^Nolte, D. D. (2010). "The tangled tale of phase space". Physics Today. 63 (4): 33–38. Bibcode:2010PhT....63d..33N. doi:10.1063/1.3397041. S2CID 17205307.
In dynamical systems theory and control theory, a phasespace or state space is a space in which all possible "states" of a dynamical system or a control...
an optical phasespace is a phasespace in which all quantum states of an optical system are described. Each point in the optical phasespace corresponds...
In applied mathematics, the phasespace method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent...
can exist Phase (matter), a region of space throughout which all physical properties are essentially uniform Phasespace, a mathematical space in which...
curve. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phasespace. This reveals...
Phasespace crystal is the state of a physical system that displays discrete symmetry in phasespace instead of real space. For a single-particle system...
complete the phase. If you land on a twist phase you can decide to play a twist phase or one of the phases on either side of the twist phasespace. If you...
product, after Hermann Weyl and Hilbrand J. Groenewold) is an example of a phase-space star product. It is an associative, non-commutative product, ★, on the...
volume of k-space, such that every possible k is "equivalent" to exactly one point in this region. Phasespace Reciprocal space Configuration space Fractional...
conditions. More specifically, given two starting trajectories in the phasespace that are infinitesimally close, with initial separation δ Z 0 {\displaystyle...
system, where bounded classical trajectories are confined onto tori in phasespace, tunnelling can be understood as the quantum transport between semi-classical...
theorem: Assume the phasespace has a finite Liouville volume and let F be a phasespace volume-preserving map and A a subset of the phasespace. Then almost...
The phases of ice are all possible states of matter for water as a solid. Currently, 19 phases, including both crystalline and amorphous ice, have been...
each xi is a point in 3-dimensional space. This has analogies with the classical phasespace. A classical phasespace contains a real-valued function in...
({\boldsymbol {p}},{\boldsymbol {q}})} is called phasespace coordinates. (Also canonical coordinates). In phasespace coordinates ( p , q ) {\displaystyle ({\boldsymbol...
variables). It is a two-dimensional case of the general n-dimensional phasespace. The phase plane method refers to graphically determining the existence of...
transients and settle the system into its typical behavior. The subset of the phasespace of the dynamical system corresponding to the typical behavior is the...
infinitesimally close trajectories. Quantitatively, two trajectories in phasespace with initial separation vector δ Z 0 {\displaystyle \delta \mathbf {Z}...
long periods of time, the time spent by a system in some region of the phasespace of microstates with the same energy is proportional to the volume of...
conserved quantities on the phasespace. More explicitly, suppose that the energy E is fixed, and let ΩE be the subset of the phasespace consisting of all states...
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