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Thephase space of aphysical system is the set of all possiblephysical states of the system when described by a given parameterization. Each possible state corresponds uniquely to apoint in the phase space. Formechanical systems, the phase space usually consists of all possible values of theposition andmomentum parameters. It is thedirect product of direct space andreciprocal space.[clarification needed] The concept of phase space was developed in the late 19th century byLudwig Boltzmann,Henri Poincaré, andJosiah Willard Gibbs.[1]
In a phase space, everydegree of freedom orparameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called aphase line, while a two-dimensional system is called aphase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (aphase-space trajectory for the system) through the high-dimensional space. The phase-space trajectory represents the set of states compatible with starting from one particularinitial condition, located in the full phase space that represents the set of states compatible with starting fromany initial condition. As a whole, 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 number of dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle'sx,y andz positions and momenta (6 dimensions for an idealized monatomic gas), and for more complex molecular systems additional dimensions are required to describe vibrational modes of the molecular bonds, as well as spin around 3 axes. Phase spaces are easier to use when analyzing the behavior of mechanical systems restricted to motion around and along various axes of rotation or translation – e.g. in robotics, like analyzing the range of motion of arobotic arm or determining the optimal path to achieve a particular position/momentum result.
In classical mechanics, any choice ofgeneralized coordinatesqi for the position (i.e. coordinates onconfiguration space) definesconjugate generalized momentapi, which together define co-ordinates on phase space. More abstractly, in classical mechanics phase space is thecotangent bundle of configuration space, and in this interpretation the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural localDarboux coordinates for the standardsymplectic structure on a cotangent space.
The motion of anensemble of systems in this space is studied by classicalstatistical mechanics. The local density of points in such systems obeysLiouville'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 dynamic 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.
For simple systems, there may be as few as one or two degrees of freedom. One degree of freedom occurs when one has anautonomousordinary differential equation in a single variable, with the resulting one-dimensional system being called aphase line, and the qualitative behaviour of the system being immediately visible from the phase line. The simplest non-trivial examples are theexponential growth model/decay (one unstable/stable equilibrium) and thelogistic growth model (two equilibria, one stable, one unstable).
The phase space of a two-dimensional system is called aphase plane, which occurs in classical mechanics for a single particle moving in one dimension, and where the two variables are position and velocity. In this case, a sketch of thephase portrait may give qualitative information about the dynamics of the system, such as thelimit cycle of theVan der Pol oscillator shown in the diagram.
Here the horizontal axis gives the position, and vertical axis the velocity. As the system evolves, its state follows one of the lines (trajectories) on the phase diagram.
A plot of position and momentum variables as a function of time is sometimes called aphase plot or aphase diagram. However the latter expression, "phase diagram", is more usually reserved in thephysical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists ofpressure,temperature, and composition.
Inmathematics, aphase portrait is ageometric representation of theorbits of adynamical system in thephase plane. Each set of initial conditions is represented by a differentpoint orcurve.
Phase portraits are an invaluable tool in studying dynamical systems. They consist of aplot of typical trajectories in the phase space. This reveals information such as whether anattractor, arepellor orlimit cycle is present for the chosen parameter value. The concept oftopological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called a "sink". The repeller is considered as an unstable point, which is also known as a "source".
A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stablesteady states (with dots) and unstable steady states (with circles) in a phase space. The axes are ofstate variables.In classical statistical mechanics (continuous energies) the concept of phase space provides a classical analog to thepartition function (sum over states) known as the phase integral.[2] Instead of summing the Boltzmann factor over discretely spaced energy states (defined by appropriate integerquantum numbers for each degree of freedom), one may integrate over continuous phase space. Such integration essentially consists of two 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). Once the phase integral is known, it may be related to the classical partition function by multiplication of a normalization constant representing the number ofquantum energy states per unit phase space. This normalization constant is simply the inverse of thePlanck constant raised to a power equal to the number of degrees of freedom for the system.[3]
Classic examples of phase diagrams fromchaos theory are:
Inquantum mechanics, the coordinatesp andq of phase space normally becomeHermitian operators in aHilbert space.
But they may alternatively retain their classical interpretation, provided functions of them compose in novel algebraic ways (throughGroenewold's 1946 star product). This is consistent with theuncertainty principle of quantum mechanics. Every quantum mechanicalobservable corresponds to a unique function ordistribution on phase space, and conversely, as specified byHermann Weyl (1927) and supplemented byJohn von Neumann (1931);Eugene Wigner (1932); and, in a grand synthesis, byH. J. Groenewold (1946). WithJ. E. Moyal (1949), these completed the foundations of thephase-space formulation of quantum mechanics, a complete and logically autonomous reformulation of quantum mechanics.[4] (Its modern abstractions includedeformation quantization andgeometric quantization.)
Expectation values in phase-space quantization are obtained isomorphically to tracing operator observables with the density matrix in Hilbert space: they are obtained by phase-space integrals of observables, with theWigner quasi-probability distribution effectively serving as a measure.
Thus, by expressing quantum mechanics in phase space (the same ambit as for classical mechanics), theWeyl map facilitates recognition of quantum mechanics as adeformation (generalization) of classical mechanics, with deformation parameterħ/S, whereS is theaction of the relevant process. (Other familiar deformations in physics involve the deformation of classical Newtonian intorelativistic mechanics, with deformation parameterv/c;[citation needed] or the deformation of Newtonian gravity intogeneral relativity, with deformation parameterSchwarzschild radius/characteristic dimension.)[citation needed]
Classical expressions, observables, and operations (such asPoisson brackets) are modified byħ-dependent quantum corrections, as the conventional commutative multiplication applying in classical mechanics is generalized to the noncommutative star-multiplication characterizing quantum mechanics and underlying its uncertainty principle.
Inthermodynamics andstatistical mechanics contexts, the term "phase space" has two meanings: for one, it is used in the same sense as in classical mechanics. If a thermodynamic system consists ofN particles, then a point in the 6N-dimensional phase space describes the dynamic state of every particle in that system, as each particle is associated with 3 position variables and 3 momentum variables. In this sense, as long as the particles aredistinguishable, a point in phase space is said to be amicrostate of the system. (Forindistinguishable particles a microstate consists of a set ofN! points, corresponding to all possible exchanges of theN particles.)N is typically on the order of theAvogadro number, thus describing the system at a microscopic level is often impractical. This leads to the use of phase space in a different sense.
The phase space can also refer to the space that is parameterized by themacroscopic states of the system, such as pressure, temperature, etc. For instance, one may view thepressure–volume diagram ortemperature–entropy diagram as describing part of this phase space. A point in this phase space is correspondingly called a macrostate. There may easily be more than one microstate with the same macrostate. For example, for a fixed temperature, the system could have many dynamic configurations at the microscopic level. When used in this sense, a phase is a region of phase space where the system in question is in, for example, theliquid phase, orsolid phase, etc.
Since there are many more microstates than macrostates, the phase space in the first sense is usually amanifold of much larger dimensions than in the second sense. Clearly, many more parameters are required to register every detail of the system down to the molecular or atomic scale than to simply specify, say, the temperature or the pressure of the system.
Phase space is extensively used innonimaging optics,[5] the branch of optics devoted to illumination. It is also an important concept inHamiltonian optics.
In medicine andbioengineering, the phase space method is used to visualizemultidimensional physiological responses.[6][7]
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