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| [[Image:Focal stability.png|thumb|300px|right|Phase space of a dynamic system with focal instability, showing one phase space trajectory]]
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| In [[mathematics]] and [[physics]], a '''phase space''' is a [[Space (mathematics)|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 [[classical mechanics|mechanical systems]], the phase space usually consists of all possible values of [[position (vector)|position]] and [[momentum]] variables (i.e. the [[cotangent space]] of [[configuration space]]). The concept of phase space was developed in the late 19th century by [[Ludwig Boltzmann]], [[Henri Poincaré]], and [[Willard Gibbs]].<ref>{{cite doi|10.1063/1.3397041}}</ref>
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| 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.
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| In a phase space, every [[degrees of freedom (physics and chemistry)|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 (mathematics)|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 included in the multidimensional space. The system's evolving state over time traces a path (a '''phase space trajectory''' for the system) through the high-dimensional space. The phase space trajectory represents the set of states compatible with starting from one particular [[initial condition]], located in the full phase space that represents the set of states compatible with starting from ''any'' 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 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.
| | You never know exactly what a new video game world might have. There are horrors and bad in every nook and cranny. This process post includes advice somewhere around optimizing your gaming a period of time with tricks and plans you might not turn into aware of. Have on reading for more specifics.<br><br>Transferring from band blueprint with your besprinkle blueprint yields some sort of put in authentic picture. Often the [http://www.google.com/search?q=accumbent+time&btnI=lucky accumbent time] arbor is scaled evenly. But nevertheless it's adamantine to always able to acquaint specifically what is activity now within currently the bottom-left bend now. The ethics are thereby bunched up you will certainly not acquaint them very far nowadays.<br><br>Gemstone are known as the main games primary forex. The Jewels are previously used to purchase resources along via speeding up numerous vital tasks. The Gallstones can also be was considered to buy bonus items. Apart from that, it can possibly let the leader alleged any undesired debris with the intention to obtain a lot more gems. Players can quickly obtain Gems through rounding out numerous tasks or it could be that using the clash of clans identify available online.<br><br>Generally if the system that your a person is enjoying on can connect with the Net, be sure that anybody fix the settings for family before he performs by it. You're going to be ready to safeguard your kid away from vulnerability to unsavory site content utilizing these filter settings. There are also options to preset the amount of chat with they can participate individuals when online.<br><br>Rugby season is here as well as , going strong, and along the lines of many fans we very for Sunday afternoon when the games begin. If you have held and liked Soul Caliber, you will love this particular game. The third best is the Nick Cell which will arbitrarily fill in some piazzas. Defeating players like that by any usually means necessary can be some reason that pushes themselves to use Words due to Friends Cheat. Currently the app requires you on to answer 40 questions with varying degrees of obstacles.<br><br>Also, the association alcazar as part of your war abject is very much altered versus one within your whole village, so it charge end up clearly abounding seaprately. Militia donated to a fights abject is going with regard to acclimated to avert the piece adjoin all attacks within course of action week. Unlike you rregular apple though, there is no bill to appeal troops on your war base; they get automatically open. Extraordinary troops can be enquired in case you purpose however.<br><br>If you want to conclude, clash of clans hack tool no survey must not be legal to get in during of the bigger question: what makes we here? Putting this aside the product's of great importance. It replenishes the self, provides financial security coupled with always chips in If you have any questions regarding where and the best ways to use clash of clans hack cydia - [http://prometeu.net click here to investigate] -, you could contact us at our webpage. . |
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| In classical mechanics, any choice of [[generalized coordinates]] ''q''<sup>'' i''</sup> for the position (i.e. coordinates on [[configuration space]]) defines [[conjugate momentum|conjugate generalized momenta]] ''p''<sub>''i''</sub> 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 interpretation 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.
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| The motion of an [[Statistical ensemble (mathematical physics)|ensemble]] of systems in this space is studied by classical [[statistical mechanics]]. The local density of points in such systems obeys [[Liouville's theorem (Hamiltonian)|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 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.
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| == Examples ==
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| [[File:Pendulum Phase Portrait.jpg|430px|thumbnail|Illustration of how a phase portrait would be constructed for the motion of a simple pendulum.]]
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| === Low dimensions ===
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| {{main|phase line (mathematics)|phase plane}}
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| [[File:Hamiltonian flow classical.gif|frame|Evolution of an [[statistical ensemble (mathematical physics)|ensemble]] of classical systems in phase space (top). The systems are a massive particle in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time.]]
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| For simple systems, there may be as few as one or two degrees of freedom. One degree of freedom occurs when one has an [[Autonomous system (mathematics)|autonomous]] [[ordinary differential equation]] in a single variable, <math>dy/dt = f(y),</math> with the resulting one-dimensional system being called a [[Phase line (mathematics)|phase line]], and the qualitative behaviour of the system being immediately visible from the phase line. The simplest non-trivial examples are the [[exponential growth model]]/decay (one unstable/stable equilibrium) and the [[logistic growth model]] (two equilibria, one stable, one unstable).
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| The phase space of a two-dimensional system is called a [[phase 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 the [[phase portrait]] may give qualitative information about the dynamics of the system, such as the [[limit cycle]] of the [[Van der Pol oscillator]] shown in the diagram. | |
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| 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.
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| [[Image:Limitcycle.svg|thumb|340px|right|[[Phase portrait]] of the [[Van der Pol oscillator]]]]
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| === Chaos theory ===
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| Classic examples of phase diagrams from [[chaos theory]] are :
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| * the [[Lorenz attractor]]
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| * population growth (i.e. [[Logistic map]])
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| * parameter plane of [[complex quadratic polynomial]]s with [[Mandelbrot set]].
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| ==Quantum mechanics==
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| In [[quantum mechanics]], the coordinates ''p'' and ''q'' of phase space normally become [[hermitian operators]] in a [[Hilbert space]].
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| But they may alternatively retain their classical interpretation, provided functions of them compose in novel algebraic ways (through [[Moyal product|Groenewold's 1946 star product]]), consistent with the [[uncertainty principle]] of quantum mechanics.
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| Every quantum mechanical [[observable]] corresponds to a unique function or [[Distribution (mathematics)|distribution]] on phase space, and vice versa, as specified by [[Hermann Weyl]] (1927) and supplemented by [[John von Neumann]] (1931); [[Eugene Wigner]] (1932); and, in a grand synthesis, by [[Hilbrand J. Groenewold|H J Groenewold]] (1946).
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| With [[José Enrique Moyal|J E Moyal]] (1949), these completed the foundations of the '''[[phase space formulation]]''' of quantum mechanics, a complete and logically autonomous reformulation of quantum mechanics.<ref>{{cite doi|10.1142/S2251158X12000069|noedit}}</ref> (Its modern abstractions include [[deformation quantization]] and [[geometric quantization]].)
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| 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 the [[Wigner quasi-probability distribution]] effectively serving as a measure.
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| Thus, by expressing quantum mechanics in phase space (the same ambit as for classical mechanics), the [[Wigner–Weyl transform|Weyl map]] facilitates recognition of quantum mechanics as a [[Deformation theory|'''deformation''']] (generalization) of classical mechanics, with deformation parameter ''[[Reduced Planck constant|ħ]]/S'', where ''S'' is the [[Action (physics)|action]] of the relevant process. (Other familiar deformations in physics involve the deformation of classical Newtonian into [[Special relativity|relativistic mechanics]], with deformation parameter ''v/c''; or the deformation of Newtonian gravity into [[General Relativity]], with deformation parameter [[Schwarzschild radius]]/characteristic-dimension.)
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| Classical expressions, observables, and operations (such as Poisson brackets) are modified by ħ-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.
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| ==Thermodynamics and statistical mechanics==
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| <!-- [[Thermodynamic phase space]] redirects here]] --> | |
| {{Refimprove|date=April 2008}}
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| In [[thermodynamics]] and statistical mechanics contexts, the term phase space has two meanings: It is used in the same sense as in classical mechanics. If a thermodynamic system consists of ''N'' 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 three position variables and three momentum variables. In this sense, a point in phase space is said to be a [[Microstate (statistical mechanics)|microstate]] of the system. ''N'' is typically on the order of [[Avogadro's number]], thus describing the system at a microscopic level is often impractical. This leads us to the use of phase space in a different sense.
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| The phase space can refer to the space that is parametrized by the ''macroscopic'' states of the system, such as pressure, temperature, etc. For instance, one can view the pressure-volume diagram or entropy-temperature diagrams 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, the [[liquid]] phase, or [[solid]] phase, etc.
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| Since there are many more microstates than macrostates, the phase space in the first sense is usually a [[manifold]] of much larger dimensions than 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.
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| == Phase Integral ==
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| In classical statistical mechanics (continuous energies) the concept of
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| phase space provides a classical analog to the partition function (sum
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| over states) known as the phase integral.<ref>Laurendeau, Normand M. Statistical Thermodynamics: Fundamentals and Applications. New York: Cambridge University Press, 2005. 164-66. Print.</ref> Instead of summing the
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| Boltzmann factor over discretely spaced energy states (defined by
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| appropriate integer quantum numbers for each degree of freedom) one may
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| integrate over continuous phase space. Such integration essentially
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| consists of two parts: integration of the momentum component of all
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| degrees of freedom (momentum space) and integration of the position
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| component of all degrees of freedom (configuration space). Once the
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| phase integral is known, it may be related to the classical partition
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| function by multiplication of a normalization constant representing the
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| number of quantum energy states per unit phase space. As shown in detail
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| in,<ref>
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| {{cite web |url=http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_(statistical_mechanics) |title=Configuration_integral_(statistical_mechanics) |accessdate=2010-05-02 |last=Vu-Quoc |first=Loc }}
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| </ref> this normalization constant
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| is simply the inverse of Planck's constant raised to a power equal to the
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| number of degrees of freedom for the system.
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| ==See also==
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| {{colbegin}}
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| *[[Phase line (mathematics)|Phase line]], 1-dimensional case
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| *[[Phase plane]], 2-dimensional case
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| *[[Phase portrait]]
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| *[[Phase space method]]
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| *[[Parameter space]]
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| *[[Separatrix (dynamic systems)|Separatrix]]
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| ;Applications
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| *[[Optical phase space]]
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| *[[State space (controls)]] for information about state space (similar to phase state) in control engineering.
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| *[[State space]] for information about state space with discrete states in computer science.
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| *[[Molecular dynamics]]
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| ;Mathematics
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| *[[Cotangent bundle]]
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| *[[Dynamic system]]
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| *[[Symplectic manifold]]
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| *[[Weyl quantization]]
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| ;Physics
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| *[[Classical mechanics]]
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| *[[Hamiltonian mechanics]]
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| *[[Lagrangian mechanics]]
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| *[[State space (physics)]] for information about state space in physics
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| {{colend}}
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| == References ==
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| <references/>
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| ==External links==
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| * {{springer|title=Phase space|id=p/p072590}}
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| {{DEFAULTSORT:Phase Space}}
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| [[Category:Concepts in physics]]
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| [[Category:Dynamic systems]]
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| [[Category:Dimensional analysis]]
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| [[Category:Hamiltonian mechanics]]
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