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Encyclopedia > Phase space
Phase space of a dynamical system with focal stability.
Phase space of a dynamical system with focal stability.

In mathematics and physics, a phase space 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. A plot of position and momentum variables as a function of time is sometimes called a phase diagram. This term, 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, as a function of pressure, temperature, and composition. Image File history File links Download high resolution version (880x715, 27 KB) This Math image (or all images in this article or category) should be recreated using vector graphics as an SVG file. ... Image File history File links Download high resolution version (880x715, 27 KB) This Math image (or all images in this article or category) should be recreated using vector graphics as an SVG file. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article needs additional references or sources for verification. ... Space has been an interest for philosophers and scientists for much of human history. ... System (from Latin systēma, in turn from Greek systēma) is a set of entities, real or abstract, comprising a whole where each component interacts with or is related to at least one other component and they all serve a common objective. ... Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In physical chemistry, mineralogy, and materials science, a phase diagram is a type of graph used to show the equilibrium conditions between the thermodynamically-distinct phases. ... Physical science is the branch of science including chemistry and physics, usually contrasted with the social sciences and sometimes including and sometimes contrasted with natural or biological science. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin in Canberra, Australia. ... This article includes a list of works cited or a list of external links, but its sources remain unclear because it lacks in-text citations. ... The composition of a material refers to the proportion and combination of elements or components to form a substance. ...

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space. 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 very many dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle's x, y and z positions and velocities as well as any number of other properties. Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... The factual accuracy of this article is disputed. ...

In classical mechanics the phase space co-ordinates are the generalized coordinates qi and their conjugate generalized momenta pi. 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. Furthermore, because each point in phase space lies on exactly one phase trajectory, no two phase trajectories can intersect. Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ... This article or section does not cite any references or sources. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ...

For simple systems, such as a single particle moving in one dimension for example, there may be as few as two degrees of freedom, (typically, position and velocity), and a sketch of the phase portrait may give qualitative information about the dynamics of system, such as the limit-cycle of the Van der Pol oscillator shown in the diagram. A limit-cycle is a closed trajectory in phase space exhibited by nonlinear systems. ... Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle. ...

Phase portrait of the Van der Pol oscillator

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. The image illustrates a limitcycle. ... The image illustrates a limitcycle. ...

Classic examples of phase diagrams from chaos theory are the Lorenz attractor and Mandelbrot set. A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3 A trajectory of Lorenzs equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, noted for its butterfly shape. ... Initial image of a Mandelbrot set zoom sequence with continuously colored environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ...

Quantum mechanics

In quantum mechanics, the coordinates p and q of phase space become hermitian operators in a Hilbert space, but may alternatively retain their classical interpretation, provided functions of them compose in novel algebraic ways (through Groenewold's 1946 star product). Every quantum mechanical observable corresponds to a unique function or 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 H J Groenewold (1946). With José Enrique Moyal (1949), these completed the foundations of phase-space quantization, a logically autonomous reformulation of quantum mechanics. Its modern abstractions include deformation quantization and geometric quantization. Fig. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, the Moyal product is an example for an associative, non-commutative product on the functions of a Poisson manifold. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... John von Neumann (Hungarian Margittai Neumann János Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics... Eugene Wigner Eugene Paul Wigner (Hungarian Wigner Pál Jenő) (November 17, 1902 – January 1, 1995) was a Hungarian physicist and mathematician who received the Nobel Prize in Physics in 1963 for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and... José Enrique, Jo or Joe, Moyal (October 1, 1910 - May 22, 1998) was an aeronautical and electrical engineer, statistician, and mathematical physicist who established the phase space formulation of quantum mechanics in 1949 by bringing together some previous works of Hermann Weyl, John von Neumann, Eugene Wigner, and H. Groenewold. ... In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for associating a quantum mechanical Hermitian operator with a classical distribution in phase space. ... In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for associating a quantum mechanical Hermitian operator with a classical distribution in phase space. ... In mathematical physics, geometric quantization is a mathematical approach to define a quantum theory corresponding to a given classical theory in such a way that certain analogies between the classical theory and the quantum theory remain manifest, for example the similarity between the Heisenberg equation in the Heisenberg picture of...

Thermodynamics and statistical mechanics

In thermodynamics and statistical mechanics contexts, the term phase space has two meanings: Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

  • It is used in the same sense as in classical mechanics. If a thermodynamical system consists of N particles, then a point in the 6N-dimensional phase space describes the dynamical state of every particle in that system. In this sense, a point in phase space is said to be a 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.
  • 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.

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 up to the molecular or atomic scale than to simply specify, say, the temperature or the pressure of the system. In physics, the term state is used in several related senses, each of which expresses something about the way a physical system is. ... Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon-12 atoms in 12 grams (0. ... In its most common usage, the term phase change indicates that a substance has changed among the three classical phases of matter: solid, liquid and gas. ... A liquid will usually assume the shape of its container A liquid is one of the main states of matter. ... For other uses, see Solid (disambiguation). ...

See also

  Results from FactBites:
Phase space - definition of Phase space in Encyclopedia (284 words)
A phase diagram or phase space is a useful construct used in mathematics and physics to demonstrate and visualise the changes in a given system.
A phase space may contain very many dimensions, For instance a gas containing many molecules may require a separate dimension for each particle's x, y and z positions and velocities as well as any number of other properties.
A classic example of a phase diagram from chaos theory is the Lorenz attractor.
  More results at FactBites »



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