Phase space of a dynamical system with focal stability.

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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, as a function of pressure, temperature, and composition. In Physical chemistry, Mineralogy, and Materials science, a phase diagram is a type of graph used to show the equilibrium conditions Physical science is an encompassing term for the branches of Natural science and Science that study non-living systems in contrast to the biological sciences Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space. For information on degrees of freedom in other sciences see Degrees of freedom. In Mathematics, Statistics, and the mathematical Sciences a parameter ( G auxiliary measure) is a quantity that defines certain characteristics 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.

In classical mechanics the phase space co-ordinates are the generalized coordinates q_i and their conjugate generalized momenta p_i. By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent The motion of an ensemble of systems in this space is studied by classical statistical mechanics. In Mathematical physics, especially as introduced into Statistical mechanics and Thermodynamics by J Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics The local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant. In Physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian 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.

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 the system, such as the limit-cycle of the Van der Pol oscillator shown in the diagram. In Mathematics, in the area of Dynamical systems, a limit-cycle on a plane or a Two-dimensional manifold In dynamics, the Van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a type of nonconservative Oscillator with nonlinear

Phase portrait of the Van der Pol oscillator

Here, the horizontal axis gives the position and vertical axis the velocity. In dynamics, the Van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a type of nonconservative Oscillator with nonlinear As the system evolves, its state follows one of the lines (trajectories) on the phase diagram.

Classic examples of phase diagrams from chaos theory are the Lorenz attractor and Mandelbrot set. The Lorenz attractor, named for Edward N Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its Butterfly In Mathematics, the Mandelbrot set, named after Benoît

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). Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product an associative non-commutative 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). 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 In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a With José Enrique Moyal (1949), these completed the foundations of phase-space quantization, a logically autonomous reformulation of quantum mechanics. José Enrique Moyal (also found as "Jo" or "Joe" Moyal (b In Mathematics and Physics, in the area of Quantum mechanics, Weyl quantization is a method for associating a "quantum mechanical" Hermitian Its modern abstractions include deformation quantization and geometric quantization. In Mathematics and Physics, in the area of Quantum mechanics, Weyl quantization is a method for associating a "quantum mechanical" Hermitian In Mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given Classical theory.

Thermodynamics and statistical mechanics

In thermodynamics and statistical mechanics contexts, the term phase space has two meanings: It is used in the same sense as in classical mechanics. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " 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. The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms 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. In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another Liquid is one of the principal States of matter. A liquid is a Fluid that has the particles loose and can freely form a distinct surface at the boundaries of A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume.

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.

To explain why there are 6N dimensions: Start out with a single particle. How many numbers does it take to specify the state of the particle? In three dimensions it takes six numbers: the 3 components of position and the 3 components of momentum (or equivalently velocity). It is important that you remember to tell how fast the particle is going as well as where it is to completely specify the state of the particle. Now imagine a six dimensional space where each point in this space is a list of six numbers. The first 3 numbers are components of the position of the particle and the second three numbers are the components of the momentum of the particle. If you now jump to N particles then phase becomes much larger. It now has 6N dimensions, 6 dimensions for each of the N particles. A single point in phase space is a list of 6N numbers that tells you the state (position and momentum) of all N particles in your system.

See also