Phase space

In low dimensions

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 ordinary differential equation in a single variable, ${\displaystyle dy/dt=f(y),}$ with the resulting one-dimensional system being called a 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).

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.

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.

Related concepts

Phase plot

A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. However the latter expression, "phase diagram", 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.

Phase portrait

In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve.

Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological 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 stable steady states (with dots) and unstable steady states (with circles) in a phase space. The axes are of state variables.

Phase integral

In classical statistical mechanics (continuous energies) the concept of phase space provides a classical analog to the partition function (sum over states) known as the phase integral. Instead of summing the Boltzmann factor over discretely spaced energy states (defined by appropriate integer quantum 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 of quantum energy states per unit phase space. This normalization constant is simply the inverse of the Planck constant raised to a power equal to the number of degrees of freedom for the system.

Applications

Chaos theory

Classic examples of phase diagrams from chaos theory are:

• the Lorenz attractor
• population growth (i.e. logistic map)
• parameter plane of complex quadratic polynomials with Mandelbrot set.

Quantum mechanics

In quantum mechanics, the coordinates p and q of phase space normally become Hermitian operators in a Hilbert space.

But they may alternatively retain their classical interpretation, provided functions of them compose in novel algebraic ways (through Groenewold's 1946 star product). This is consistent with the uncertainty principle of quantum mechanics. Every quantum mechanical observable corresponds to a unique function or distribution on phase space, and conversely, 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 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. (Its modern abstractions include deformation quantization and geometric 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 the Wigner quasi-probability distribution effectively serving as a measure.

Thus, by expressing quantum mechanics in phase space (the same ambit as for classical mechanics), the Weyl map facilitates recognition of quantum mechanics as a deformation (generalization) of classical mechanics, with deformation parameter ħ/S, where S is the action of the relevant process. (Other familiar deformations in physics involve the deformation of classical Newtonian into relativistic mechanics, with deformation parameter v/c; or the deformation of Newtonian gravity into general relativity, with deformation parameter Schwarzschild radius/characteristic dimension.)

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.

Thermodynamics and statistical mechanics

In thermodynamics and statistical 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 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 3 position variables and 3 momentum variables. In this sense, as long as the particles are distinguishable, a point in phase space is said to be a microstate of the system. (For indistinguishable particles a microstate consists of a set of N! points, corresponding to all possible exchanges of the N particles.) N is typically on the order of the Avogadro 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 the macroscopic states of the system, such as pressure, temperature, etc. For instance, one may view the pressure–volume diagram or temperature–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, 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 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.

Optics

Phase space is extensively used in nonimaging optics, the branch of optics devoted to illumination. It is also an important concept in Hamiltonian optics.

Medicine

In medicine and bioengineering, the phase space method is used to visualize multidimensional physiological responses.