phase space

English

Alternative forms

• phase-space

Etymology

The concept was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré and Josiah Willard Gibbs.

Noun

phase space (countable and uncountable, plural phase spaces)

1. (mathematics, physics, dynamical system theory) Given a (dynamical) system, any topological space such that every point in the space's underlying set uniquely represents a state of the system and every possible state is represented by some point;
   (less formally) the underlying set of such a topological space.

   For mechanical systems, the phase space usually consists of all possible values of position and momentum variables.

   • 1991, Jan J. Slawianowski, Geometry of Phase Spaces, Wiley, page 485:

     The phase space of a single material point is a manifold M×V*, and the Newton state-space is identifical^[sic] with M×V.

   • 2006, Matthew Charles Cargo, Phase Space Approaches to Higher Order WKB Approximations, University of California, Berkeley, page 8,

     Their energies are those of the level sets $p^{2}/2+V(x)=E$ that enclose an area of $2\pi \hbar (n+1/2)$ in the classical phase space. Despite embracing a configuration space point of view, these methods have produced a phase space result.

   • 2007, Helmut Wiedemann, Particle Accelerator Physics, Springer, page 377:

     The distribution of particles in phase space is given either by the injector characteristics and injection process or in the case of electron beams by the equilibrium of quantum excitation due to synchrotron radiation and damping.

Hyponyms

• phase line
• phase plane

Derived terms

• phase space analysis
• phase space trajectory

Related terms

• configuration space
• parameter space
• state space
• phase diagram
• phase integral
• phase portrait

Translations

topological space of states of a system

• Catalan: espai de fases m, espai fàsic m
• Chinese:

  Mandarin: 相空間／相空间 (zh) (xiàngkōngjiān)

• Czech: fázový prostor m
• Danish: faserum n
• Dutch: faseruimte (nl) f
• Finnish: faasiavaruus
• French: espace des phases m
• German: Phasenraum m
• Indonesian: ruang fase
• Italian: spazio delle fasi m
• Japanese: 位相空間 (ja) (いそう くうかん, isō kūkan)
• Korean: 위상공간(位相空間) (wisanggonggan)
• Polish: przestrzeń fazowa f
• Portuguese: espaço de fases m (Brazil), espaço de fase m (Portugal), espaço fásico m
• Romanian: spațiu de fază n
• Russian: фа́зовое простра́нство (ru) n (fázovoje prostránstvo)
• Spanish: espacio fásico m, espacio de fases m
• Swedish: fasrum n
• Ukrainian: фа́зовий про́стір m (fázovyj próstir)

Further reading

• Phase portrait on Wikipedia.Wikipedia
• Phase space method on Wikipedia.Wikipedia
• Parameter space on Wikipedia.Wikipedia
• Separatrix (mathematics) on Wikipedia.Wikipedia