Phase space volume is the volume of a region of possible states in the phase space of a dynamical system. In Euclidean coordinates it is ordinary Lebesgue measure. A region transported by a smooth flow has volume

where is the matrix of first derivatives of the flow and its determinant is the Jacobian. If the generating vector field has constant divergence , then . Thus negative divergence contracts phase space volume, as in the Chen-Lee system. The flow of a Hamiltonian system preserves phase space volume in canonical coordinates by Liouville's phase space theorem (Arnold 1989).

See also

Divergence, Liouville's Phase Space Theorem, Phase Space

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References

Arnold, V. I. Mathematical Methods of Classical Mechanics, 2nd ed. New York: Springer-Verlag, 1989.

Cite this as:

Weisstein, Eric W. "Phase Space Volume." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PhaseSpaceVolume.html

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