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Are there any other important practical and theoretical consequences of Liouville's theorem on the conservation of phase space volume besides the calculation of the microcanonical potential in Thermodynamics?

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4 Answers 4

1) On a symplectic manifold $(M,\omega)$, Liouville's theorem is often stated as that every Hamiltonian vector field $X_f=\{f,\cdot\}$ is divergence-free

$$ {\rm div}_{\rho} X_f~=~0 ,$$

where the volume density $\rho$ comes from the canonical volume form

$$\Omega~=~\rho dx^1 \wedge \ldots \wedge dx^{2n}.$$

Here the canonical volume form

$$\Omega~:=~\omega^{\wedge n}$$

is top exterior power of the symplectic 2-form $\omega$.

Equivalently, the Lie derivative of the canonical volume form

$$ {\cal L}_{X_f}\Omega~=~0, $$

wrt. a Hamiltonian vector field, vanishes.

2) It is interesting to note that Liouville's theorem generalizes to symplectic supermanifolds $(M,\omega)$ with Grassmann-even symplectic structure.

3) However Liouville's theorem fails for Grassmann-odd symplectic manifolds, also known as antisymplectic manifolds, or Batalin-Vilkovisky (BV) manifolds. Such manifolds do not have a canonical volume density $\rho$ either. Nevertheless, let us assume that there at least exists some volume density $\rho$. Then the failure of Hamiltonian vector fields to be divergence-free is measured by the odd Laplacian

$$ \Delta_{\rho} f ~=~\frac{(-1)^{|f|}}{2}{\rm div}_{\rho} X_f, $$

where $|f|$ denotes the Grassmann parity of the function $f$.

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Conservation of étendue is essentially the same thing as Liouville's theorem applied to the space of rays of light in geometric optics. This is central to non-imaging optics, for example in the design of car headlamps or in concentrating sunlight in photovoltaic cells.

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Although I know little about this, other applications I have seen are in beam physics where one can make arguments about beam emittance. In particular one can argue that emittance remains constant in certain situations, using Liouville's theorem. See e.g. this article.

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This theorem is concerned with defining two parts: one is conservation of density in phase space and other is conservation of extension in phase space.

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