Liouville's theorem (physics)

The set of Liouville (also Liouville theorem , named after Joseph Liouville ) belongs to the realm of theoretical mechanics. The theorem states that the (multidimensional) volume enclosed by neighboring trajectories in phase space is constant as a function of time. The theorem applies to all systems described by the Hamilton formalism , the Hamilton function can also explicitly depend on time.

The Liouville equation is closely related to and easily derived from Liouville's theorem .

Derivation

In the context of the Hamilton formalism , the instantaneous state of a mechanical system is given by canonical coordinates and canonical impulses with , i. H. through a point in phase space . The Hamilton function of the system defines a vector field in phase space, which describes the time evolution of the system. The solution to the equation is the flow . Applying the divergence operator to the vector field yields ${\displaystyle q_{i}}$ ${\displaystyle p_{i}}$ ${\displaystyle 1\leq i\leq N}$ ${\displaystyle X=(q_{i},p_{i})}$ ${\displaystyle H\left(q,p,t\right)}$ ${\displaystyle {\dot {X}}=({\frac {dq_{i}}{dt}},{\frac {dp_{i}}{dt}})=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q_{i}}}\right)}$ ${\displaystyle X\left(t\right)}$ ${\displaystyle \operatorname {div} =\left({\frac {\partial }{\partial q_{i}}},{\frac {\partial }{\partial p_{i}}}\right)}$

${\displaystyle \operatorname {div} \cdot {\dot {X}}=\sum _{i=1}^{N}\left({\frac {\partial ^{2}H}{\partial p_{i}\partial q_{i}}}-{\frac {\partial ^{2}H}{\partial q_{i}\partial p_{i}}}\right)=0.}$

The velocity field and the flow are therefore source-free. The dynamics in phase space can therefore be visualized as the flow of an incompressible liquid (that the disappearance of the divergence of a velocity field implies incompressibility can be shown formally with the help of Gauss theorem , similar to hydrodynamics). ${\displaystyle {\dot {X}}}$ ${\displaystyle X\left(t\right)}$

On a more mathematical level, Liouville's theorem is a consequence of the invariance of the -form under symplectic transformations , i.e. also under canonical transformations and in time development. Thus the -form and its powers are also invariant, is the phase space volume. ${\displaystyle 1}$ ${\displaystyle \omega ^{\left(1\right)}=\sum p_{i}dq_{i}}$ ${\displaystyle 2}$ ${\displaystyle \omega =d\omega ^{\left(1\right)}=\sum dp_{i}\wedge dq_{i}}$ ${\displaystyle \omega ^{N}}$

Applications

Liouville's theorem plays a fundamental role in statistical mechanics, but there are also simpler applications.

One application relates to the transverse expansion of particle or light beams, for example in particle accelerators or optical instruments. One can focus a beam with lens systems, i. H. reduce its transversal extent. But this is only possible at the expense of the transversal impulses. The expansion of the transverse impulses must increase so that the total volume, i.e. H. the product of the expansion in the spatial and momentum space, remains constant. The name for the corresponding phase space volume in geometric optics is Etendue .

Another example is a monatomic ideal gas with volume and temperature . With adiabatic compression, the product remains constant. Since the temperature is proportional to the momentum square of the particles, the constant corresponds to the phase space volume of a particle (in an ideal gas the particles are independent of each other). ${\displaystyle V}$ ${\displaystyle T}$ ${\displaystyle VT^{3/2}}$

In a polyatomic gas, degrees of freedom of translation, rotation and vibration are not relevant for a gas molecule as a whole , but degrees of freedom. The momentum component of the phase space of a molecule is therefore the product of momentum. With adiabatic compression is therefore constant (the rotation and vibration coordinates themselves are cyclical and are not affected by the compression). ${\displaystyle 3}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle VT^{f/2}}$

literature

Franz Schwabl : Statistical Mechanics . Springer 2006, ISBN 978-3-540-31095-2

Theodore Frankel: The Geometry of Physics . Cambridge University Press 2012, ISBN 978-1-107-60260-1

See also

• Liouville's theorem (function theory)
• Liouville's theorem (differential geometry)