- Liouville's theorem (Hamiltonian)
In

physics ,**Liouville's theorem**, named after the French mathematicianJoseph Liouville , is a key theorem in classical statistical andHamiltonian mechanics . It asserts that the phase-space distribution function is constant along the trajectories of the system - that is that the density of system points in the vicinity of a given system point travelling through phase-space is constant with time.There are also related mathematical results in

symplectic topology andergodic theory .**Liouville equation**The Liouville equation describes the time evolution of phase space

distribution function (while measure is the correct term frommathematics , physicists generally call it a distribution). Consider a dynamical system withcanonical coordinates $q^\; i$ andconjugate momenta $p\_i$, where $i=1,dots,d$. Then the phase space distribution $ho(p,q)$ determines the probability $ho(p,q),d^dq,d^dp$ that a particle will be found in the infinitesimal phase space volume $d^dq,d^dp$. The**Liouville equation**governs the evolution of $ho(p,q;t)$ in time $t$::$frac\{d\; ho\}\{dt\}=frac\{partial\; ho\}\{partial\; t\}+sum\_\{i=1\}^dleft(frac\{partial\; ho\}\{partial\; q^i\}dot\{q\}^i+frac\{partial\; ho\}\{partial\; p\_i\}dot\{p\}\_i\; ight)=0.$

Time derivatives are denoted by dots, and are evaluated according to

Hamilton's equations for the system. This equation demonstrates the conservation of density in phase space (which was Gibbs's name for the theorem). Liouville's theorem states that:"The distribution function is constant along any trajectory in phase space."

A simple proof of the theorem is to observe that the evolution of $ho$ is "defined" by the

continuity equation ::$frac\{partial\; ho\}\{partial\; t\}+sum\_\{i=1\}^dleft(frac\{partial(\; hodot\{q\}^i)\}\{partial\; q^i\}+frac\{partial(\; hodot\{p\}\_i)\}\{partial\; p\_i\}\; ight)=0.$

That is, the tuplet $(\; ho,\; hodot\{q\}^i,\; hodot\{p\}\_i)$ is a

conserved current . Notice that the difference between this and Liouville's equation are the terms:$hosum\_\{i=1\}^dleft(frac\{partialdot\{q\}^i\}\{partial\; q^i\}+frac\{partialdot\{p\}\_i\}\{partial\; p\_i\}\; ight)=\; hosum\_\{i=1\}^dleft(frac\{partial^2\; H\}\{partial\; q^i,partial\; p\_i\}-frac\{partial^2\; H\}\{partial\; p\_i\; partial\; q^i\}\; ight)=0,$

where $H$ is the Hamiltonian, and Hamilton's equations have been used. That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem that the

convective derivative of the density $d\; ho/dt$ is zero follows from the equation of continuity by noting that the 'velocity field' $(dot\; p\; ,\; dot\; q)$ in phase space has zero divergence (which follows from Hamilton's relations).Another illustration is to consider the trajectory of a cloud of points through phase space. It is straightforward to show that as the cloud stretches in one coordinate – $p\_i$ say – it shrinks in the corresponding $q^i$ direction so that the product $Delta\; p\_i\; Delta\; q^i$ remains constant.

Equivalently, the existence of a conserved current implies, via

Noether's theorem , the existence of asymmetry . The symmetry is invariance under time translations, and the generator (orNoether charge ) of the symmetry is the Hamiltonian.**Physical interpretation**The expected total number of particles is the integral over phase space of the distribution::$N=int\; d^dq,d^dp,\; ho(p,q).$A normalizing factor is conventionally included in the phase space measure but has here been omitted. In the simple case of a

nonrelativistic particle moving inEuclidean space under aforce field $mathbf\{F\}$ with coordinates $mathbf\{x\}$ and momenta $mathbf\{p\}$, Liouville's theorem can be written:$frac\{partial\; ho\}\{partial\; t\}+frac\{mathbf\{p\{m\}cdot\; abla\_mathbf\{x\}\; ho+mathbf\{F\}cdot\; abla\_mathbf\{p\}\; ho=0.$This is different from theVlasov equation , or sometimes theCollisionless Boltzmann Equation , inastrophysics . The latter, which has a 6-D phase space, is used to describe the evolution of a large number of collisionless particles moving under the influence ofgravity and/orelectromagnetic field .In classical

statistical mechanics , the number of particles $N$ is very large, (typically of orderAvogadro's number , for a laboratory-scale system). Setting $partial\; ho/partial\; t=0$ gives an equation for the stationary states of the system and can be used to find the density ofmicrostates accessible in a givenstatistical ensemble . Thestationary states equation is satisfied by $ho$ equal to any function of the Hamiltonian $H$: in particular, it is satisfied by theMaxwell-Boltzmann distribution $hopropto\; e^\{-H/kT\}$, where $T$ is thetemperature and $k$ theBoltzmann constant .See also

canonical ensemble andmicrocanonical ensemble **Other formulations****Poisson bracket**The theorem is often restated in terms of the

Poisson bracket as:$frac\{partial\; ho\}\{partial\; t\}=-\{,\; ho,H,\}$or in terms of the**Liouville operator**or**Liouvillian**,:$hat\{mathbf\{L=sum\_\{i=1\}^\{d\}left\; [frac\{partial\; H\}\{partial\; p\_\{ifrac\{partial\}\{partial\; q^\{i-frac\{partial\; H\}\{partial\; q^\{ifrac\{partial\; \}\{partial\; p\_\{i\; ight]\; ,$as:$frac\{partial\; ho\; \}\{partial\; t\}+\{hat\{L\; ho\; =0.$**Ergodic theory**In

ergodic theory anddynamical systems , motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. InHamiltonian mechanics , the phase space is a smooth manifold that comes naturally equipped with a smooth measure (locally, this measure is the 6"n"-dimensionalLebesgue measure ). The theorem says this smooth measure is invariant under theHamiltonian flow . More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow. The Hamiltonian case then becomes a corollary.**Symplectic geometry**In terms of

symplectic geometry , the theorem states that the d-power of the symplectic structure (2-form , formed by summation of thewedge product s of $Delta\; p\_i$ and $Delta\; q\_i$) has a zeroLie derivative for its Hamiltonian evolution. (The d-power of the symplectic structure is just the measure on the phase space described above.)In fact, the symplectic structure itself is preserved (not only the d-power). For this reason, in this context, symplectic structure is also called Poincaré invariant. Hence the theorem about Poincaré invariant is a generalization of the Liouville's theorem.

Further generalization is also possible. In the frame of [

*http://daarb.narod.ru/wircq-eng.html#ihf invariant Hamiltonian formalism*] , the theorem about existence of symplectic structure on invariant phase space turns out to be a deep generalization of the theorem about Poincaré invariant.**Quantum mechanics**The analog of Liouville equation in

quantum mechanics describes the time evolution of amixed state .Canonical quantization yields a quantum-mechanical version of this theorem. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced bycommutator s. In this case, the resulting equation is:$frac\{partial\}\{partial\; t\}\; ho=-frac\{i\}\{hbar\}\; [H,\; ho]$where ρ is thedensity matrix .When applied to the

expectation value of anobservable , the corresponding equation is given byEhrenfest's theorem , and takes the form:$frac\{d\}\{dt\}langle\; A\; angle\; =\; frac\{i\}\{hbar\}langle\; [H,A]\; angle$

where $A$ is an observable. Note the sign difference, which follows from the assumption that the operator is stationary and the state is time-dependent.

**Remarks***The Liouville equation is valid for both equilibrium and nonequilibrium systems. It is a fundamental equation of nonequilibrium statistical mechanics. Its approximation to collisional systems is called the

Boltzmann equation .

*The Liouville equation is integral to the proof of thefluctuation theorem from which thesecond law of thermodynamics can be derived. It is also the key component of the derivation ofGreen-Kubo relations for linear transport coefficients such as shearviscosity ,thermal conductivity orelectrical conductivity .

* Virtually any textbook onHamiltonian mechanics , advancedstatistical mechanics , orsymplectic geometry will derive the Liouville theorem.**ee also***

reversible reference system propagation algorithm (r-RESPA)

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