Liouville's theorem (Hamiltonian)

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian 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 and ergodic theory.

Liouville equations

These Liouville equations describe the time evolution of the phase space distribution function. Although the equation is usually referred to as the "Liouville equation", this equation was in fact first published by Gibbs in 1902. It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838. Consider a Hamiltonian dynamical system with canonical coordinates $q_i$ and conjugate momenta $p_i$ , where $i=1,\dots,n$ . Then the phase space distribution $\rho(p,q)$ determines the probability $\rho(p,q)\,d^nq\,d^n p$ that the system will be found in the infinitesimal phase space volume $d^nq\,d^n p$ . The Liouville equation governs the evolution of $\rho(p,q;t)$ in time $t$ :

$\frac{d\rho}{dt}= \frac{\partial\rho}{\partial t} +\sum_{i=1}^n\left(\frac{\partial\rho}{\partial q_i}\dot{q}_i +\frac{\partial\rho}{\partial p_i}\dot{p}_i\right)=0.$

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 $\rho$ is defined by the continuity equation:

$\frac{\partial\rho}{\partial t}+\sum_{i=1}^n\left(\frac{\partial(\rho\dot{q}^i)}{\partial q^i}+\frac{\partial(\rho\dot{p}_i)}{\partial p_i}\right)=0.$

That is, the tuplet $(\rho, \rho\dot{q}^i,\rho\dot{p}_i)$ is a conserved current. Notice that the difference between this and Liouville's equation are the terms

$\rho\sum_{i=1}^n\left( \frac{\partial\dot{q}^i}{\partial q^i} +\frac{\partial\dot{p}_i}{\partial p_i}\right) =\rho\sum_{i=1}^n\left( \frac{\partial^2 H}{\partial q^i\,\partial p_i} -\frac{\partial^2 H}{\partial p_i \partial q^i}\right)=0,$

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, $d \rho/dt$ , 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 a symmetry. The symmetry is invariant under time translations, and the generator (or Noether charge) of the symmetry is the Hamiltonian.

Physical interpretation

The expected total number of particles is the integral over phase space of the distribution:

$N=\int \rho(p,q) \,d^n q\,d^n p.$

A normalizing factor is conventionally included in the phase space measure but has here been omitted. In the simple case of an onrelativistic particle moving in Euclidean space under a force field $\mathbf{F}$ with coordinates $\mathbf{x}$ and momenta $\mathbf{p}$ , Liouville's theorem can be written

$\frac{\partial\rho}{\partial t}+\frac{\mathbf{p}}{m}\cdot\nabla_\mathbf{x}\rho+\mathbf{F}\cdot\nabla_\mathbf{p}\rho=0.$

This is similar to the Vlasov equation, or the collisionless Boltzmann equation, in astrophysics. 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 of gravity and/or electromagnetic field.

In classical statistical mechanics, the number of particles $N$ is very large, (typically of order Avogadro's number, for a laboratory-scale system). Setting $\frac{ \partial \rho }{ \partial t} = 0$ gives an equation for the stationary states of the system and can be used to find the density of microstates accessible in a given statistical ensemble. The stationary states equation is satisfied by $\rho$ equal to any function of the Hamiltonian $H$ : in particular, it is satisfied by the Maxwell-Boltzmann distribution $\rho\propto e^{-H/kT}$ , where $T$ is the temperature and $k$ the Boltzmann constant.

See also canonical ensemble and microcanonical ensemble.

Black Holes

Friday, 21 June 2013

Liouville's theorem (Hamiltonian)

Liouville equations

Physical interpretation

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