Liouville's theorem (Hamiltonian)

From Academic Kids

A separate article is about Liouville's theorem in complex analysis: see Liouville's theorem (complex analysis).

In mathematical 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..

Liouville's theorem is also important in the study of symplectic topology, where it is formulated rather differently.


Liouville equation

The Liouville equation describes the time evolution of phase space distribution function (while density is the correct term from mathematics, physicists generally call it a distribution). Consider a dynamical system with coordinates <math>q_i<math> and conjugate momenta <math>p_i<math>, where <math>i=1,\dots,d<math>. Then the phase space distribution <math>\rho(p,q)<math> determines the probability <math>\rho(p,q)\,d^dq\,d^dp<math> that a particle will be found in the infinitesimal phase space volume <math>d^dq\,d^dp<math>. The Liouville equation governs the evolution of <math>\rho(p,q;t)<math> in time <math>t<math>:

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

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 <math>\rho<math> is defined by the continuity equation:

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

and notice that the difference between this and Liouville's equation are the terms

<math>\rho\sum_{i=1}^d\left(\frac{\partial\dot{q}_i}{\partial q_i}+\frac{\partial\dot{p}_i}{\partial p_i}\right)=\rho\sum_{i=1}^d\left(\frac{\partial H}{\partial q_i\,\partial p_i}-\frac{\partial H}{\partial p_i \partial q_i}\right)=0,<math>

where <math>H<math> 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 <math>d \rho/dt<math> is zero follows from the equation of continuity by noting that the 'velocity field' <math>(\dot p , \dot q)<math> 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 – <math>p_i<math> say – it shrinks in the corresponding <math>q_i <math> direction so that the product <math>\Delta p_i \Delta q_i <math> remains constant.

Physical interpretation

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

<math>N=\int d^dq\,d^dp\,\rho(p,q)<math>

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

<math>\frac{\partial\rho}{\partial t}+\mathbf{v}\cdot\nabla_\mathbf{x}\rho+\frac{\mathbf{F}}{m}\cdot\nabla_\mathbf{p}\rho=0,<math>

where <math>\mathbf{v}=\dot{\mathbf{x}}<math> is the velocity. In astrophysics this is called the Vlasov equation, and is used to describe the evolution of a large number of collisionless particles moving in a gravitational potential.

In classical statistical mechanics, the number of particles <math>N<math> is very large, (typically of order Avogadro's number, for a laboratory-scale system). Setting <math>\partial\rho/\partial t=0<math> 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 <math>\rho<math> equal to any function of the Hamiltonian <math>H<math>: in particular, it is satisfied by the Maxwell-Boltzmann distribution <math>\rho\propto e^{-H/kT}<math>, where <math>T<math> is the temperature and <math>k<math> the Boltzmann constant.

See also canonical ensemble and microcanonical ensemble

Other formulations

The theorem is often restated in terms of the Poisson bracket:

<math>\frac{\partial\rho}{\partial t}=-\{\,\rho,H\,\}<math>

or the Liouville operator or Liouvillian,

<math>\hat{\mathbf{L}}=\sum_{i=1}^{d}\left[\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial q_{i}}-\frac{\partial H}{\partial q_{i}}\frac{\partial }{\partial p_{i}}\right],<math>


<math>\frac{\partial \rho }{\partial t}+{\hat{L}}\rho =0.<math>

Another way to formulate Liouville's theorem is to say that a phase-space volume <math>\Gamma<math> is conserved under time translation. If

<math>\int_\Gamma d^dq\,d^dp = C,<math>

and <math>\Gamma(t)<math> is the set of points in phase-space which the points of <math>\Gamma<math> can evolve into at time <math>t<math>, then

<math>\int_{\Gamma(t)} d^dq\,d^dp = C,<math>

for all times <math>t<math>. That is, phase space volumes are conserved. Since time-evolution in Hamiltonian mechanics is a canonical transformation this can be proved by showing that all canonical transformations have unit Jacobian.

In terms of a symplectic geometry the theorem states that the 2-form S, formed from the wedge product of <math>\Delta p_i<math> and <math>\Delta q_i<math> has a Lie derivative for its Hamiltonian evolution (given by the Poisson bracket with respect to the vector field {H, } which vanishes).

See also

  • 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 by commutators. In this case, the resulting equation is
<math>\frac{\partial}{\partial t}\rho=-\frac{i}{\hbar}[\rho,H]<math>
where ρ is the density matrix.



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