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Liouville's_theorem_(Hamiltonian)

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.

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Liouville equation

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

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

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 $ρ$ is defined by the continuity equation:

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

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

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

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 $d ρ / d t$ 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 $Δ p i Δ q i$ remains constant.

Equivalently, the existence of a conserved current implies, via Noether's theorem, the existence of a symmetry. The symmetry is invariance 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:

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

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 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 different from the Vlasov equation, or sometimes 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 $\partial\rho/\partial t=0$ 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 $ρ$ 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.

Other formulations

Poisson bracket

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

$\frac{\partial\rho}{\partial t}=-\{\,\rho,H\,\}$

or in terms of the Liouville operator or Liouvillian,

$\hat{\mathbf{L}}=\sum_{i=1}^{d}\left[\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{\partial }{\partial p_{i}}\right],$

$\frac{\partial \rho }{\partial t}+{\hat{L}}\rho =0.$

Ergodic theory

In ergodic theory and dynamical systems, motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. In Hamiltonian mechanics, the phase space is a smooth manifold that comes naturally equipped with a smooth measure (locally, this measure is the 6n-dimensional Lebesgue measure). The theorem says this smooth measure is invariant under the Hamiltonian 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 the wedge products of $Δ p i$ and $Δ q i$ ) has a zero Lie 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 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 a mixed 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 by commutators. In this case, the resulting equation is

$\frac{\partial}{\partial t}\rho=-\frac{i}{\hbar}[H,\rho]$

where ρ is the density matrix.

When applied to the expectation value of an observable, the corresponding equation is given by Ehrenfest's theorem, and takes the form

$\frac{d}{dt}\langle A\rangle = \frac{i}{\hbar}\langle [H,A] \rangle$

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. Generalized to collisional systems it is called the Boltzmann equation.
The Liouville equation is integral to the proof of the fluctuation theorem from which the second law of thermodynamics can be derived. It is also the key component of the derivation of Green-Kubo relations for linear transport coefficients such as shear viscosity, thermal conductivity or electrical conductivity.
Virtually any textbook on Hamiltonian mechanics, advanced statistical mechanics, or symplectic geometry will derive the Liouville theorem.