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.. Mathematical physics is a scientific discipline aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... A phase diagram or phase space is a useful construct used in mathematics and physics to demonstrate and visualise the changes in a given system. ...

Liouville's theorem is also important in the study of symplectic topology, where it is formulated rather differently. In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

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 q_i and conjugate momenta p_i, where . Then the phase space distribution ρ(p,q) determines the probability that a particle will be found in the infinitesimal phase space volume . The Liouville equation governs the evolution of ρ(p,q;t) in time t: Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... Mathematics is the study of quantity, structure, space and change. ...

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 In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ... Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American physical chemist. ...

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: Note that all the examples given below express the same idea (i. ...

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

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 dρ / dt is zero follows from the equation of continuity by noting that the 'velocity field' 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 Δp_iΔq_i remains constant.

Physical interpretation

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

(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 with coordinates and momenta , Liouville's theorem can be written

where is the velocity. In astrophysics this is called the Vlasov equation (or sometimes the Collisionless Boltzmann Equation), and is used to describe the evolution of a large number of collisionless particles moving in a gravitational potential. Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ...

In classical statistical mechanics, the number of particles N is very large, (typically of order Avogadro's number, for a laboratory-scale system). Setting 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 , where T is the temperature and k the Boltzmann constant. Avogadros number, also called Avogadros constant (NA) is a large constant used in chemistry and physics. ... In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

See also canonical ensemble and microcanonical ensemble A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... The microcanonical ensemble is the simplest of the ensembles of statistical mechanics, in which many replicas of a system are assumed to be confined to a region of phase space of constant energy. ...

Other formulations

The theorem is often restated in terms of the Poisson bracket: In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...

or the Liouville operator or Liouvillian,

as

Another way to formulate Liouville's theorem is to say that a phase-space volume Γ is conserved under time translation. If

and Γ(t) is the set of points in phase-space which the points of Γ can evolve into at time t, then

for all times t. 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 mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

In terms of a symplectic geometry the theorem states that the 2-form S, formed from the wedge product of Δp_i and Δq_i has a Lie derivative for its Hamiltonian evolution (given by the Poisson bracket with respect to the vector field {H, } which vanishes). In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...

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

where ρ is the density matrix.

• 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.