Liouville's theorem (Hamiltonian)

Encyclopedia

In physics

,

, 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

.

. 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 dynamical system

with canonical coordinates

and conjugate momenta , where . Then the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume . The

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

A simple proof of the theorem is to observe that the evolution of is

:

That is, the tuplet is a conserved current

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

where 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 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 – say – it shrinks in the corresponding direction so that the product 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.

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

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

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 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 equal to any function of the Hamiltonian : in particular, it is satisfied by the Maxwell-Boltzmann distribution , where is the temperature

and the Boltzmann constant.

See also canonical ensemble

and microcanonical ensemble

.

as

or in terms of the

as

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 6

). 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.

. The theorem then states that the natural volume form

on a symplectic manifold is invariant under the Hamiltonian flows. The symplectic structure is represented as a 2-form, given as a sum of wedge products of d

of this volume form is zero along every Hamiltonian vector field.

In fact, the symplectic structure itself is preserved, not only its top exterior power. For this reason, in this context, the 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.

describes the time evolution of a mixed state. Canonical quantization

yields a quantum-mechanical version of this theorem, the Von Neumann equation. 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 commutator

s. In this case, the resulting equation is

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

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

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

,

**Liouville's theorem**, named after the French mathematician Joseph LiouvilleJoseph Liouville

- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...

, is a key theorem in classical statistical

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

and Hamiltonian mechanics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

. It asserts that the phase-space

Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in 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

Symplectic topology

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...

and ergodic theory

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

.

## Liouville equation

The Liouville equation describes the time evolution of the phase space distribution functionDistribution function

In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...

. 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 dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

with canonical coordinates

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics...

and conjugate momenta , where . Then the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume . The

**Liouville equation**governs the evolution of in time :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 is

*defined*by the continuity equationContinuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

:

That is, the tuplet is a conserved current

Conserved current

In physics a conserved current is a current, j^\mu, that satisfies the continuity equation \partial_\mu j^\mu=0. The continuity equation represents a conservation law, hence the name....

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

where 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

Convective derivative

The material derivative is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and classical mechanics...

of the density 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 – say – it shrinks in the corresponding direction so that the product remains constant.

Equivalently, the existence of a conserved current implies, via Noether's theorem

Noether's theorem

Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

, the existence of a symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

. 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: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

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

under a force field

Force field

A force field, sometimes known as an energy shield, force shield, or deflector shield is a concept of a field tightly bounded and of significant magnitude so that objects affected by the particular force relating to the field are unable to pass through the central axis of the field and reach the...

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

This is similar to the Vlasov equation

Vlasov equation

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction...

, or the collisionless Boltzmann equation, in astrophysics

Astrophysics

Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...

. 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

Electromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

.

In classical statistical mechanics

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, the number of particles is very large, (typically of order Avogadro's number

Avogadro's number

In chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e...

, 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

Microstate (statistical mechanics)

In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...

accessible in a given statistical ensemble. The stationary states equation is satisfied by equal to any function of the Hamiltonian : in particular, it is satisfied by the Maxwell-Boltzmann distribution , where is the temperature

Temperature

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

and the Boltzmann constant.

See also canonical ensemble

Canonical ensemble

The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...

and microcanonical ensemble

Microcanonical ensemble

In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...

.

### Poisson bracket

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

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...

as

or in terms of the

**Liouville operator**or**Liouvillian**,as

### Ergodic theory

In ergodic theoryErgodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

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

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

, the phase space is a smooth manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

that comes naturally equipped with a smooth measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

(locally, this measure is the 6

*n*-dimensional Lebesgue measureLebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

). 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 phase space is represented as a symplectic manifoldSymplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

. The theorem then states that the natural volume form

Volume form

In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...

on a symplectic manifold is invariant under the Hamiltonian flows. The symplectic structure is represented as a 2-form, given as a sum of wedge products of d

*p*_{i}with d*q*^{i}. The volume form is the top exterior power of the symplectic 2-form, and is just another representation of the measure on the phase space described above. One formulation of the theorem states that the Lie derivativeLie derivative

In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

of this volume form is zero along every Hamiltonian vector field.

In fact, the symplectic structure itself is preserved, not only its top exterior power. For this reason, in this context, the 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 Liouville equation

The analog of Liouville equation in quantum mechanicsQuantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

describes the time evolution of a mixed state. Canonical quantization

Canonical quantization

In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...

yields a quantum-mechanical version of this theorem, the Von Neumann equation. 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 commutator

Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

s. In this case, the resulting equation is

where ρ is the density matrix

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

.

When applied to the expectation value of an observable

Observable

In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

, the corresponding equation is given by Ehrenfest's theorem, and takes the form

where 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 application to dilute gases is called the Boltzmann equationBoltzmann equationThe Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...

. - The Liouville equation is integral to the proof of the fluctuation theoremFluctuation theoremThe fluctuation theorem , which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium will increase or decrease over a given amount of time...

from which the second law of thermodynamicsSecond law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

can be derived. It is also the key component of the derivation of Green-Kubo relationsGreen-Kubo relationsThe Green–Kubo relations give the exact mathematical expression for transport coefficients in terms of integrals of time correlation functions.-Thermal and mechanical transport processes:...

for linear transport coefficients such as shear viscosityViscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

, thermal conductivityThermal conductivityIn physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

or electrical conductivity. - Virtually any textbook on Hamiltonian mechanicsHamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

, advanced statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, or symplectic geometry will derive the Liouville theorem.