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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, Liouville's theorem, named after the French mathematician
Joseph Liouville Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...
, is a key theorem in classical
statistical Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
and
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
. 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 traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical
a priori probability A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
. Liouville's theorem applies to
conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink o ...
s, that is, systems in which the effects of
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system. Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general setting for conjugate position and momentum coordinates is available in the mathematical setting of
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
. Liouville's theorem ignores the possibility of
chemical reaction A chemical reaction is a process that leads to the chemistry, chemical transformation of one set of chemical substances to another. When chemical reactions occur, the atoms are rearranged and the reaction is accompanied by an Gibbs free energy, ...
s, where the total number of particles may change over time, or where energy may be transferred to internal degrees of freedom. The non-squeezing theorem, which applies to all symplectic maps (the Hamiltonian is a symplectic map) implies further restrictions on phase-space flows beyond volume/density/measure conservation. There are extensions of Liouville's theorem to cover these various generalized settings, including stochastic systems.


Liouville equation

The Liouville equation describes the time evolution of the ''phase space distribution function''. Although the equation is usually referred to as the "Liouville equation",
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynami ...
was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics. 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 In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
q_i and conjugate momenta p_i, where i=1,\dots,n. Then the phase space distribution \rho(p,q,t) determines the probability \rho(p,q,t)\; \mathrm^nq\,\mathrm^n p that the system will be found in the infinitesimal phase space volume \mathrm ^nq\,\mathrm^n p at time t. The Liouville equation is \frac + \sum_^n \left(\frac \dot_i +\frac \dot_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: A proof of Liouville's theorem uses the ''n''-dimensional divergence theorem. The proof is based on the fact that the evolution of \rho obeys an ''2n''-dimensional version of the
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity ...
: \frac + \vec \nabla \cdot (\rho \vec u) = 0 with \vec u = (\dot_1, \dot_2, \dots, \dot_n, \dot_1,\dot_2,...,\dot_n ) being the "velocity" vector of p_i and q_i. The above equation means that change of the total probability within a small volume in phase space is equal to the net flux of probability density into or out of the volume. After inserting \vec u in the above equation, we reach \frac+\sum_^n\left(\frac+\frac\right)=0. That is, the 3-tuple (\rho, \rho\dot_i,\rho\dot_i) is a
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. Indeed, integrating the continuity equation over a volume V, ...
. The above equation can be reduced to the Liouville equation based on the following identity \rho\sum_^n \left(\frac + \frac\right) = \rho\sum_^n \left(\frac - \frac\right)=0, where H is the Hamiltonian, and we have used the relationships \dot_i = \partial H / \partial p_i and \dot_i = - \partial H / \partial. The derivation of the Liouville equation can be viewed as 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).


Other formulations


Poisson bracket

The theorem above is often restated in terms of the
Poisson 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. Th ...
as \frac = \ or, in terms of the linear Liouville operator or Liouvillian, \mathrm\widehat=\sum_^n \left frac\frac-\frac\frac\right-\ as \frac+\rho =0.


Ergodic theory

In
ergodic theory Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
and
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. In
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
, the phase space is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
that comes naturally equipped with a smooth measure (locally, this measure is the 6''n''-dimensional
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
). 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

We can also formulate Liouville's Theorem in terms of
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
. For a given system, we can consider the phase space (q^\mu, p_\mu) of a particular Hamiltonian H as a manifold (M,\omega) endowed with a symplectic 2-form \omega = dp_\mu\wedge dq^\mu. The
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...
of our manifold is the top
exterior power In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
of the symplectic 2-form, and is just another representation of the measure on the phase space described above. On our phase space
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
we can define a
Hamiltonian vector field Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
generated by a function f(q,p) as X_f = \frac\frac - \frac\frac. Specifically, when the
generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
is the Hamiltonian itself, f(q,p) = H, we get X_H = \frac\frac - \frac\frac = \frac\frac + \frac\frac = \frac where we utilized Hamilton's equations of motion and the definition of the chain rule. In this formalism, Liouville's Theorem states that the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of the volume form is zero along the flow generated by X_H. That is, for (M,\omega) a 2n-dimensional symplectic manifold, \mathcal_(\omega^n) = 0. In fact, the symplectic structure \omega itself is preserved, not only its top exterior power. That is, Liouville's Theorem also gives Proves Liouville's theorem using the language of modern differential geometry. \mathcal_(\omega) = 0.


Quantum Liouville equation

The analog of Liouville equation in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
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 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 The commutator of two elements, ...
s. In this case, the resulting equation is \frac = \frac , \rho where ρ is the
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...
. When applied to the
expectation value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected va ...
of an
observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
, the corresponding equation is given by Ehrenfest's theorem, and takes the form \frac\langle A\rangle = -\frac\langle
, A The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
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. In the
phase-space formulation The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
of quantum mechanics, substituting the
Moyal bracket In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a l ...
s for
Poisson 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. Th ...
s in the phase-space analog of the von Neumann equation results in compressibility of the probability fluid, and thus violations of Liouville's theorem incompressibility. This, then, leads to concomitant difficulties in defining meaningful quantum trajectories.


Examples


Simple harmonic oscillator phase-space volume

Consider an N-particle system in three dimensions, and focus on only the evolution of \mathrm\mathcal particles. Within phase space, these \mathrm\mathcal particles occupy an infinitesimal volume given by \mathrm\Gamma = \displaystyle\prod_^N d^3p_i d^3q_i. We want \frac to remain the same throughout time, so that \rho(\Gamma, t) is constant along the trajectories of the system. If we allow our particles to evolve by an infinitesimal time step \delta t, we see that each particle phase space location changes as \begin q_i' = q_i + \dot\delta t,\\ p_i' = p_i + \dot\delta t, \end where \dot and \dot denote \frac and \frac respectively, and we have only kept terms linear in \delta t. Extending this to our infinitesimal hypercube \mathrm\Gamma, the side lengths change as \begin dq_i' = dq_i + \tfrac dq_i \delta t,\\ ptdp_i' = dp_i + \tfrac dp_i \delta t. \end To find the new infinitesimal phase-space volume \mathrm\Gamma', we need the product of the above quantities. To first order in \delta t, we get the following: dq_i' dp_i' = dq_i dp_i \left + \left( \frac + \frac\right) \delta t\right So far, we have yet to make any specifications about our system. Let us now specialize to the case of N 3-dimensional isotropic harmonic oscillators. That is, each particle in our ensemble can be treated as a
simple harmonic oscillator In mechanics and physics, simple harmonic motion (sometimes abbreviated as ) is a special type of periodic function, periodic motion an object experiences by means of a restoring force whose magnitude is directly proportionality (mathematics), ...
. The Hamiltonian for this system is given by H = \sum_^\left(\fracp_i^2 + \fracq_i^2\right). By using Hamilton's equations with the above Hamiltonian we find that the term in parentheses above is identically zero, thus yielding dq_i' dp_i' = dq_i dp_i. From this we can find the infinitesimal volume of phase space: \mathrm\Gamma' = \prod_^N d^3q_i'd^3p_i' = \prod_^N d^3q_i d^3p_i = \mathrm\Gamma. Thus we have ultimately found that the infinitesimal phase-space volume is unchanged, yielding \rho(\Gamma', t + \delta t) = \frac = \frac = \rho(\Gamma, t), demonstrating that Liouville's theorem holds for this system. The question remains of how the phase-space volume actually evolves in time. Above we have shown that the total volume is conserved, but said nothing about what it looks like. For a single particle we can see that its trajectory in phase space is given by the ellipse of constant H. Explicitly, one can solve Hamilton's equations for the system and find \begin q_i(t) &= Q_i\cos + \frac\sin,\\ p_i(t) &= P_i\cos - m\omega Q_i\sin, \end where Q_i and P_i denote the initial position and momentum of the i-th particle. For a system of multiple particles, each one will have a phase-space trajectory that traces out an ellipse corresponding to the particle's energy. The frequency at which the ellipse is traced is given by the \omega in the Hamiltonian, independent of any differences in energy. As a result, a region of phase space will simply rotate about the point (\mathbf, \mathbf) = (0, 0) with frequency dependent on \omega. This can be seen in the animation above.


Damped harmonic oscillator

To see an example where Liouville's theorem does ''not'' apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping. Consider again the system of N particles each in a 3-dimensional isotropic harmonic potential, the Hamiltonian for which is given in the previous example. This time, we add the condition that each particle experiences a frictional force -\gamma p_i, where \gamma is a positive constant dictating the amount of friction. As this is a
non-conservative force In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done ...
, we need to extend Hamilton's equations as \begin \dot &= \hphantom\frac, \\ pt\dot &= -\frac - \gamma p_i. \end Unlike the equations of motion for the simple harmonic oscillator, these modified equations do not take the form of Hamilton's equations, and therefore we do not expect Liouville's theorem to hold. Instead, as depicted in the animation in this section, a generic phase space volume will shrink as it evolves under these equations of motion. To see this violation of Liouville's theorem explicitly, we can follow a very similar procedure to the undamped harmonic oscillator case, and we arrive again at dq_i' dp_i' = dq_i dp_i\left + \left( \frac + \frac\right) \delta t\right Plugging in our modified Hamilton's equations, we find \begin dq_i' dp_i' &= dq_i dp_i\left + \left( \frac - \frac - \gamma\right) \delta t\right \\ ex&= dq_i dp_i\left -\gamma \delta t\right \end Calculating our new infinitesimal phase space volume, and keeping only first order in \delta t we find the following result: \mathrm\Gamma' = \prod_^N d^3q_i' d^3p_i' = \left - \gamma\delta t\right \prod_^N d^3q_i d^3p_i = \mathrm\Gamma \left - 3N\gamma\delta t\right We have found that the infinitesimal phase-space volume is no longer constant, and thus the phase-space density is not conserved. As can be seen from the equation as time increases, we expect our phase-space volume to decrease to zero as friction affects the system. As for how the phase-space volume evolves in time, we will still have the constant rotation as in the undamped case. However, the damping will introduce a steady decrease in the radii of each ellipse. Again we can solve for the trajectories explicitly using Hamilton's equations, taking care to use the modified ones above. Letting \alpha \equiv / for convenience, we find \begin q_i(t) &= e^ \left _i\cos + B_i\sin\right& &\omega_1 \equiv \sqrt, \\ exp_i(t) &= e^ \left _i\cos - m(\omega_1 Q_i + 2\alpha B_i)\sin\right& &B_i \equiv \frac\left(\frac + 2\alpha Q_i\right), \end where the values Q_i and P_i denote the initial position and momentum of the i-th particle. As the system evolves the total phase-space volume will spiral in to the origin. This can be seen in the figure above.


Remarks

* The Liouville equation is valid for both equilibrium and nonequilibrium systems. It is a fundamental equation of
non-equilibrium statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. * The Liouville equation is integral to the proof of the
fluctuation theorem The fluctuation theorem (FT), which originated from statistical mechanics, deals with the relative probability that the Entropy (statistical thermodynamics), entropy of a system which is currently away from thermodynamic equilibrium (i.e., maxim ...
from which the
second law of thermodynamics The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
can be derived. It is also the key component of the derivation of
Green–Kubo relations The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for a transport coefficient \gamma in terms of the integral of the equilibrium time correlation function of the time derivative of a c ...
for linear transport coefficients such as shear
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
,
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...
or
electrical conductivity Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity in ...
. * Virtually any textbook on
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
, advanced
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, or
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
will derive the Liouville theorem. Uses the ''n''-dimensional divergence theorem (without proof). * In plasma physics, the Vlasov equation can be interpreted as Liouville's theorem, which reduces the task of solving the Vlasov equation to that of single particle motion.Schwartz, S. J., Daly, P. W., and Fazakerley, A. N., 1998, Multi-Spacecraft Analysis of Plasma Kinetics, in ''Analysis Methods for Multi-Spacecraft Data'', edited by G. Paschmann and P. W. Daly, no. SR-001 in ISSI Scientific Reports, chap. 7, pp. 159–163, ESA Publ. Div., Noordwijk, Netherlands. By using Liouville's theorem in this way with energy or
magnetic moment In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
conservation, for example, one can determine unknown fields using known particle distribution functions, or vice versa. This method is known as Liouville mapping.


See also

* Boltzmann transport equation * Reversible reference system propagation algorithm (r-RESPA)


References


Further reading

* *


External links

*{{cite web , title=Phase space distribution functions and Liouville's theorem , url=https://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_1/node7.html Eponymous theorems of physics Hamiltonian mechanics Theorems in dynamical systems Statistical mechanics theorems