Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named aer 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 traveling through phase-space is constant with time. is time-independent density is in statistical mechanics known as the classical a priori probability.[1]
ere are related mathematical results in symplectic topology and ergodic theory.
Contents
Liouville equations
Other formulations Poisson bracket Ergodic theory Symplectic geometry Quantum Liouville equation
Remarks
See also
References
External links
Liouville equations
e 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 was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics.[2][3] It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838.[4] Consider a Hamiltonian dynamical system with canonical coordinates and conjugate momenta , where . en the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume . e 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. is 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 proof of Liouville's theorem uses the n-dimensional divergence theorem. is proof is based on the fact that the evolution of obeys an n-dimensional version of the continuity equation:
at is, the tuplet is a conserved current. Notice that the difference between this and Liouville's equation are the terms
Evolution of an ensemble of classical systems in phase space (top). Each system consists of one massive particle in a one-dimensional potential well (red curve, lower figure). Whereas the motion of an individual member of the ensemble is given by Hamilton's equations, Liouville's equations describe the flow of the whole distribution. The motion is analogous to a dye in an incompressible fluid.
where is the Hamiltonian, and Hamilton's equations as well as conservation of the Hamiltonian along the flow have been used. at 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. e symmetry is invariance under time translations, and the generator (or Noether charge) of the symmetry is the Hamiltonian.
Other formulations
Poisson bracket e theorem is oen restated in terms of the Poisson bracket as
or in terms of the Liouville operator or Liouvillian,
as
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). e 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. e Hamiltonian case then becomes a corollary.
Symplectic geometry In terms of symplectic geometry, the phase space is represented as a symplectic manifold. e theorem then states that the natural volume form on a symplectic manifold is invariant under the Hamiltonian flows. e symplectic i structure is represented as a 2-form, given as a sum of wedge products of dpi with dq . e 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 derivative 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, namely a symplectomorphism.
Quantum Liouville equation e 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, the Von Neumann equation. is procedure, oen 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[5][6]
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.
Remarks
The Liouville equation is valid for both equilibrium and nonequilibrium systems. It is a fundamental equation of non-equilibrium statistical mechanics. 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[7] the Liouville theorem.[8][9][10][11]
See also
Reversible reference system propagation algorithm (r-RESPA)
References
Modern Physics, by R. Murugeshan, S. Chand publications Liouville's theorem in curved space-time : Gravitation § 22.6, by Misner,Thorne and Wheeler, Freeman
1. Harald J.W. Müller-Kirsten, Basics of Statistical Physics, 2nd ed., World Scientific (Singapore, 2013) 2. J. W. Gibbs, "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics." Proceedings of the American Association for the Advancement of Science, 33, 57-58 (1884). Reproduced in The Scientific Papers of J. Willard Gibbs, Vol II (1906), pp. 16 (https://archive.org/stream/scientificpapers02gibbuoft#page/16/mode/2up). 3. Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons. 4. [J. Liouville, Journ. de Math., 3, 349(1838)]. 5. The theory of open quantum systems, by Breuer and Petruccione, p110 (https://books.google.com/books?id=0Yx5VzaMYm8C&pg=PA110). 6. Statistical mechanics, by Schwabl, p16 (https://books.google.com/books?id=o-HyHvRZ4VcC& pg=PA16). 7. [for a particularly clear derivation see "The Principles of Statistical Mechanics" by R.C. Tolman , Dover(1979), p48-51]. 8. http://hepweb.ucsd.edu/ph110b/110b_notes/node93.html Nearly identical to proof in this Wikipedia article. Assumes (without proof) the n-dimensional continuity equation. Retrieved 01/06/2014. 9. http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_2/node2.html A rigorous proof based on how the Jacobian volume element transforms under Hamiltonian mechanics. Retrieved 01/06/2014. 10. http://www.pmaweb.caltech.edu/~mcc/Ph127/a/Lecture_3.pdf Uses the n-dimensional divergence theorem (without proof) Retrieved 01/06/2014. 11. http://www.maths.tcd.ie/~onash/liouville_pedants_files/notes.pdf Proves Liouville's theorem using the language of modern differential geometry Retrieved 10/01/2015.
External links
Phase space distribution functions and Liouville's theorem→http://www.nyu.edu/classes /tuckerman/stat.mech/lectures/lecture_1/node7.html
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This page was last edited on 10 March 2018, at 16:25.
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