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 traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.[1]

There are related mathematical results in symplectic topology and ergodic theory.

There are extensions of Liouville's theorem to stochastic systems.[2]

Liouville equationsEdit

 
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.

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 was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics.[3][4] It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838.[5] Consider a Hamiltonian 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 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 proof of Liouville's theorem uses the n-dimensional divergence theorem. This proof is based on the fact that the evolution of   obeys an n-dimensional version of the continuity equation:

 

That is, the 3-tuple   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 as well as conservation of the Hamiltonian along the flow 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).[6]

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.

Other formulationsEdit

Poisson bracketEdit

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

 

or in terms of the Liouville operator or Liouvillian,

 

as

 

Ergodic theoryEdit

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[citation needed]. The Hamiltonian case then becomes a corollary.

Symplectic geometryEdit

We can also formulate Liouville's Theorem in terms of symplectic geometry. For a given system, we can consider the phase space   of a particular Hamiltonian   as a manifold   endowed with a symplectic 2-form

 

The volume form of our manifold is the top exterior power 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 we can define a Hamiltonian vector field generated by a function   as

 

Specifically, when the generating function is the Hamiltonian itself,  , we get

 

where we utilized Hamilton's equations of motion and the definition of the chain rule.[7]

In this formalism, Liouville's Theorem states that the Lie derivative of the volume form is zero along the flow generated by  . That is, for   a 2n-dimensional symplectic manifold,

 

In fact, the symplectic structure   itself is preserved, not only its top exterior power. That is, Liouville's Theorem also gives [8]

 

Quantum Liouville equationEdit

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, 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 commutators. In this case, the resulting equation is[9][10]

 

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.

In the Phase space formulation of Quantum mechanics, supplanting the Moyal brackets for Poisson brackets 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.

ExamplesEdit

SHO Phase Space VolumeEdit

 
The time evolution of phase space for the simple harmonic oscillator (SHO). Here we have taken   and are considering the region  .

Consider an   particle system in three dimensions, and focus on only the evolution of   particles. Within phase space, these   particles occupy an infinitesimal volume given by

 

We want   to remain the same throughout time, so that   is constant along the trajectories of the system. If we allow our particles to evolve by an infinitesimal time step  , we see that each particle phase space location changes as

 

where   and   denote   and   respectively, and we have only kept terms linear in  . Extending this to our infinitesimal hypercube  , the side lengths change as

 

To find the new infinitesimal phase space volume  , need the product of the above quantities. To first order in  , we get the following.

 

So far, we have yet to make any specifications about our system. Let us now specialize to the case of    -dimensional isotropic harmonic oscillators. That is, each particle in our ensemble can be treated as a simple harmonic oscillator. The Hamiltonian for this system is given by

 

By using Hamilton's equations with the above Hamiltonian we find that the term in parentheses above is identically zero, thus yielding

 

From this we can find the infinitesimal volume of phase space.

 

Thus we have ultimately found that the infinitesimal phase space volume is unchanged, yielding

 

demonstrating Liouville's Theorem holds for this system.[11]

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  . Explicitly, one can solve Hamilton's equations for the system and find

 

where   and   denote the initial position and momentum of the   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   in the Hamiltonian, independent of any differences in energy. As a result a region of phase space will simply rotate about the point   with frequency dependent on   [12]. This can be seen in the animation above.


Damped Harmonic OscillatorEdit

 
The evolution of phase space volume for the damped harmonic oscillator. The same values of parameters are used as in the SHO case, with  .

One of the foundational assumptions of Liouville's Theorem is that the system obeys the conservation of energy. In the context of phase space, this is to say that   is constant on phase space surfaces of constant energy  . If we break this requirement by considering a system in which energy is not conserved, we find that   also fails to be constant.

As an example of this, consider again the system of   particles each in a  -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. As this is a non-conservative force, we need to extend Hamilton's equations as

 

where   is a positive constant dictating the amount of friction. Following a very similar procedure to the undamped harmonic oscillator case, we arrive again at

 

Plugging in our modified Hamilton's equations, we find

 

Calculating our new infinitesimal phase space volume, and keeping only first order in   we find the following result.

 

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   for convenience, we find

 

where the values   and   denote the initial position and momentum of the   particle. As the system evolves the total phase space volume will spiral in to the origin. This can be seen in the figure above.

RemarksEdit

See alsoEdit

ReferencesEdit

  1. ^ Harald J. W. Müller-Kirsten, Basics of Statistical Physics, 2nd ed., World Scientific (Singapore, 2013)
  2. ^ Kubo, Ryogo (1963-02-01). "Stochastic Liouville Equations". Journal of Mathematical Physics. 4 (2): 174–183. doi:10.1063/1.1703941. ISSN 0022-2488.
  3. ^ 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), p. 16.
  4. ^ Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.
  5. ^ J. Liouville, Journ. de Math., 3, 342 (1838), [1].
  6. ^ Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012).
  7. ^ Nakahara, Mikio (2003). Geometry, Topology, and Physics (2 ed.). Taylor & Francis Group. pp. 201–204. ISBN 978-0-7503-0606-5.
  8. ^ Nash, Oliver (8 January 2015). "Liouville's theorem for pedants" (PDF).
  9. ^ The theory of open quantum systems, by Breuer and Petruccione, p 110.
  10. ^ Statistical mechanics, by Schwabl, p 16.
  11. ^ Kardar, Mehran (2007). Statistical Physics of Particles. University of Cambridge Press. pp. 59–60. ISBN 978-0-521-87342-0.
  12. ^ Eastman, Peter (2014–2015). "Evolution of Phase Space Probabilities".CS1 maint: date format (link)
  13. ^ For a particularly clear derivation see Tolman, R. C. (1979). The Principles of Statistical Mechanics. Dover. pp. 48–51. ISBN 9780486638966.
  14. ^ "Phase Space and Liouville's Theorem". Retrieved January 6, 2014. Nearly identical to proof in this Wikipedia article. Assumes (without proof) the n-dimensional continuity equation.
  15. ^ "Preservation of phase space volume and Liouville's theorem". Retrieved January 6, 2014. A rigorous proof based on how the Jacobian volume element transforms under Hamiltonian mechanics.
  16. ^ "Physics 127a: Class Notes" (PDF). Retrieved January 6, 2014. Uses the n-dimensional divergence theorem (without proof).
  17. ^ Nash, Oliver (January 8, 2015). "Liouville's theorem for pedants" (PDF). Retrieved October 1, 2015. Proves Liouville's theorem using the language of modern differential geometry.

Further readingEdit

External linksEdit