Preservation of phase space volume and Liouville's theorem next up previous
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Preservation of phase space volume and Liouville's theorem

Consider a phase space volume element tex2html_wrap_inline507 at t=0, containing a small collection of initial conditions on a set of trajectories. The trajectories evolve in time according to Hamilton's equations of motion, and at a time t later will be located in a new volume element tex2html_wrap_inline513 as shown in the figure below:

   figure67
Figure 1:

How is tex2html_wrap_inline507 related to tex2html_wrap_inline513 ? To answer this, consider a trajectory starting from a phase space vector tex2html_wrap_inline519 in tex2html_wrap_inline507 and having a phase space vector tex2html_wrap_inline523 at time t in tex2html_wrap_inline513 . Since the solution of Hamilton's equations depends on the choice of initial conditions, tex2html_wrap_inline523 depends on tex2html_wrap_inline519 :

eqnarray75

Thus, the phase space vector components can be viewed as a coordinate transformation on the phase space from t=0 to time t. The phase space volume element then transforms according to

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where tex2html_wrap_inline537 is the Jacobian of the transformation:

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where n=6N. The precise form of the Jacobian can be determined as will be demonstrated below.

The Jacobian is the determinant of a matrix M,

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whose matrix elements are

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Taking the time derivative of the Jacobian, we therefore have

eqnarray98

The matrices M tex2html_wrap_inline541 and tex2html_wrap_inline543 can be seen to be given by

eqnarray117

Substituting into the expression for dJ/dt gives

eqnarray126

where the chain rule has been introduced for the derivative tex2html_wrap_inline547 . The sum over i can now be performed:

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Thus,

eqnarray155

or

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The initial condition on this differential equation is tex2html_wrap_inline551 . Moreover, for a Hamiltonian system tex2html_wrap_inline553 . This says that dJ/dt=0 and J(0)=1. Thus, tex2html_wrap_inline559 . If this is true, then the phase space volume element transforms according to

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which is another conservation law. This conservation law states that the phase space volume occupied by a collection of systems evolving according to Hamilton's equations of motion will be preserved in time. This is one statement of Liouville's theorem.

Combining this with the fact that df/dt=0, we have a conservation law for the phase space probability:

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which is an equivalent statement of Liouville's theorem.


next up previous
Next: Liouville's theorem for non-Hamiltonian Up: No Title Previous: The Liouville operator and

Mark Tuckerman
Mon Jan 28 09:08:52 EST 2002