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Phase space distribution functions and Liouville's theorem

Given an ensemble with many members, each member having a different phase space vector tex2html_wrap_inline612 corresponding to a different microstate, we need a way of describing how the phase space vectors of the members in the ensemble will be distributed in the phase space. That is, if we choose to observe one particular member in the ensemble, what is the probability that its phase space vector will be in a small volume tex2html_wrap_inline634 around a point tex2html_wrap_inline612 in the phase space at time t. This probability will be denoted


where tex2html_wrap_inline640 is known as the phase space probability density or phase space distribution function. It's properties are as follows:


Liouville's Theorem: The total number of systems in the ensemble is a constant. What restrictions does this place on tex2html_wrap_inline640 ? For a given volume tex2html_wrap_inline644 in phase space, this condition requires that the rate of decrease of the number of systems from this region is equal to the flux of systems into the volume. Let tex2html_wrap_inline646 be the unit normal vector to the surface of this region.

Figure 1:

The flux through the small surface area element, dS is just tex2html_wrap_inline650 . Then the total flux out of volume is obtained by integrating this over the entire surface that encloses tex2html_wrap_inline644 :


which follows from the divergence theorem. tex2html_wrap_inline654 is the 6N dimensional gradient on the phase space


On the other hand, the rate of decrease in the number of systems out of the volume is


Equating these two quantities gives


But this result must hold for any arbitrary choice of the volume tex2html_wrap_inline644 , which we may also allow to shrink to 0 so that the result holds locally, and we obtain the local result:




This equation resembles an equation for a ``hydrodynamic'' flow in the phase space, with tex2html_wrap_inline640 playing the role of a density. The quantity tex2html_wrap_inline662 , being the divergence of a velocity field, is known as the phase space compressibility, and it does not, for a general dynamical system, vanish. Let us see what the phase space compressibility for a Hamiltonian system is:


However, by Hamilton's equations:


Thus, the compressibility is given by


Thus, Hamiltonian systems are incompressible in the phase space, and the equation for tex2html_wrap_inline640 becomes


which is Liouville's equation, and it implies that tex2html_wrap_inline640 is a conserved quantity when tex2html_wrap_inline612 is identified as the phase space vector of a particular Hamiltonian system. That is, tex2html_wrap_inline670 will be conserved along a particular trajectory of a Hamiltonian system. However, if we view tex2html_wrap_inline612 is a fixed spatial label in the phase space, then the Liouville equation specifies how a phase space distribution function tex2html_wrap_inline640 evolves in time from an initial distribution tex2html_wrap_inline676 .

next up previous
Next: About this document Up: No Title Previous: Classical microscopic states or

Mark Tuckerman
Wed Jan 8 22:51:23 EST 2003