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Consider phase space for the *micro-*canonical
ensemble of the *N*-particle systems. Evolution of the ensemble is shown
below:

As the number of systems in the ensemble is constant, we can write the continuity equation for density in phase space:

(1) |

For the flux in phase space, coordinates are 's and 's and velocities are 's and 's:

(2) |

(3) |

(4) |

(5) |

(6) |

The left hand side of is actually the full derivative
of the distribution function, describing its change along the
trajectories (flux lines). We proved *Liouville theorem*:

In conservative system the distribution function is constant along the trajectories, being an integral of motion.

© 1997

Tue Dec 2 20:24:47 EST 1997