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Unformatted text preview: 1 Homework 1 (due September 27) Problem 1-2 (3 pts.) A useful way of calculating entropy for a known distribution function is: S = h log i = Z d ( q , p ) log ( q , p ) Starting with Liuoville theorem, show that classical mechanics preserves the overall value of S . Hint: try to prove continuity equation for entropy density, log . Solution: ( log ) t = (log + 1) t Let v be the "velocity" in phase space. By applying Liuoville theorem ( v = 0 ), we obtain: ( log ) t = (log + 1) ( v ) = ( v ) ( log ) = ( v log ) We can now use the Gauss theorem: S = Z d ( v log ) = I log ( v d A ) = Here we have assumed that the motion is limited to a f nite region of the phase space, and therefore = 0 at its boundary Problem 1-2 (4 pts.) a) What is the average number of molecules in volume v , if their overall concentration (in a big container) is ....
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