hw01soln - 1 Homework 1(due September 27 Problem 1-2(3 pts...

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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 ρ i = − Z d Γ ρ ( q , p ) log ρ ( 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, − ρ log ρ . Solution: ∂ ( ρ log ρ ) ∂t = (log ρ + 1) ∂ρ ∂t Let v be the "velocity" in phase space. By applying Liuoville theorem ( ∇ v = 0 ), we obtain: ∂ ( ρ log ρ ) ∂t = − (log ρ + 1) ( v ∇ ) ρ = − ( v ∇ ) ( ρ log ρ ) = −∇ ( v ρ log ρ ) We can now use the Gauss theorem: ˙ S = Z d Γ ∇ ( v ρ log ρ ) = I ρ log ρ ( v · d A ) = Here we have assumed that the motion is limited to a f nite region of the phase space, and therefore ρ = 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 ρ ....
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hw01soln - 1 Homework 1(due September 27 Problem 1-2(3 pts...

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