p212midtermsols

p212midtermsols - Physics 212 Statistical mechanics II Fall...

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Physics 212: Statistical mechanics II, Fall 2005 Midterm solutions 1. (a) Liouville’s theorem is that f N is constant along particle trajectories: ∂f N ∂t + Nd X i =1 ± dx i dt ∂f N ∂x i + dp i dt ∂f N ∂p i ² = 0 . (1) (b) The collision term in the Boltzmann equation for f ( t, x , p 1 ) is C ( f ) = Z w ( p 1 , p 2 ; p 0 1 , p 0 2 )( f ( t, x , p 0 1 ) f ( t, x , p 0 2 ) - f ( t, x , p 1 ) f ( t, x , p 2 )) d p 2 d p 0 1 d p 0 2 . (2) The collision term will vanish if for any incoming and outgoing momenta, f ( t, x , p 0 1 ) f ( t, x , p 0 2 ) = f ( t, x , p 1 ) f ( t, x , p 2 ) . (3) This in turn is satisfied if log f is a sum of conserved quantities. Normalization in three dimensions then gives f ( t, x , p ) = n ( x ) e - a ( p - p 0 ) 2 / 2 / (2 π/a ) 3 / 2 , (4) where a and p 0 are undetermined constants. (c) The volume of phase space in which f N is nonzero does not change by the following argument. Liouville’s theorem implies that at any later time t > 0, f N still takes either the value 0 or 1 /V . Conservation of probability then requires that the volume in which f N
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p212midtermsols - Physics 212 Statistical mechanics II Fall...

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