Phys761/F10/Hassam/Problem 1 At t=0, a neutral gas is configured so that f(x, v ,0) = (2 π v th 2 )-3/2 exp[-v 2 /2v th 2 ] [n0 + n 1 sin(2 π x/L)], where v th is the thermal speed and n 1 < n0 . Assuming that L << λ , where λ is the mean free path, and that gravity can be neglected, the collisionless Boltzmann equation ∂ f/ ∂ t + v. ∂ f/ ∂ x = 0applies. 1. Calculate the density n(x,0). Make a sketch of n(x,0) in x-y space. Make a sketch of f(x,v x ,0) in x-v x space (looks like baguettes). 2. Solve the PDE to find f(x, v ,t) for all t. The PDE can be solved by the method of characteristics; or you may use the fact that f(x-v x t,v y ,v z ) is a solution – prove this. Make a sketch of the f(x,v x ,t) baguettes at some t in x-v x space. 3. Calculate n(x,t). What happens to the ripples in x as t goes to infinity?
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