Phys761/F10/Hassam/Problem 1At t=0, a neutral gas is configured so that f(x,v,0) = (2πvth2)-3/2 exp[-v2/2vth2] [n0+ n1sin(2πx/L)], where vthis the thermal speed and n1< n0. Assuming that L << λ, where λis 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,vx,0) in x-vxspace (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-vxt,vy,vz) is a solution – prove this. Make a sketch of the f(x,vx,t) baguettes at some t in x-vxspace.3.Calculate n(x,t). What happens to the ripples in x as t goes to infinity?
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