phys210bhw1

phys210bhw1 - Phys 210B Fall 2010 Problem Set #1: Kinetic...

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Unformatted text preview: Phys 210B Fall 2010 Problem Set #1: Kinetic Theory & the Boltzmann Equation Due date: Wednesday October 13 1. Evolution of the density function in phase space A thermalized classical gas is confined to a one-dimensional trap. Suppose the particles of this gas do not collide so that the gas can be described by an ensemble of 1d boxes each containing a single thermalized gas particle. Let the initial density function of this gas be ( r,p,t = 0) = ( r ) f ( p ), where f ( p ) exp(- p 2 / 2 mk B T ). (a) Starting from Liouvilles equation, derive ( r,p,t ) and sketch it in the ( r,p ) plane for several t . (b) Derive the expressions for the average h r 2 i and h p 2 i for t > 0. (c) Suppose that hard walls are placed at r = R . Sketch ( r,p,t ) again for several t and describe its behavior for t , where is an appropriately large relaxation time. (d) A coarse-grained density e is obtained by ignoring variations of below some small resolution in the ( r,p ) plane; for example, by averaging over cells of the resolution area....
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This note was uploaded on 11/20/2010 for the course PHYS 250a taught by Professor Hwa during the Spring '10 term at UCSD.

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phys210bhw1 - Phys 210B Fall 2010 Problem Set #1: Kinetic...

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