homework10 - Physics 262 Statistical Physics Fall 2008 Problem Set 10 Not to be turned in Reading Kardar Chapter 3 1 A dilute gas of molecules of mass m

# homework10 - Physics 262 Statistical Physics Fall 2008...

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Physics 262 : Statistical Physics Fall 2008 Problem Set 10 Not to be turned in Reading: Kardar, Chapter 3 1. A dilute gas of molecules of mass m is enclosed in a container and maintained in the presence of a small temperature gradient ∂T/∂z in the z direction. We want to find the thermal conductivity, κ , of this gas at a temperature T . (a) Argue that a general solution of the condition ∂f p /∂τ | coll = 0 for the Boltzmann equation is of the form f 0 p ( z ) = n ( z ) 1 (2 πmk B T ( z )) 3 / 2 e - p 2 / (2 mk B T ( z )) (1) where both the local density, n , and the temperature T , are allowed to be functions of position (in the present case, only z ). We assume that the collisions relax the distribution function towards such a local equilibrium, and so write ∂f p ∂τ coll = - f p - f 0 p ( z ) τ (2) (b) Solve the Boltzmann equation for the steady state distribution function f p to linear order in the gradients