3. The solution of the collisionless kinetic equation at equilibrium has a Gaussian speed distribution of the form f(v)
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3. The solution of the collisionless kinetic equation at equilibrium has a Gaussian speed distribution of the form f(v) = Ceaʼmu? 2" < 2 The two constants C and a can be obtained by using the thermodynamic relation of the average kinetic energy of free particle in three dimensions 1 3 >s(u)=žkBT, and by using a suitable normalization of the distribution f(u) to the number of particles per unit volume n. (a) find the the constants C and a, (b) find the average speed <u>f(v), (c) The directed flux r, in the +z direction is given by n < V>f(u), where the average is taken over vz > 0 only. Writing v; = v cos in spherical 0 coordinates, show that T2 T= in <v >f(v) 1