Example 2.10: Probability of one particle picked out from among N Show that the probability for one particle of a total of N noninteracting particles in a volume V to have a velocity component u1 in the interval (u1, u1 +du1) is given by the Maxwell distribution law Πu1du1. Start from the following expression du1



du2 . . . duNdv1 . . . dvNdw1 . . . dwNδ



1 2

m(u22

+ · · · + w2N

) −



E −

1 2

mu21



∼ θ



E −

1 2

mu21



du1



duu3N−2δ



1 2

mu2 −



E −

1 2

mu21



, (2.68)

where the second expression results from integration over the angles of the (3N − 1)-dimensional unit sphere with u the modulus of (u2, . . . , wN). [Hint: Set the argument of the delta function equal to y (say) and integrate and take N and E as large, so that 1 − α/E exp(−α/E)].