How to prove the Bose enhancement factor $(1+f)$ and the Pauli blocking factor $(1-f)$ in Boltzmann equation? $$1+2\rightarrow3+4+\cdots$$
For the collision integral in the Boltzmann equation for particles obeying different statistic,  the factor is 1 for classical final particles , 1-f for final fermions, 1+f for final Boson.
However why it's exactly this form $1\pm f$? For fermion it can be understood because if there is a particle in one point of phase space, this kind of particle cannot be created again. And certainly it's not a proof. But why for boson the effect is exactly the $1+f$ not like $1+2f$. So how to prove these effects from fundamental principle? Or where can I find the proof.
 A: A proper derivation of the Boltzmann equation from non-equilibrium quantum field theory (which will give the factors $1\pm f$ in the weak coupling, quasi-particle dominated, limit) is a difficult problem. The standard reference is Kadanoff and Baym, Quantum Statistical Mechanics. 
The standard approach in introductory text books (and indeed, historically, Landau's approach), is to observe that since the collision term is a rate, the Bose-enhacement/Pauli-blocking factors should ``obviously'' be included. What is indeed easy to see is that if a rate is computed at non-zero temperature or density using the Matsubara formalism (or second quantized fields), then the factors $(1\pm f)$ will appear. This is described in any text book on thermal field theory, see, for example, Le Bellac's book. 
A: I can't exactly answer your question, but perhaps the book of Ichimaru "Statistical Plasma Physics" can help. 
As in the end the difference between classical kinetics and degenerate kinetics must arise from the phase space density, this could help, because Ichimaru derives in Chap2 the collision operator starting from the phase-space-distribution.
