The Pauli master equation (see Kreuzer, 1981, ch. 10) is derived under the assumption that the density matrix is and remains diagonal in the basis of eigenstates of the Hamiltonian:
where is the probability of the system to be in state i. The master equation is then
where the are the golden-rule transition rates. Consider transitions from a state i to a state j which have different spatial distributions: . Then the rate of change of the density is:
However, i and j are eigenstates of the Hamiltonian, which means that and are constant (for scattering states) or even zero (for bound states). In either case,
Now, the rate of change of the density will be zero if either of the two bracketed terms in (10.137) are zero. In thermal equilibrium the first term is zero by the principle of detailed balance, but away from equilibrium it is, in general, nonzero. The second term will be zero if the probability distributions of the eigenstates i and j are identical. This happens in only a very few cases, most notably for the plane-wave states of a free particle.
Thus, the assumption that the density matrix has the form (10.135) for far-from-equilibrium systems will lead, in general, to a violation of the continuity equation.