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.

Thu Jun 8 17:53:37 CDT 1995