Irreversible or energy-dissipating processes always involve transitions between quantum states. Such processes are described, at the simplest level, by master or rate equations [32]. The operators which generate the time-evolution in such equations are of a very different form from that of the Liouville operator.
If the state of a system is described by an array of probabilities or occupation
factors 
for a
particle to occupy a (stationary) quantum level i, the time evolution of that
system is determined by the rates of transition between the levels i. These rates
are usually estimated using the ``Fermi golden rule:''
where 
is the Hamiltonian describing the interaction that causes
the transitions, and 
is the transition rate from state j to state i.
The 
-function ensures energy conservation, but it must be remembered that
and 
are the total energy of each state, including, for example, the energy
in an emitted phonon. Thus (82) can describe energy-dissipating
processes despite its appearance. If one assumes that these transitions occur
independently within any small time interval (the Markov assumption), the transition
from state j to state i will produce changes in the corresponding occupation factors:
The occupation of state i increases and that of state j decreases as a result of this particular process, and the amount of change depends only upon the occupation of the initial state. (We neglect here the Pauli exclusion principle, which leads to nonlinear master equations.) If we sum (83) over all possible transition processes, we obtain the master equation:
where M is the master operator, whose matrix elements are given by
Notice the form of this operator. The off-diagonal elements are all positive and the diagonal elements are all negative, with a magnitude equal to the sum of the off-diagonal elements in the same column. (If one considers an open system, the coupling to external reservoirs can lead to master operators in which the magnitude of the diagonal elements exceeds the sum of the off-diagonal elements [5].) The eigenvalues of an operator of this form will all have real parts less than or equal to zero [5,32]. Thus the solutions of (84) will consist of a linear combination of terms with a decaying exponential time-dependence, and so will always show a stable approach to some steady state.
The Pauli master equation [33] is the most commonly used model of irreversible processes in simple quantum systems. It can be derived from elementary quantum mechanics plus a Markov assumption. Within these assumptions, the density matrix has the form (73) with the states i being the eigenstates of the system Hamiltonian, and remains of this form at all times. The Pauli master equation is then just (84) with the Fermi golden-rule rates (82). There are a number of conceptual problems with the Pauli equation [33] , not the least of which is that it produces violations of the continuity equation [5]. It is nevertheless employed, either explicitly or implicitly, in almost all semi-classical treatments of electron transport in semiconductors.
Master operators most often occur in the description of stochastic (random) processes,
where they describe the average behavior of the system. In such cases there will
always be fluctuations (noise) about the solution of the master equation. Diffusion
phenomena are the classic example of this. The master operator in the classical
diffusion equation 
is just the laplacian
. By examining the form of the finite-difference approximation to the
second derivative (44), it is easy to see that this has the form of a
master operator (85).
Another case of particular importance (and a source of some confusion) is the
gradient operator, which appears in the classical Liouville equation (79)
and in the drift term of the drift-diffusion equation (
),
among many other contexts. The unique property of this operator is that, depending
upon the boundary conditions imposed, 
can be an anti-Hermitian operator
(generating unitary time-evolution), or
can be a master operator (generating an approach to a steady state).
If one applies periodic boundary conditions,
the eigenstates of 
are of the form 
, with real-valued eigenvalues
k. The finite-difference approximation appropriate to this situation is the
centered-difference form
which (if written in matrix form) is clearly anti-Hermitian. On the other hand, if one
applies initial conditions to 
, a single boundary condition should be
imposed on the left if v > 0 or on the right if v < 0. The appropriate discretization
in this case is the upwind difference [5,34]
which clearly has the form of a master operator. The upwind difference is known to produce excellent stability in fluid dynamic calculations [34], and the master-operator form is the ultimate explanation of its success.
