When a system such as an electron device is driven far from equilibrium by the application of an external voltage, both coherent and incoherent processes will generally occur within the device. Coherent processes include tunneling and ballistic transport, and incoherent processes include dissipative scattering via phonons, for example. Coherent effects are described by adding complex-valued amplitudes (that is, values of the wavefunction), which is done implicitly in the solution of Schrödinger 's equation above. Incoherent effects are described by superposition of real-valued probabilities. An example of such incoherent superposition is the summation of the current density over energies and transverse modes to obtain the total current density, as discussed in Subsection 2. We can formalize the statistical summation procedure described there into a mathematical object known as the density matrix [30,31]. In terms of the continuum position variable x, the density matrix is actually a complex-valued function of two arguments, and has the general form:
where the form a complete set of states ( not necessarily the eigenstates of the Hamiltonian), and the are real-valued probabilities for finding an electron in each state . With this definition, the expectation value of any physical observable represented by an operator A is given by:
where A is taken to operate with respect to the first argument of . Inserting (73) into (74) and rearranging the expression, we get the more familiar form for the expectation value:
In particular, the particle density is given by
and the current density is
If is non-parabolic, a more complicated expression is required for the current density.
If the motion of the particles described by the density matrix is purely ballistic (no energy loss) and defined by a Hamiltonian H, the equation describing the evolution of the density matrix may be derived by substituting Schrödinger 's equation into (73). The result is the Liouville-von Neumann equation:
where is a linear operator which operates upon the density matrix and is called the Liouville operator. (Since it operates upon , which is itself a quantum-mechanical operator, is technically a superoperator .) The Liouville equation acts upon the density matrix by evolving the wavefunctions, but does not change the probabilities . This is a characteristic of ballistic, or conservative, motion. Irreversible, or dissipative, processes involve transitions between quantum states, and are described by operators which modify the probabilities . Such operators are discussed below.
In the classical systems, the quantity which describes the state of the system corresponding to is the phase-space distribution function where r is now the position and p is the momentum. The classical Liouville equation is
where v is the velocity and V is the potential in which the particles are moving. The particle and current densities are obtainable from the classical distribution function by
The Liouville equation, in either the classical (79) or quantum (78) context, describes only ideal, conservative motion. Within the scope of these equations, particles can only oscillate within the system, unless one applies boundary conditions which permit particles to escape from it. The form of the equations (for closed systems) does not describe an approach to a steady-state, neither equilibrium nor non-equilibrium. The reason for this involves the eigenvalue spectrum of and . The solutions of (78) will consist of a linear combination of terms with time dependence , where are the eigenvalues of . The Liouville operator [as defined in (78), including the imaginary factor] is anti-Hermitian, so the frequencies are purely real. Thus the transformation which maps the state of the system at some initial time into some later time is a unitary linear transformation, and we will call the behavior described by such equations ``unitary time evolution.'' Devices of course usually approach a steady state after a sufficiently long time. To describe this behavior, we must incorporate irreversible processes into the equations.