To investigate the eigenvalue spectrum of the Liouville operator subject to the boundary conditions (4.36) we again construct a small, discrete model. The position variable q will take the same set of discrete values that x did in the previous section: . The values of p are also restricted to a discrete, bounded set: . The mesh spacing in the p direction is thus . The choice of the discrete values for p follows from a desire to avoid the point p = 0 and the need to satisfy a Fourier completeness relation, which will be discussed later. The discrete Wigner distribution is then related to the discrete density matrix of Section 3.2 by
where j indexes position q, and k indexes momentum p.
The discrete version of the potential term is readily defined. Using (4.42) the discrete potential kernel becomes
[Notice that (4.43) invokes values of which are outside the domain . This expresses the nonlocality of quantum phenomena and is one way in which the environment of an open system influences the system's behavior. The values which one assumes for , where or , depend upon the nature of the environment. If ideal reservoirs are assumed, then setting these values equal to the potential at the appropriate boundary appears to be an adequate procedure.] The elements of are then:
where the notation is introduced to shorten the expressions to be derived from the discrete Liouville equation. Note that the elements of are real and that so is an imaginary Hermitian superoperator.
The boundary conditions (4.36) affect the form of the drift term because they determine the proper finite-difference form for the gradient. On a discrete mesh a first derivative can be approximated by either a left-hand difference
or a right-hand difference
(There is also a centered difference form, , which has poor stability properties when used to approximate a drift term.) The boundary conditions determine which of the above difference forms must be used simply because one or the other will not couple the boundary value into the domain. Again, let us imagine that the boundary conditions (4.36) are implemented by fixing the value of f on mesh points just outside the domain:
This scheme is illustrated in Fig. 8.
Figure 8. Discretization scheme for the kinetic-energy superoperator (drift term) in the Wigner representation. The flow of probability between mesh points is indicated by the arrows, which also define the sense of the finite-difference approximation for the gradient. A flow toward the right requires a left-hand difference and vice versa. This is the ``upwind'' difference scheme and is uniquely determined by the form of the boundary conditions (4.36).Consider . The boundary conditions are specified for , and if this value is to be coupled into the domain, we must use the left-hand difference formula (4.45) for the gradient at . Consistency then requires that we use the left-hand difference for all (for ). Similarly, we must use the right-hand difference (4.46) for . In the context of hydrodynamic calculations such a difference scheme is called an ``upwind'' or ``upstream'' difference and is known to enormously enhance the stability of a computation (Roache, 1976, pp 4--5). It has also been used in neutron transport calculations at the kinetic (phase space) level (Duderstadt and Martin, 1979). The elements of are thus:
The terms and couple to the fixed boundary values of f and are thus the coefficients of inhomogeneous terms and are not strictly elements of . (In particular, these terms are not included in the eigenvalue calculation because eigenvalues are properties of homogeneous linear operators.) It is convenient to group these terms into a boundary contribution :
The discrete form of the Liouville equation then becomes
with the inhomogeneous terms explicitly displayed. Expanding the definitions of the operators, the Liouville equation can be written as
This provides a more convenient starting point for many of the manipulations which will be described below.
Figure 9. Eigenvalue spectrum for a model open system with irreversible boundary conditions. All eigenvalues have negative imaginary parts, verifying that the model is stable, despite the fact that no damping is yet included.
The eigenvalue spectrum for constructed from (4.37), (4.44), and (4.48) is shown in Fig. 9. The potential of Fig. 4 was used, with and . All the eigenvalues of have negative imaginary parts. (Note in particular that there is no eigenvalue equal to zero, and thus is nonsingular.) Because the eigenvalues have negative imaginary parts, the time-dependence of f contains only decaying exponentials, so the model is stable. The stability of this model follows from the boundary conditions (4.36) and does not depend upon discretization (Frensley, 1986). To demonstrate this, let us consider the expectation value of with respect to an arbitrary distribution f: . If we demonstrate that this is nonpositive for any f, we will have shown that no eigenvalue of has a positive real part, because the operator itself is purely real. In the Wigner-Weyl representation the operator inner product (2.4) becomes simply (Wigner, 1971; Hillery, O'Connell, Scully, and Wigner, 1984):
The expectation value can be rewritten
because from the antisymmetry of . For the mathematically homogeneous problem (source terms set to zero) the boundary conditions are for p>0 and for p<0. With this we can integrate the expectation value for and simplify it to obtain
Thus, the stability of the solutions to the Liouville equation using follow from the boundary conditions alone. The physical significance of this argument is that the particles in an open system will eventually escape and the density will approach zero if there is no inward current flow from the environment. However, if the potential has a local minimum within the system deep enough to create one or more bound states, any particles in those states will not escape. Their contribution to f will be zero at the boundaries, and this is the significance of the case in which (4.54) is equal to zero. Such states should correspond to eigenvalues of which are equal to zero, although I have not observed such a situation in the models which I have examined. In an open system of finite extent and with potentials of finite depth, the tunneling tail of a bound-state wavefunction will be nonzero at the system boundaries, perhaps leading to a finite rate of escape from that state within the present model.
Let us examine how this open system model can be used. The methods of calculation are more readily visualized if we write (4.50) in a block-matrix notation:
Here and represent column vectors, and and represent matrices, whose internal indices range over the allowed values of k. The are diagonal matrices, whereas the are dense. The block-tridiagonal form of greatly reduces the computational labor required to solve the Liouville equation compared to that required to work with superoperators of a more general form.
Now suppose that we wish to find the nonequilibrium steady state (). Can we simply move the column vector over to the other side of the equation and solve for the ? The answer is yes, provided that the operator is nonsingular. If there are no bound states, all the eigenvalues of are nonzero (see Fig. 9), so is a nonsingular operator and its inverse exists. This steady-state solution for the Wigner function may be written:
where refers to the ``direct current'' case. Equation (4.55) is also used to solve time-dependent problems, as will be described in the following section.
Let us compare this approach to the most commonly studied problem in transport theory, transport in a spatially homogeneous system with a uniform driving field (as is done to evaluate transport coefficients such as mobilities) (Dresden, 1961; Conwell, 1967). This generates a mathematically homogeneous problem, and the solution corresponds to the null space of that superoperator which appears in the transport equation (Aubert, Vaissiere, and Nougier, 1984). Thus, the superoperator must be singular and, if the transport equation is linear, the solution is not unique (the total density is not determined). What the present model demonstrates is that this formulation of transport through a spatially inhomogeneous system leads to a mathematically inhomogeneous problem, which is in many respects a good deal simpler than a similar homogeneous problem. For example, because is nonsingular, there is no problem of compatibility relations for the boundary conditions (Lanczos, 1961). Any choice of distribution function on the boundary will generate a unique steady-state solution. The same considerations apply to the evaluation of the transient response of an open system by integrating (4.33) with respect to t. The solution is unique and, as we have seen, stable.
These considerations clarify a point discussed by Kluksdahl et al. (1989), concerning the role of the initially assumed Wigner function in a calculation in which the steady state is found by simulating the time evolution. Kluksdahl et al. assert that the initial state must be quantum-mechanically correct. The only components of the initial state which remain through the time-evolution calculation are those which lie in the null space of the Liouville operator. All other components will approach steady-state values which are independent of the initial condition. Thus, if there is no null space (the operator is nonsingular) the initial condition makes no difference whatsoever. A concern about the correctness of the initial state is warranted only if there are bound states within the system, and possibly in the continuum limit where the smallest eigenvalue approaches zero.