To provide the motivation for the first model, let us consider a spatially uniform particle gas of infinite extent, , and take the open system to be the finite region . The thermal equilibrium density matrix for a uniform gas may be obtained by integrating the Bloch equation (Feynman, 1972)
The solution (for free particles in equilibrium) is
where the normalization is such that gives the number of particles per unit length, is the chemical potential, and is a thermal coherence length given by
Now if we arbitrarily impose boundaries along the lines x = 0, x = l, , and , what boundary conditions would satisfy? Note that the dependence is only upon , so that . Thus, in this particular case obeys the homogeneous boundary condition
In other words, the directional derivative of in a direction parallel to the principal diagonal is set to zero at the boundaries.
Is (3.19) the appropriate boundary condition for a general open system? Let us explore some of its consequences. Suppose at time t = 0 we apply a uniform force field F to the particle gas. The solution to the Liouville equation (2.3) over the infinite domain and with initial condition (3.17) describes an accelerating gas and is given by
Now also obeys (3.19), so it is also the solution to (2.3) over the finite domain subject to boundary condition (3.19).
A more general consequence of boundary condition (3.19) is that the particle densities at the boundaries, and , remain constant as the density matrix evolves with time. To demonstrate this, note that we can factor the hyperbolic operator in the Liouville equation (2.3) derived from the kinetic energy terms as
The boundary condition assures that the second factor in (3.21) is zero along the boundaries, and along the diagonal the potential term is zero. Thus, and . This might be interpreted as the behavior of a large reservoir with a fixed particle density (or fixed pressure if the temperature is also fixed). Thus, the boundary condition (3.19) provides a plausible model for an open system.
In fact, the Liouville equation (2.3) subject to the boundary condition (3.19) generates an unphysical solution in the form of exponentially growing particle densities when it is applied to more general potentials which do not have the symmetry of the uniform field (Frensley, 1985). The nature of the time-dependent solutions (whether they are growing, decaying, or oscillating) depends upon the eigenvalue spectrum of the Liouville superoperator (the definition of which requires both the differential operator and the boundary conditions). The problem with the growing densities (and ultimately the identification of the correct model) is a consequence of opening the system, which violates the Hermiticity of the Hamiltonian operator and of the Liouville superoperator. Recall the proof (Messiah, 1962) of the Hermiticity of the Hamiltonian (2.2). It proceeds by invoking Green's identity to transpose the Laplace operator, which leaves a surface term. The precise expression is
where refers to the volume of the domain, S is its surface, and is the current-density operator. One maintains the Hermiticity of the Hamiltonian by choosing basis functions for which the the surface integral is identically zero: states well localized within the domain, and stationary scattering states (or periodic boundary conditions) for which the incoming and outgoing currents cancel. Because the total number of particles in an open system can change in response to externally imposed conditions, such a basis set is too restrictive.
The violation of the Hermiticity of the Liouville superoperator follows directly from that of the Hamiltonian. This leads to eigenvalues of the Liouville superoperator which have nonzero imaginary parts, leading to real exponential behavior in the time dependence of . As mentioned previously, the inclusion of dissipative interactions will introduce decaying exponential behavior. It is thus quite enlightening to observe both the separate and combined effects of dissipation and open-system boundary conditions on the eigenvalue spectrum of the Liouville superoperator (though technically it is no longer the Liouville operator when dissipation is included). For this purpose let us consider an extremely simple model of dissipation. This model is simple Brownian motion as described by the Fokker-Planck or Kramers equation (Kubo, Toda, and Hashitsume, 1985). It is classically valid in the limit that the particles of interest are weakly coupled to an ideal reservoir. Caldeira and Leggett (1983) have studied the quantum-mechanical derivation of this equation and have shown it to be valid at higher temperatures ( smaller than or comparable to the response time of the reservoir to which the particles are coupled). In terms of the Fokker-Planck equation may be written in the form (2.15) with the collision operator given by
where is the damping rate. The first term in (3.23) describes dissipation and corresponds to a frictional force equal to , where p is the linear momentum. The second term describes the thermal fluctuations. An important property of is that , which is required for consistency with the continuity equation. will be used below to add dissipative interactions to our open-system models.