A Analysis of Petrov-Galerkin Techniques Applied to Lattice Gas Models for Quantum Transport Processes in Open Systems.

Quantum transport processes in open systems are analyzed in terms of generalized master equations using a representation-independent operator approach and projection superoperators. Boundary conditions associated with ideal reservoirs are imposed to derive simplified master equations in the steady state limit. Conserved quantities and corresponding rate observables are defined for interacting subsystems. Galerkin techniques for finite element models are extended to develop discrete models for quantum transport processes, yielding the Galerkin form of the simplified master equation. It is shown using an oblique projection that an appropriately chosen test space can completely account for the dynamical effects of the residual interaction, and that the test space can be chosen so that the Galerkin solution yields exact expectation values for the observables of interest. A condition relating the test and trial spaces to these observables is derived and used to develop heuristic techniques for choosing the test space. The resulting discrete model is described as a set of coupled continuity and transport equations. Lattice gas models are extended to describe nonequilibrium transport processes in finite, open systems, and exact analytic solutions are obtained for small systems described at the microscopic level. It is shown that generalized reservoir interactions and inhomogeneous external magnetic fields lead to a nonvanishing residual interaction and field dependent spin current. Petrov-Galerkin techniques are applied, and test spaces are obtained that yield exact expectation values for the observables of interest in each case. Macroscopic lattice gas models are introduced to evaluate the scaling of heuristic Petrov-Galerkin techniques to large problems. It is shown that the form of the transport equations implies contact limited behavior determined by long range correlations for collisionless devices. Mesoscopic models are introduced with a number of spins that can simultaneously allow efficient numerical solutions and represent large scale behavior. Device behavior is analyzed for a series configuration with a variety of inhomogeneous field profiles. A nonlocal "echo" phenomenon results from the interaction of long range correlations with the reservoirs for a localized field inhomogeneity. Extrapolation from the solutions for mesoscopic models is justified by rapid convergence in the macroscopic limit.