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We contrast Dirac’s theory of transition probabilities and the theory of nonadiabatic transition probabilities, applied to a perturbed system that is coupled to a bath. In Dirac’s analysis, the presence of an excited state |k0⟩ in the time-dependent wave function constitutes a transition. In the nonadiabatic theory, a transition occurs when the wave function develops a term that is not adiabatically connected to the initial state. Landau and Lifshitz separated Dirac’s excited-state coefficients into a term that follows the adiabatic theorem of Born and Fock and a nonadiabatic term that represents excitation across an energy gap. If the system remains coherent, the two approaches are equivalent. However, differences between the two approaches arise when coupling to a bath causes dephasing, a situation that was not treated by Dirac. For two-level model systems in static electric fields, we add relaxation terms to the Liouville equation for the time derivative of the density matrix. We contrast the results obtained from the two theories. In the analysis based on Dirac’s transition probabilities, the steady state of the system is not an equilibrium state; also, the steady-state population ρkk,s increases with increasing strength of the perturbation and its value depends on the dephasing time T2. In the nonadiabatic theory, the system evolves to the thermal equilibrium with the bath. The difference is not simply due to the choice of basis because the difference remains when the results are transformed to a common basis. 

In this paper, we describe how we extended the Pegasus Workflow Management System to support edge-to-cloud workflows in an automated fashion. We discuss how Pegasus and HTCondor (its job scheduler) work together to enable this automation. We use HTCondor to form heterogeneous pools of compute resources and Pegasus to plan the workflow onto these resources and manage containers and data movement for executing workflows in hybrid edge-cloud environments. We then show how Pegasus can be used to evaluate the execution of workflows running on edge only, cloud only, and edge-cloud hybrid environments. Using the Chameleon Cloud testbed to set up and configure an edge-cloud environment, we use Pegasus to benchmark the executions of one synthetic workflow and two production workflows: CASA-Wind and the Ocean Observatories Initiative Orcasound workflow, all of which derive their data from edge devices. We present the performance impact on workflow runs of job and data placement strategies employed by Pegasus when configured to run in the above three execution environments. Results show that the synthetic workflow performs best in an edge only environment, while the CASA - Wind and Orcasound workflows see significant improvements in overall makespan when run in a cloud only environment. The results demonstrate that Pegasus can be used to automate edge-to-cloud science workflows and the workflow provenance data collection capabilities of the Pegasus monitoring daemon enable computer scientists to conduct edge-to-cloud research.