This paper presents a comprehensive review of the wave-function approach for derivation of the numberresolved Master equations, used for description of transport and measurement in mesoscopic systems. The review contains important amendments, clarifying subtle points in derivation of the Master equations and their validity. This completes the earlier works on the subject. It is demonstrated that the derivation does not assume weak coupling with the environment and reservoirs, but needs only high bias condition. This condition is very essential for validity of the Markovian Master equations, widely used for a phenomenological description of different physical processes.
This paper presents a comprehensive review of the wave-function approach for derivation of the numberresolved Master equations, used for description of transport and measurement in mesoscopic systems. The review contains important amendments, clarifying subtle points in derivation of the Master equations and their validity. This completes the earlier works on the subject. It is demonstrated that the derivation does not assume weak coupling with the environment and reservoirs, but needs only high bias condition. This condition is very essential for validity of the Markovian Master equations, widely used for a phenomenological description of different physical processes.
For quantum transport through mesoscopic systems, a quantum master-equation approach is developed in terms of compact expressions for the transport current and the reduced density matrix of the system. The present work is an extension of Gurvitz's approach for quantum transport and quantum measurement, namely, to finite temperature and arbitrary bias voltage. Our derivation starts from a second-order cumulant expansion of the tunneling Hamiltonian; then follows the conditional average over the electrode reservoir states. As a consequence, in the usual weak-tunneling regime, the established formalism is applicable for a wide range of transport problems. The validity of the formalism and its convenience in application are well illustrated by a number of examples.
We derive Markovian master equations for single and interacting harmonic systems in different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding master equations with numerical simulations of the global system's evolution, we delimit their validity regimes and assess the robustness of the assumptions usually made in the process of deriving the reduced Markovian dynamics. The results of these illustrative examples serve to clarify the general properties of other open quantum system scenarios subject to treatment within a Markovian approximation.
We derive Markovian master equations for single and interacting harmonic systems in different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding master equations with numerical simulations of the global system's evolution, we delimit their validity regimes and assess the robustness of the assumptions usually made in the process of deriving the reduced Markovian dynamics. The results of these illustrative examples serve to clarify the general properties of other open quantum system scenarios subject to treatment within a Markovian approximation.
A comprehensive approach to modeling open quantum systems consistent with thermodynamics is presented. The theory of open quantum systems is employed to define system bath partitions. The Markovian master equation defines an isothermal partition between the system and bath. Two methods to derive the quantum master equation are described: the weak coupling limit and the repeated collision model. The role of the eigenoperators of the free system dynamics is highlighted, in particular, for driven systems. The thermodynamical relations are pointed out. Models that lead to loss of coherence, i.e., dephasing are described. The implication of the laws of thermodynamics to simulating transport and spectroscopy is described. The indications for self-averaging in large quantum systems and thus its importance in modeling are described. Basic modeling by the surrogate Hamiltonian is described, as well as thermal boundary conditions using the repeated collision model and their use in the stochastic surrogate Hamiltonian. The problem of modeling with explicitly time dependent driving is analyzed. Finally, the use of the stochastic surrogate Hamiltonian for modeling ultrafast spectroscopy and quantum control is reviewed.
This paper achieves a fundamental reconstruction of the quantum theory paradigm based on the energy distribution framework. This framework radically steps outside the traditional description based on probability amplitude evolution, reinterpreting physical processes as deterministic energy flow among quantum units (Q), an interfacial defect network (D), and bulk dissipation channels (B). A series of previous works have laid a solid foundation for this new paradigm: the first paper rigorously proved the exact relation between internal dissipation power and energy relaxation rate, providing a mathematical link for the energy perspective; the second paper established a single-body energy distribution theory, defined the core order parameter—the energy distribution ratio f, and proved that the relaxation rate is entirely determined by the microstructure of D; the third paper extended the theory to many-body systems and, under the Markovian approximation, derived a mesoscopic diffusion equation describing energy transport within the D network. As the fourth paper in this series, this work completely abandons the Markovian approximation and constructs a universal theory of non-Markovian mesoscopic dynamics from first principles. Taking the energy density u(r,t) as the fundamental field variable, this paper derives an integro-differential equation containing memory kernels K0 and K2 by tracing over the degrees of freedom describing the defect network D and implementing spatial coarse-graining—termed herein the \textbf{generative master equation}. The essence of this equation lies in the fact that the memory kernels are completely determined by the spatial correlation function CD(+)(s,t) of the defect network D, which, as the system's "microscopic fingerprint," uniquely encodes all dynamical information of D. By selecting different forms of the correlation function, the master equation can naturally reduce to or transform into almost all known macroscopic transport equations in physics: in the Markovian limit, it returns to the diffusion equation; exponential correlation yields the telegraph equation and a finite propagation speed v=D0/τc; power-law correlation gives rise to fractional diffusion equations; nonlinear correlation leads to nonlinear diffusion equations. This unified mapping reveals the deep-seated role of the theory presented herein: it does not seek to replace existing theories but rather to provide them with a common microscopic origin. This paper also establishes a quantitative criterion for energy backflow and directly links it to the energy distribution ratio f, opening new avenues for the experimental inversion of memory kernels. This work marks the evolution of the energy distribution framework from a specific model into a generative mother theory capable of uniformly describing various transport behaviors. \textbf{Keywords}: Generative master equation; Energy distribution framework; Non-Markovian dynamics; Mesoscopic dynamics; Memory kernel; Green's function; Telegraph equation; Finite propagation speed
The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application.