Master equation

About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.

Interrelations between stochastic equations for systems with pair interactions

Abstract: Several types of stochastic equations are important in thermodynamics, chemistry, evolutionary biology, population dynamics and quantitative social science. For systems with pair interactions four different types of equations are derived, starting from a master equation for the state space: First, general mean value and (co)variance equations. Second, Boltzmann-like equations. Third, a master equation for the configuration space allowing transition rates which depend on the occupation numbers of the states. Fourth, a Fokker-Planck equation and a “Boltzmann-Fokker-Planck equation”. The interractions of these equations and the conditions for their validity are worked out clearly. A procedure for a self-consistent solution of the nonlinear equations is proposed. Generalizations to interactions between an arbitrary number of systems are discussed.

A phase-space study of Bloch–Redfield theory

Abstract: A phase-space representation of Bloch–Redfield theory is used to describe the dynamical evolution of quantum dissipative systems. The resulting Liouville operator equations are capable of incorporating both the master equation in eigenstate space and the stochastic equation in classical phase space, and thus provide a useful framework for mixing classical, semiclassical, and quantum dynamics for simulating complicated dissipative systems. In addition, the proper limit of quantum dissipation, the approximate nature of the second-order cumulant truncation, the detailed balance of quantum correlation functions, and the reduction of dissipation by a transformation of the bath Hamiltonian are investigated within the framework of phase-space Bloch–Redfield theory.

Mean-field stochastic theory for species-rich assembled communities

Abstract: A dynamical model of an ecological community is analyzed within a ‘‘mean-field approximation’’ in which one of the species interacts with the combination of all of the other species in the community. Within this approximation the model may be formulated as a master equation describing a one-step stochastic process. The stationary distribution is obtained in closed form, and is shown to reduce to a log-series or log-normal distribution, depending on the values that the parameters describing the model take on. A hyperbolic relationship between the connectance of the matrix of interspecies interactions and the average number of species exists for a range of parameter values. The time evolution of the model at short and intermediate times is analyzed using van Kampen’s approximation, which is valid when the number of individuals in the community is large. Good agreement with numerical simulations is found. The large time behavior, and the approach to the stationary state, is obtained by solving the equation for the generating function of the probability distribution. The analytical results which follow from the analysis are also in good agreement with direct simulations of the model.

Local vs global master equation with common and separate baths: superiority of the global approach in partial secular approximation

Abstract: Open systems of coupled qubits are ubiquitous in quantum physics. Finding a suitable master equation to describe their dynamics is therefore a crucial task that must be addressed with utmost attention. In the recent past, many efforts have been made toward the possibility of employing local master equations, which compute the interaction with the environment neglecting the direct coupling between the qubits, and for this reason may be easier to solve. Here, we provide a detailed derivation of the Markovian master equation for two coupled qubits interacting with common and separate baths, considering pure dephasing as well as dissipation. Then, we explore the differences between the local and global master equation, showing that they intrinsically depend on the way we apply the secular approximation. Our results prove that the global approach with partial secular approximation always provides the most accurate choice for the master equation when Born-Markov approximations hold, even for small inter-system coupling constants. Using different master equations we compute the stationary heat current between two separate baths, the entanglement dynamics generated by a common bath, and the emergence of spontaneous synchronization, showing the importance of the accurate choice of approach.

Density-operator approaches to transport through interacting quantum dots: Simplifications in fourth-order perturbation theory

Abstract: Various theoretical methods address transport effects in quantum dots beyond single-electron tunneling while accounting for the strong interactions in such systems. In this paper we report a detailed comparison between three prominent approaches to quantum transport: the fourth-order Bloch-Redfield quantum master equation (BR), the real-time diagrammatic technique (RT), and the scattering rate approach based on the T-matrix (TM). Central to the BR and RT is the generalized master equation for the reduced density matrix. We demonstrate the exact equivalence of these two techniques. By accounting for coherences (nondiagonal elements of the density matrix) between nonsecular states, we show how contributions to the transport kernels can be grouped in a physically meaningful way. This not only significantly reduces the numerical cost of evaluating the kernels but also yields expressions similar to those obtained in the TM approach, allowing for a detailed comparison. However, in the TM approach an ad hoc regularization procedure is required to cure spurious divergences in the expressions for the transition rates in the stationary (zero-frequency) limit. We show that these problems derive from incomplete cancellation of reducible contributions and do not occur in the BR and RT techniques, resulting in well-behaved expressions in the latter two cases. Additionally, we show that a standard regularization procedure of the TM rates employed in the literature does not correctly reproduce the BR and RT expressions. All the results apply to general quantum dot models and we present explicit rules for the simplified calculation of the zero-frequency kernels. Although we focus on fourth-order perturbation theory only, the results and implications generalize to higher orders. We illustrate our findings for the single impurity Anderson model with finite Coulomb interaction in a magnetic field.