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Quantum-Dot Cellular Automata: The Prospective Technology for Digital Telecommunication Systems
Published in Anwar Sohail, Raja M Yasin Anwar Akhtar, Raja Qazi Salahuddin, Ilyas Mohammad, Nanotechnology for Telecommunications, 2017
Shahram Mohammad Nejad, Ehsan Rahimi
In the above equations, P stands for the state probabilities of charge configurations and is the ensemble average of the charge in the islands. In (12.9), Г denotes a time-dependent transition matrix. In this matrix, the diagonal elements are -∑i≠jΓii and other elements are the transpose of Гi,j [23]. All the states of the system have to be tracked in order to solve (12.9) indirectly in master equation method.One of the drawbacks of this method is that when the number of the states is too large, it is difficult and totally impossible to exploit it. The interesting point is that the QCA generally operates near the ground state, thus the master equation is tractable considering just a few states.
Thin-Film Solar Cells
Published in Joachim Piprek, Handbook of Optoelectronic Device Modeling and Simulation, 2017
Matthias Auf der Maur, Tim Albes, Alessio Gagliardi
The probability pi(t) to find the system in state i at a certain time t changes in a way that is determined by the rate constants aij and the probabilities of only the initial state pi(t) and the possible final states pj(t). Finding solutions of the master equation, be it by analytical or numerical methods, is often complex and not feasible. The kMC method is based on a stochastical framework and provides a numerical approach to obtain the dynamic system evolution based on the time-dependent propagations from state to state. By choosing a transition pathway through a chain of subsequent states, one possible dynamic system evolution is obtained. Such a walk through a pathway of transitions forms a Markov chain. Averaging over a large number of such Markov chains provides an equivalent system behavior as described by the master equation. Therefore, the kMC algorithm is essentially emulating the master equation.
Diffusion processes, stochastic differential equations and applications
Published in Henry C. Tuckwell, Elementary Applications of Probability Theory, 2018
Equation (12.4) is also called a Fokker-Planck equation, especially by physical scientists, who sometimes refer to it as a ‘Master equation’, to emphasize its generality. Proof that this equation follows from the Chapman-Kolmogorov equation (11.14) and the relations (12.1)-(12.3), though not difficult, is rather long and is hence omitted here. Interested readers may refer to, for example, Jaswinski (1970).
Unified theory of nucleation and coarsening in the solid state
Published in Philosophical Magazine, 2023
The evolution of a system is governed by a rate equation, which is called the master equation. Considering a one-step process, the equation takes the following form [14]: This equation can be rewritten as a balance equation: where the flux J in the size space is given by: The zero-flux condition, J = 0 implies that: For a finite system, the zero-flux condition means that the system is in equilibrium because of the constraint balance between each pair of states n and n+1. Moreover, it is common practice to consider the size space as a continuous variable, which is fine when the clusters have more than 100 atoms. Then, for large n, the continuity of the size distribution makes that and consequently . This is likely the reason why transforming the master equation into a differential equation brings some difficulty, since in the continuous size space, when Δr → 0. So, using the detailed balance, the zero-flux condition forces the drift term to be zero, or way too small, preventing a suitable prediction of the nucleation and coarsening processes.
Understanding the temperatures of H3 + and H2 in diffuse interstellar sightlines
Published in Molecular Physics, 2023
Jacques Le Bourlot, Evelyne Roueff, Franck Le Petit, Florian Kehrein, Annika Oetjens, Holger Kreckel
In practice, the 24 energetically lowest rotational levels of are included in the model. Those are the levels for which radiative emission rates as well as collisional excitation/de-excitation rates are known or have been estimated. The highest level in this framework is the ortho level, located above the ground state. We find that the solution to the set of equations (14) recovers very nicely the full results obtained with the Meudon PDR code for diffuse line of sight conditions. The advantage of the master equation approach is that it is much less computationally expensive, and allows for a rapid exploration of the parameter space and the various possible hypotheses concerning the less well-known physical processes.
Numerical calculation of interstitial dumbbell-mediated transport coefficients in dilute crystalline systems with non-truncated correlations
Published in Philosophical Magazine, 2022
Soham Chattopadhyay, Dallas R. Trinkle
To do this, the Green's function method in [42] starts with the ‘Master Equation’ for mass transport, Here, denotes the thermodynamic probability of a state n, and denotes the rate of transition of the system from state n to state per unit time. The Master Equation stems from the assumption that diffusion is a Markovian process, where the residence time of the system at each state is large compared to the time it takes for the system to transition from one state to another. This means that the transition to the next state depends only on the current state.