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Master equation
About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.
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TL;DR: In this paper, the authors study the Deffuant et al. model for continuous-opinion dynamics under the influence of noise and derive the master equation of this process.
Abstract: We study the Deffuant et al. model for continuous--opinion dynamics under the influence of noise. In the original version of this model, individuals meet in random pairwise encounters after which they compromise or not depending of a confidence parameter. Free will is introduced in the form of noisy perturbations: individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside the whole opinion space. We derive the master equation of this process. One of the main effects of noise is to induce an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered state a set of well defined opinion groups are formed, although with some opinion spread inside them. Using a linear stability analysis we can derive approximate conditions for the transition between opinion groups and the disordered state. The master equation analysis is compared with direct Monte-Carlo simulations. We find that the master equation and the Monte-Carlo simulations do not always agree due to finite-size induced fluctuations that we analyze in some detail.
78 citations
01 Jun 1988
TL;DR: In this article, a theoretical study of vibrational excitations and dissociations of nitrogen undergoing a nonequilibrium relaxation process upon heating and cooling is reported, and the rate coefficients for collisional induced vibrational transitions and transitions from a bound vibrational state into a dissociative state are calculated using an extension of the theory originally proposed by Schwarz (SSH) et al. (1952).
Abstract: A theoretical study of vibrational excitations and dissociations of nitrogen undergoing a nonequilibrium relaxation process upon heating and cooling is reported. The rate coefficients for collisional induced vibrational transitions and transitions from a bound vibrational state into a dissociative state have been calculated using an extension of the theory originally proposed by Schwarz (SSH) et al. (1952). High-lying vibrational states and dissociative states were explicitly included but rotational energy transfer was neglected. The transition probabilities calculated from the SSH theory were fed into the master equation, which was integrated numerically to determine the population distribution of the vibrational states as well as bulk thermodynamic properties. The results show that: (1) the transition rates have a minimum near the middle of the bound vibrational levels, causing a bottleneck in the vibrational relaxation and dissociation rates; (2) high vibrational states are always in equilibrium with the dissociative state; (3) for the heating case, only the low vibrational states relax according to the Landau-Teller theory; (4) for the cooling case, vibrational relaxation cannot be described by a rate equation; (5) Park's (1985, 1988) two-temperature model is approximately valid; and (6) the average vibrational energy removed in dissociation is about 30 percent of themore » dissociation energy. 29 references.« less
78 citations
TL;DR: It is shown that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models, and identifies that exchanging the limits of infinite system size and infinite time is problematic.
78 citations
TL;DR: In this paper, the authors present a computational study of elementary bifurcations that occur in a driven and damped cavity quantum electrodynamics (cavity QED) model at low intracavity photon number.
Abstract: The interplay of quantum fluctuations with nonlinear dynamics is a central topic in the study of open quantum systems, connected to fundamental issues (such as decoherence and the quantum-classical transition) and practical applications (such as coherent information processing and the development of mesoscopic sensors and amplifiers) With this context in mind, we here present a computational study of some elementary bifurcations that occur in a driven and damped cavity quantum electrodynamics (cavity QED) model at low intracavity photon number In particular, we utilize the single-atom cavity QED master equation and associated stochastic Schrodinger equations to characterize the equilibrium distribution and dynamical behavior of the quantized intracavity optical field in parameter regimes near points in the semiclassical (mean-field, Maxwell-Bloch) bifurcation set Our numerical results show that the semiclassical limit sets are qualitatively preserved in the quantum stationary states, although quantum fluctuations apparently induce phase diffusion within periodic orbits and stochastic transitions between attractors We restrict our attention to an experimentally realistic parameter regime
78 citations
TL;DR: In this paper, the effects of phase noise and particle loss on the dynamics of a Bose-Einstein condensate in an optical lattice were investigated, and the applicability of generalized mean-field approximations in the presence of dissipation as well as methods to simulate quantum effects beyond mean field by including higher-order correlation functions.
Abstract: We investigate the effects of phase noise and particle loss on the dynamics of a Bose-Einstein condensate in an optical lattice. Starting from the many-body master equation, we discuss the applicability of generalized mean-field approximations in the presence of dissipation as well as methods to simulate quantum effects beyond mean field by including higher-order correlation functions. It is shown that localized particle dissipation leads to surprising dynamics, as it can suppress decay and restorethe coherence of a Bose-Einstein condensate. These effects can be applied to engineer coherent structures such as stable discrete breathers and dark solitons.
78 citations