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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 article, the authors developed methods for calculating the zero-frequency noise for quantum shuttles, i.e., nanoelectromechanical devices where the mechanical motion is quantized.
Abstract: We develop methods for calculating the zero-frequency noise for quantum shuttles, i.e., nanoelectromechanical devices where the mechanical motion is quantized. As a model system we consider a three-dot array, where the internal electronic coherence both complicates and enriches the physics. Two different formulations are presented: (i) quantum regression theorem and (ii) the counting variable approach. It is demonstrated, both analytically and numerically, that the two formulations yield identical results, when the conditions of their respective applicability are fulfilled. We describe the results of extensive numerical calculations for current and current noise (Fano factor), based on a solution of a Markovian generalized master equation. The results for the current and noise are further analyzed in terms of Wigner functions, which help to distinguish different transport regimes (in particular, shuttling versus cotunneling). In the case of weak interdot coupling, the electron transport proceeds via sequential tunneling between neighboring dots. A simple rate equation with the rates calculated analytically from the $P(E)$ theory is developed and shown to agree with the full numerics.
99 citations
TL;DR: In this paper, it was shown that the behavior at very low and very high frequencies can be described in terms of apparent relaxation times, which differ by approximately 50% from the temperature dependence predicted by Parker, but the magnitudes are larger by a factor of 2.
Abstract: In his well‐known theory, Parker assumes that the rotational relaxation of diatomic gases may be described by a single relaxation time which is calculated for the special case of initially nonrotating molecules. In the present theory the evolution of the rotational distribution function is described by a diffusion‐equation approximation to the master equation. This equation is linearized and solved for the case of acoustic waves. The results indicate that the absorption and dispersion of acoustic waves cannot be described by a single relaxation time. However, the behavior at very low and very high frequencies can be described in terms of “apparent relaxation times” which differ by approximately 50%. The temperature dependence of the apparent relaxation times is similar to that predicted by Parker, but the magnitudes are larger by approximately a factor of 2.
98 citations
98 citations
TL;DR: It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard--linear Fokker-Planck equation--may be also related to a family of nonlinear Foksker- planck equations, which is equivalent to the maximum-entropy principle.
Abstract: A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The $H$ theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard---linear Fokker-Planck equation---may be also related to a family of nonlinear Fokker-Planck equations.
98 citations
TL;DR: In this article, the authors analyzed the quantum regime of the dynamical backaction cooling of a mechanical resonator assisted by a driven harmonic oscillator (cavity) and derived the corresponding motional master equation using the Nakajima-Zwanzig formalism.
Abstract: We analyze the quantum regime of the dynamical backaction cooling of a mechanical resonator assisted by a driven harmonic oscillator (cavity). Our treatment applies to both optomechanical and electromechanical realizations and includes the effect of thermal noise in the driven oscillator. In the perturbative case, we derive the corresponding motional master equation using the Nakajima-Zwanzig formalism and calculate the corresponding output spectrum for the optomechanical case. Then we analyze the strong optomechanical coupling regime in the limit of small cavity linewidth. Finally we consider the steady state covariance matrix of the two coupled oscillators for arbitrary input power and obtain an analytical expression for the final mechanical occupancy. This is used to optimize the drive's detuning and input power for an experimentally relevant range of parameters that includes the "ground state cooling" regime.
98 citations