Expansion Formulas in Nonequilibrium Statistical Mechanics

Shibata, Fumiaki; Arimitsu, Toshihico

doi:10.1143/jpsj.49.891

Cited by 250 publications

(133 citation statements)

References 15 publications

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“…Even though the units of time can be arbitrary, by doing so we do not lose generality, since we will be working in the interaction picture where only the frequencies Ω n appear in relation to the state of the bath [Eq. (12)]. Since the ratios of these frequencies and the temperature of the bath occur in the equations, only their values relative to the temperature are of interest.…”

Section: A the Modelmentioning

confidence: 99%

“…Because the model we consider is exactly solvable, we are able to accurately assess the performance of the approximation techniques that we study. In particular, we study the Born-Markov and Born master equations, and the perturbation expansions of the Nakajima-Zwanzig (NZ) [9,10] and the timeconvolutionless (TCL) master equations [11,12] up to fourth order in the coupling constant. We also study the post-Markovian (PM) master equation proposed in [13].…”

Section: Introductionmentioning

confidence: 99%

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Non-Markovian dynamics of a qubit coupled to an Ising spin bath

et al. 2007

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We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed post-Markovian master equation performs comparably to the time-convolutionless master equation for a properly chosen memory kernel, and outperforms all the approximation methods considered here at long times. Our findings shed light on the applicability of master equations to the description of reduced system dynamics in the presence of spin-baths.

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Section: A the Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-Markovian dynamics of a qubit coupled to an Ising spin bath

et al. 2007

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“…[13,14]. The Nakajima-Zwanzig formalism [15,16] and the timeconvolutionless projection operator method [17,18,19] have proved to be useful in deriving approximations based on projection operator techniques. The latter method, employing correlated projection superoperators, was recently used to derive a non-Markovian generalization of the Lindblad equation [20].…”

Section: Introductionmentioning

confidence: 99%

Stochastic wave-function unraveling of the generalized Lindblad master equation

Moodley

Petruccione

2009

Phys. Rev. A

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Recently a generalized master equation was derived that extends the Lindblad theory to highly non-Markovian quantum processes (H.-P. Breuer, Phys. Rev. A 75, 022103 (2007)). We perform a stochastic unravelling of this master equation by considering n random state vectors that satisfy the corresponding stochastic differential equation for a piecewise deterministic process. As an application we consider a two-state system randomly coupled to an environment consisting of two energy bands with finite number of levels. Our numerical results are compared to results obtained from the time-convolutionless (TCL) projection operator method using correlated projection superoperators and the exact solution of the Schrödinger equation for this system.

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“…Much effort is directed towards extending the class of systems which can be solved analytically, but only little attention is paid to fully exploit the power of a perturbative treatment: Usually the expansion of the equation of motion is done to second order only and the Born-Markov approximation is applied [10,11]. Higher order approximations of the equation of motion which are valid beyond the BornMarkov approximation can be obtained in a systematic way by exploiting the time-convolutionless projection operator technique [12][13][14]. Especially the expansion to fourth order can be used for a quantitative and qualitative analysis of a system in the vicinity of the Markovian regime.…”

Section: Introductionmentioning

confidence: 99%