Steady state solutions of master equations without detailed balance

Görtz, R.; Walls, D. F.

doi:10.1007/bf01315258

Cited by 42 publications

(13 citation statements)

References 6 publications

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“…A more refined expression for ~ can be obtained This expression just coincides with the result obtained by GOrtz and Walls in terms of the q-method [5]. But in contrast to the q-method we do not need to consider a quartic equation in q.…”

Section: N(n-1)+(n+ L)n~+ Lsupporting

confidence: 60%

“…In particular, we confine ourselves to a study of systems in which two-particle jumps with rates r/~ + = v i and r~--=co i may occur. Examples are given by the stochastic modeliugs of interacting social groups 1-14-15] or chemical reactions introducing bimolecular reaction steps [3,5,16]. For these cases the stationary birth and death master equation By use of (4.1c) and induction on n we find the recursion relation…”

Section: Stationary Solution Of a Discrete Master Equation With Two-pmentioning

confidence: 99%

“…This reaction has been studied previously by a number of authors [5,16,17]. For the following we denote by n the number of particles of a chemical species X which is the stochastic variable of the model.…”

Section: Application To a Simple Nonlinear Modelmentioning

confidence: 99%

“…However, exact solutions of master equations are known in relatively simple systems only; mainly because exact solutions in terms of the spectral properties are in general too complicated for an explicit evaluation. As a consequence, various tractable approximative solution schemes [5,6] and the transition factor method [7] have been developed. Nevertheless, it is of great interest to find classes of stochastic processes for which the exact stationary as well as time dependent solutions can be obtained in an analytical form which in addition is also appropriate for a computer evaluation [8].…”

Section: Introductionmentioning

confidence: 99%

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Exact solutions of discrete master equations in terms of continued fractions

Haag

Hänggi

1979

Z Physik B

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We present the continued fraction solution for the stationary probability of discrete master equations of one-variable processes. After we elucidate the method for simple birth and death processes we focus the study on processes which introduce at least two-particle jumps. Consequently, these processes do in general not obey a detailed balance condition. The outlined method applies as well to solutions of eigenmodes of the stochastic operator. Further we derive explicit continued fraction solutions for the Laplace transform of conditional probabilities. All the various continued fraction coefficients are given directly in terms of the transition rates and they obey recursion relations. The method is illustrated for the stationary solution of a simple nonlinear chemical reaction scheme originated by Nicolis.

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Section: N(n-1)+(n+ L)n~+ Lsupporting

confidence: 60%

Section: Stationary Solution Of a Discrete Master Equation With Two-pmentioning

confidence: 99%

Section: Application To a Simple Nonlinear Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact solutions of discrete master equations in terms of continued fractions

Haag

Hänggi

1979

Z Physik B

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“…Chap. V), detailed balance is no longer guaranteed [26] so that other procedures are necessary for evaluating the stationary solution Poo(nx) [27]. )…”

Section: T-*oomentioning

confidence: 99%

Information Contained in Open Systems by Example of Chemical Reaction Systems

Decker

1979

Zeitschrift Für Naturforschung A

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Spatially homogeneous chemical reaction systems with one or two intermediate reaction products and with autocatalytic reactionsteps are considered. Because of their non-linearities, such open systems show already primitive forms of self-organization. In order to express the "information" contained in the structures occuring, a theory is developped for measuring that quantity by help of a fictive detector: In treating the stochastic reaction kinetics in the Fokker-Planckequation approximation, expressions are derived for the averaged amount of information one gets by doing a measurement with the detector and for the temporal conservation of the message being detected. This concept is applied to a one-component reaction scheme that exhibits a nonequilibrium phase transition of second order resulting in bistability of the steady state. When pushing this reaction system from the near equilibrium side through its critical region to bistability, a certain amount of information becomes quasi conserved, thus giving rise to a definition of the degree of order of a self-organizating system. The problem of how the reaction system can be integrated into a greater chemical network as a "bit"-generator, is discussed. To explain what is necessary for the onset of a hard mode instability giving birth to limit-cycle behaviour, a two-component reaction scheme is constructed by superposing onto reaction steps causing conservative concentration oscillations those reactions of the former model system which are responsible for the instability occuring there. By applying the information formalism, again, a quasi-conservation of information is indicated, but with respect to a much smaller time scale. The consequences for using oscillating reaction models as an information pump within a network, and the necessity of a feed-back mechanism in order to get real information storage, are shortly mentioned. Finally, a one-component reaction scheme is outlined that shows successive phase transitions, each of these instabilities bringing out a higher degree of organization.

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Photon-Statistics of Stimulated Hyper Raman Scattering in an Inhomogeneously Broadened Three-Level Gaseous System

Gupta

Dash

1993

Recent Developments in Quantum Optics

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Steady state solutions of master equations without detailed balance

Cited by 42 publications

References 6 publications

Exact solutions of discrete master equations in terms of continued fractions

Exact solutions of discrete master equations in terms of continued fractions

Information Contained in Open Systems by Example of Chemical Reaction Systems

Photon-Statistics of Stimulated Hyper Raman Scattering in an Inhomogeneously Broadened Three-Level Gaseous System

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