Field theory approach in kinetic reaction: Role of random sources and sinks

Hnatich, M.; Honkonen, Juha; Lučivjanský, T.

doi:10.1007/s11232-011-0125-8

Cited by 7 publications

(6 citation statements)

References 5 publications

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“…6 The attractiveness of this model is that it is one-dimensional and non-linear. 7 The same notation as in the original model [11] is used.…”

Section: One-step Processes Stochastization Methodsmentioning

confidence: 99%

“…As a demonstration of the method, we consider the Verhulst model [11], which describes the limited growth 6 . Initially, this model was written down as the differential equation: dϕ dt = λϕ − βϕ − γϕ 2 , where λ denotes the breeding intensity factor, β -the extinction intensity factor, γ -the factor of population reduction rate (usually the rivalry of individuals is considered) 7 .…”

Section: Verhulst Modelmentioning

confidence: 99%

“…It is worthy of note that it is a matter of taste which type of formalism to use. In a number of works [4][5][6][7] the possibility of using the formalism of the second quantization for statistical tasks was studied. However, these articles are intended for theoretical physicist and that strongly limits the audience that could use the scientific results of the articles.…”

Section: Introductionmentioning

confidence: 99%

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Operator Approach to the Master Equation for the One-Step Process

Hnatič

Eferina

Korolkova

et al. 2016

EPJ Web of Conferences

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Abstract. Background. Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. The expansion of the equation in a formal Taylor series (the so called Kramers-Moyal's expansion) is used in the procedure of stochastization of one-step processes. Purpose. However, this does not eliminate the need for the study of the master equation. Method. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). Results. This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. Conclusions. We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation.

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“…6 The attractiveness of this model is that it is one-dimensional and non-linear. 7 The same notation as in the original model [11] is used.…”

Section: One-step Processes Stochastization Methodsmentioning

confidence: 99%

Section: Verhulst Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Operator Approach to the Master Equation for the One-Step Process

Hnatič

Eferina

Korolkova

et al. 2016

EPJ Web of Conferences

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“…The computational approach allows to obtain a concrete solution for the studied model. In our methodology, this approach is associated with perturbation theory (see Hnatič et al (2013); Hnatich and Honkonen (2000); Hnatich et al (2011)). Methodologically, this method is quite simple.…”

Section: General Review Of the Methodologymentioning

confidence: 99%

Stochastization Of One-Step Processes In The Occupations Number Representation

Korolkova¹,

Eferina²,

Laneev³

et al. 2016

ECMS 2016 Proceedings Edited by Thorsten Claus, Frank Herrmann, Michael Manitz, Oliver Rose

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By the means of the method of stochastization of onestep processes we get the simplified mathematical model of the original stochastic system. We can explore these models by standard methods, as opposed to the original system. The process of stochastization depends on the type of the system under study. We want to get a unified abstract formalism for stochastization of one-step processes. This formalism should be equivalent to the previously introduced. To implement an abstract approach we use the representation of occupation numbers. In this presentation we use the operator formalism. A feature of this formalism is the use of abstract linear operators which are independent from the state vectors. We use the formalism of Green's functions in order to deal with operators. We get a fully coherent formalism by using the occupation numbers representation. With its help we can get simplified stochastic model of the original system. We demonstrate the equivalence of the occupation number representation and the state vectors representation by using a one-step process example. We have suggested a convenient formalism for unified description of stochastic systems. Also, this method can be extended for the study of nonlinear stochastic systems.

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“…Our team has developed a method of stochastization of one-step processes in order to build stochastic models from first principles. For different applications, we used different representations, namely the representation of state vectors (combinatorial approach) [3,4] and the occupation number representation (operatorial approach) [4][5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%