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

EPJ Web of Conferences 108, 0 2 02 7 (2016 ) DOI: 10.1051/epjconf/ 2016 10 8 0 2 02 7  C Owned by the authors, published by EDP Sciences, 2016 Operator Approach to the Master Equation for the One-Step Process M. Hnatič3,4,5 , a , E. G. Eferina1 , b , A. V. Korolkova1 , c , D. S. Kulyabov1,2 , d , and L. A. Sevastyanov1,3 , e 1 Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia, Miklukho-Maklaya str. 6, Moscow, 117198, Russia 2 Laboratory of Information Technologies, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna, Moscow region, 141980, Russia 3 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna, Moscow region, 141980, Russia 4 Department of Theoretical Physics, SAS, Institute of Experimental Physics, Watsonova 47, 040 01 Košice, Slovakia 5 Faculty of Science, P. J. Šafárik University, Šrobárova 2, 041 54 Košice, Slovakia 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. 1 Introduction In order to construct stochastic models of the one-step processes [1] (birth–death processes) the combinatorial methodology based on N. G. van Kampen [2] and C. W. Gardiner [3] ideology was a e-mail: b e-mail: c e-mail: d e-mail: e e-mail:                4                   Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/201610802027 EPJ Web of Conferences worked out. Under this methodology the master equation (for one-step processes) is derived by using the interaction schemes. The obtained master equation is further converted to the Fokker–Planck equation by expansion in formal series (Kramers–Moyal’s decomposition) [3]. However, it is necessary to study the possibility of using this expansion for each type of process. Thus, it is necessary not only to study the master equation but also to justify its expansion. It seems that the quantum perturbation theory best fits all the requirements. There are two types of formalism which are generally used in the quantum perturbation theory: the path integral formalism and the formalism of second quantization (canonical formalism). It is worthy of note that it is a matter of taste which type of formalism to use. In a number of works [4–7] 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. The structure of the article is as follows. Section 2 contains a brief introduction to the method of stochastization of one-step processes. Section 3 describes the algorithm of the one-step processes recording in terms of the occupation number representation. The master equation in the form of the Liouville operator equation is also presented. In the section 4 case study model for both the combinatorial and operator approaches is described. The equivalence of the combinatorial and the operator approaches is proved. 2 One–step processes stochastization The one–step processes (also known as the birth–death processes) are Markov processes with continuous time, integer state of states the transition matrix of which allows only transitions between neighbouring states. 2.1 Interaction schemes The system state is defined by the vector ϕi ∈ Rn , where n is system order1 . The operator I ij ∈ Nn0 × Nn0 describes the state of the system before the interaction, the operator F ij ∈ Nn0 × Nn0 is the state after the interaction2 . The result of the interaction is the system transition from one state to another one. The interaction of the system elements will be described by interaction schemes which are similar to the schemes of the chemical kinetics: + kα j j F iα I iα j ϕ  j ϕ , − (1) kα the Greek indices specify the number of interactions and the Latin ones the system order. The coefficients +kα and −kα have the meaning of intensity (speed) of the interaction. iα iα The state transition is given by the operator: riα j = F j − I j . Thus, the one step interaction α in the iα forward and reverse directions can be written as ϕi → ϕi + r j ϕ j , iα ϕi → ϕi − r j ϕ j . + kα j j We can also write (1) not as vector equations but as sums: I iα F iα j ϕ δi  j ϕ δi , where δi = (1, . . . , 1). − kα iα j iα iα j iα j Also the following notation will be used: I iα := I iα j δ , F := F j δ , r := r j δ . 1 For brevity, we denote the module over the field R just as R. 2 The component dimension indices take values i, j = 1, n. 02027-p.2 Mathematical Modeling and Computational Physics 2015 2.2 The master equation For the system description we will use the master equation, which describes the transition probability for a Markov process [2, 3]. If the domain of variation of ϕ is discrete, then the master equation can be written as follows (the states are numbered by n and m):  ∂pn (t)   wnm pm (t) − wmn pn (t) , = ∂t m (2) where pn is the probability to find the system in the state n at the time t, wnm is the probability of the transition from the state m to the state n per unit time. There are two types of system transitions from one state to another (based on one–step processes) as iα a result of system elements interaction: in the forward direction (ϕi +r j ϕ j ) with probability + sα (ϕk ) and iα in the opposite direction (ϕi − r j ϕ j ) with probability − sα (ϕk ). The matrix of the transition probabilities has the form: wα (ϕi |ψi , t) = + sα δϕi ,ψi +1 + − sα δϕi ,ψi −1 , where δi, j is Kronecker delta. Thus, the general form of the master equati (...truncated)