Stochastization Of One-Step Processes In The Occupations Number Representation

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. INTRODUCTION When modeling various physical and technical systems, we often can model them in the form of a one-step processes (see Demidova et al. (2013, 2014); Velieva et al. (2014); Basharin et al. (2009)). Then there is the problem of adequate representation and study of the resulting model. The formalism of stochastization of one-step processes has been developed by our group for quite a long time. But so far, our efforts have been aimed at getting more models than on their investigation. For the statistical systems in addition to representation of the state vectors (combinatorial approach) the representation of the occupation numbers (operator approach) (see Hnatič et al. (2016); Grassberger and Scheunert (1980); Täuber (2005); Janssen and Täuber (2005); Mobilia et al. (2006)) is also used. This representation is especially well suited for the system with a variable number of elements description. In addition, for this representation there are effective methods for solving equations based on the formalism of Green’s functions and perturbation theory. In this paper, we want to demonstrate the methodology of both approaches. The structure of the article is as follows. In the first section basic notations and conventions are introduced. The ideology of the method of of stochastization of one-step process and its components are described in the second section. Then the interaction schemes and master equation overview are presented in the next section. The combinatorial method of modelling is discussed in the following section. The operator model approach is presented in the last section, where, in Proceedings 30th European Conference on Modelling and Simulation ©ECMS Thorsten Claus, Frank Herrmann, Michael Manitz, Oliver Rose (Editors) ISBN: 978-0-9932440-2-5 / ISBN: 978-0-9932440-3-2 (CD) particular, the algorithm of transition to the occupation number representation is described. NOTATIONS AND CONVENTIONS 1) The abstract indices notation (see Penrose and Rindler (1987)) is used in this work. Under this notation a tensor as a whole object is denoted just as an index (e.g., x), components are denoted by underlined index (e.g., x). 2) We will adhere to the following agreements. Latin indices from the middle of the alphabet (i, j, k) will be applied to the space of the system state vectors. Latin indices from the beginning of the alphabet (a) will be related to the Wiener process space. Greek indices (α) will set a number of different interactions in kinetic equations. GENERAL REVIEW OF THE METHODOLOGY Our methodology is completely formalized in such a way that it is sufficient when the original problem is formulated accordingly. It should be noted that the most of the models under our study can be formalized as a one-step process (see van Kampen (2011); Gardiner (1985)). In fact, for this type of models we developed this methodology, but it may be expanded for other processes. First we transform our model to the one-step process (see Fig. 1). Next, we need to formalize this process in the form of interaction schemes 1 (see Demidova et al. (2013, 2014); Hnatič et al. (2016)). Each scheme has its own interaction semantics. Semantics leads directly to the master equation (see van Kampen (2011); Gardiner (1985)). However, the master equation has usually rather complex structure that makes it difficult for direct study and solution. Our technique involves two possibilities (see Fig. 2): • computational approach — the solution of the master equation with help of perturbation theory; • modeling approach — the approximate models are obtained in the form of Fokker–Planck and Langevin equations. 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. Each expansion element appears in the form of of Feynman diagrams. However, with increasing order of the expansion, the number of Feynman diagrams increases rapidly and can reach tens or hundreds of thousands. It is quite natural that this should involve high-performance computing. The model approach provides a model that is convenient to study numerically and qualitatively. In addition, this approach assumes the iterative process of research: the obtained approximate model can be specified and changed, which leads to the correction of initial interaction schemes. 1The analogs of the interaction schemes are the equations of chemical kinetics, reaction particles and etc. In this article we will describe the model approach. There are two ways of building the master equation2 • combinatorial approach (see Fig. 3); • operator approach (see Fig. 4). In the combinatorial approach, all operations are performed in the space of states of the system, so we deal with a particular system throughout manipulations with the model. For the operator approach we can abstract from the specific implementation of the system under study. We are working with abstract operators. We return to the state space only at the end of the calculations. In addition, we choose a particular operator algebra on the basis of symmetry of the problem. These two approaches are belong to different paradigms of physical theories construction. Accordingly, they are complementary. Some constructions are simpler in one approach, others are simpler in another. For example, in the combinatorial approach, the process of obtaining the approximate models is more convenient, but not more easy. For perturbation theory it is easier to expand in a series the master equation by using the operator formalism. In addition, the operator formalism is suitable for describing the transient processes and nonstationary statistical systems. Interaction schemes The system state is defined by the vector φ ∈ R, where n is system dimension 3. The operator I j ∈ N0×N0 describes the state of the system before the interaction, the operator F i j ∈ N0×N0 describes the state of the system after the interaction4. The result of interaction is the system transition from one state to another one. There are s types of interaction in our system, so instead of I j and F i j operators we will use operators I iα j ∈ N0×N0×N+ and F iα j ∈ N0 ×N0 ×N+. The interaction of the system elements will be described by interaction schemes, which are similar to schemes of chemical kinetics Waage and Gulberg (1986); Gorban and Yablonsky (2015): I iα j φ j k + α −−−⇀ ↽ − k − α F iα j φ j , α = 1, s, (1) the Greek indices specify the number of interactions and Latin are the system order. The coefficients k + α and k − α have meaning intensity (speed) of interaction. The state transition is given by the operator: r iα j = F iα j − I iα j . (2) 2In quantum field theory the path integrals approach can be considered as an analogue of the combinatorial approach and the method of second quantization as analog of the operator approach. 3For brevity, we denote the module over the field R just as R. Accordingly, N, N0, N+ are modules over rings N, N0 (cardinal numbers with 0), N+ (cardinal numbers without 0). 4The component dimension indices take on values i, j = 1, n 5The component indices of number of interactions take on values α = 1, s 0 1 2 . . . i− 1 i i+ 1 . . . n . . . s + 0