## Abstract

**Background.** By the means of the method of stochastization of one-step 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. **Purpose.** We want to get a unified abstract formalism for stochastization of one-step processes. This formalism should be equivalent to the previously introduced. **Methods.** To unify the methods of construction of the master equation, we propose to use the diagram technique. **Results.** We get a diagram technique, which allows to unify getting master equation for the system under study. We demonstrate the equivalence of the occupation number representation and the state vectors representation by using a Verhulst model. **Conclusions.** We have suggested a convenient diagram formalism for unified construction of stochastic systems.

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## Notes

- 1.
The analogs of the interaction schemes are the equations of chemical kinetics, reaction particles and etc.

- 2.
In 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.

- 3.
We denote the module over the field \(\mathbb {R}\) as \(\mathfrak {R}\). Accordingly, \(\mathfrak {N}\), \(\mathfrak {N}_{0}\), \(\mathfrak {N}_{+}\) are modules over rings \(\mathbb {N}\), \(\mathbb {N}_{0}\) (cardinal numbers with 0), \(\mathbb {N}_{+}\) (cardinal numbers without 0).

- 4.
The component dimension indices take on values \(\underline{{i}},\underline{{j}} = \overline{1,n}\).

- 5.
The component indices of number of interactions take on values \(\underline{{\alpha }} = \overline{1,s}\).

- 6.
Master equation can be considered as an implementation of the Kolmogorov equation. However, the master equation is more convenient and has an immediate physical interpretation (see [18]).

- 7.
The notation is based on the notation, proposed by G. Grassmann in 1862 (see [2, p. 134]).

- 8.
In this case, we use Hermitian conjugation \({\bullet }^{\dagger }\). The sign of the complex conjugate \({\bullet }^{*}\) in this case is superfluous.

- 9.
In fact, \(a\pi |{n}\rangle - \pi a |{n}\rangle = (n+1)|{n}\rangle - n |{n}\rangle = |{n}\rangle \).

- 10.
In order not to clutter the diagram, we have only one type of interacting entities left in these schemes.

- 11.
In normal ordering product all creation operators are moved so as to be always to the left of all the annihilation operators.

- 12.
The attractiveness of this model is that it is one-dimensional and non-linear.

- 13.
The same notation as in the original model [25] is used.

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## Acknowledgments

The work is partially supported by RFBR grants No’s. 14-01-00628, 15-07-08795, and 16-07-00556. Also the publication was supported by the Ministry of Education and Science of the Russian Federation (the Agreement No. 02.a03.21.0008).

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Eferina, E.G., Hnatich, M., Korolkova, A.V., Kulyabov, D.S., Sevastianov, L.A., Velieva, T.R. (2016). Diagram Representation for the Stochastization of Single-Step Processes. In: Vishnevskiy, V., Samouylov, K., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2016. Communications in Computer and Information Science, vol 678. Springer, Cham. https://doi.org/10.1007/978-3-319-51917-3_42

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