Programming and Computer Software

, Volume 44, Issue 2, pp 86–93 | Cite as

Implementing a Method for Stochastization of One-Step Processes in a Computer Algebra System

  • M. N. Gevorkyan
  • A. V. Demidova
  • T. R. Velieva
  • A. V. Korol’kova
  • D. S. Kulyabov
  • L. A. Sevast’yanov


When modeling such phenomena as population dynamics, controllable flows, etc., a problem arises of adapting the existing models to a phenomenon under study. For this purpose, we propose to derive new models from the first principles by stochastization of one-step processes. Research can be represented as an iterative process that consists in obtaining a model and its further refinement. The number of such iterations can be extremely large. This work is aimed at software implementation (by means of computer algebra) of a method for stochastization of one-step processes. As a basis of the software implementation, we use the SymPy computer algebra system. Based on a developed algorithm, we derive stochastic differential equations and their interaction schemes. The operation of the program is demonstrated on the Verhulst and Lotka–Volterra models.


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  1. 1.
    Gardiner, C., Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer, 2009, 4th ed.zbMATHGoogle Scholar
  2. 2.
    Van Kampen, N.G., Stochastic Processes in Physics and Chemistry, Elsevier, 2011.zbMATHGoogle Scholar
  3. 3.
    Korolkova, A.V., Eferina, E.G., Laneev, E.B., Gudkova, I.A., Sevastianov, L.A., and Kulyabov, D.S., Stochastization of one-step processes in the occupations number representation, Proc. 30th Eur. Conf. Modeling and Simulation, 2016, pp. 698–704.Google Scholar
  4. 4.
    Eferina, E.G., Hnatich, M., Korolkova, A.V., Kulyabov, D.S., Sevastianov, L.A., and Velieva, T.R., Diagram representation for the stochastization of singlestep processes, Communications in Computer and Information Science, Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V., Eds., Springer, 2016, vol. 678, pp. 483–497.CrossRefGoogle Scholar
  5. 5.
    Hnatific, M., Eferina, E.G., Korolkova, A.V., Kulyabov, D.S., and Sevastyanov, L.A., Operator approach to the master equation for the one-step process, EPJ Web of Conferences, 2015, vol. 108, pp. 58–59.Google Scholar
  6. 6.
    Gevorkyan, M.N., Demidova, A.V., Zaryadov, I.S., Sobolewski, R., Korolkova, A.V., Kulyabov, D.S., and Sevastianov, L.A., Approaches to stochastic modeling of wind turbines, Proc. 31st Eur. Conf. Modeling and Simulation (ECMS), Varadi, K., Vidovics-Dancs, A., Radics, J.P., Paprika, Z.Z., Zwierczyk, P.T., and Horak, P., Eds., Budapest: European Council for Modeling and Simulation, 2017, pp. 622–627.Google Scholar
  7. 7.
    Gevorkyan, M.N., Velieva, T.R., Korolkova, A.V., Kulyabov, D.S., and Sevastyanov, L.A., Stochastic Runge–Kutta software package for stochastic differential equations, Dependability Engineering and Complex Systems, Springer, 2016, vol. 470, pp. 169–179.Google Scholar
  8. 8.
    Grassberger, P. and Scheunert, M., Fock-space methods for identical classical objects, Fortschritte der Physik, 1980, vol. 28, no. 10, pp. 547–578.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Täuber, U.C., Field-theory approaches to nonequilibrium dynamics, Ageing Glass Transition, 2005, vol. 716, pp. 295–348.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Janssen, H.-K. and Täuber, U.C., The field theory approach to percolation processes, Ann. Phys., 2005, vol. 315, no. 1, pp. 147–192.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mobilia, M., Georgiev, I.T., and Täuber, U.C., Fluctuations and correlations in lattice models for predator–prey interaction, Phys. Rev. E, 2006, vol. 73, no. 4, p. 040903.CrossRefGoogle Scholar
  12. 12.
    Penrose, R. and Rindler, W., Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge Univ. Press, 1987.zbMATHGoogle Scholar
  13. 13.
    Demidova, A.V., Korolkova, A.V., Kulyabov, D.S., and Sevastyanov, L.A., The method of constructing models of peer to peer protocols, Proc. 6th Int. Congr. Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), IEEE Computer Society, 2015, pp. 557–562.Google Scholar
  14. 14.
    Bainov, D.D. and Hristova, S.G., Differential Equations with Maxima, Chapman and Hall/CRC, 2011.zbMATHGoogle Scholar
  15. 15.
    Timberlake, T. K. and Mixon, J. W., Classical mechanics with maxima, Undergraduate Lecture Notes in Physics, New York: Springer, 2016.CrossRefzbMATHGoogle Scholar
  16. 16.
    Jenks, R.D. and Sutor, R.S., AXIOM: The Scientific Computation System, Springer, 1992.zbMATHGoogle Scholar
  17. 17.
    Eferina, E.G., Korolkova, A.V., Gevorkyan, M.N., Kulyabov, D.S., and Sevastyanov, L.A., One-step stochastic processes simulation software package, Bull. Peoples’ Friendship Univ. Russia, Ser. Math., Inf. Sci., Phys., 2014, no. 3, pp. 46–59.Google Scholar
  18. 18.
    Hindley, R., The principal type-scheme of an object in combinatory logic, Trans. Am. Math. Soc., 1969, vol. 146, p. 29.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Milner, R., A theory of type polymorphism in programming, J. Comput. Syst. Sci., 1978, vol. 17, no. 3, pp. 348–375.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lamy, R., Instant SymPy Starter, Packt Publishing, 2013, p. 52.Google Scholar
  21. 21.
    Eferina, E.G. and Kulyabov, D.S., Implementation of diagram technique for statistical systems in Sympy, Proc. 6th Int. Conf. Problems of Mathematical Physics and Mathematical Modeling, Moscow: NRNU MEPhI, 2017, pp. 125–127.Google Scholar
  22. 22.
    Perez, F. and Granger, B.E., IPython: A system for interactive scientific computing, Comput. Sci. Eng., 2007, vol. 9, no. 3, pp. 21–29.CrossRefGoogle Scholar
  23. 23.
    Oliphant, T.E., Python for scientific computing, Comput. Sci. Eng., 2007, vol. 9, no. 3, pp. 10–20.CrossRefGoogle Scholar
  24. 24.
    Oliphant, T.E., Guide to NumPy, CreateSpace Independent Publishing, 2015, 2nd ed.Google Scholar
  25. 25.
    Verhulst, P.F., Notice sur la loi que la population suit dans son accroissement, 1838, vol. 10, pp. 113–117.Google Scholar
  26. 26.
    Feller, W., Die Grundlagen der Volter-raschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung, Acta Biotheoretica, 1939, vol. 5, no. 1, pp. 11–40.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Feller, W., On the theory of stochastic processes, with particular reference to applications, Proc. 1st Berkeley Symp. Mathematical Statistics and Probability, 1949, pp. 403–432.Google Scholar
  28. 28.
    Lotka, A.J., Contribution to the theory of periodic reaction, J. Phys. Chem. A, 1910, vol. 14, no. 3, pp. 271–274.CrossRefGoogle Scholar
  29. 29.
    Lotka, A.J., Elements of Physical Biology, Williams and Wilkins Company, 1925.zbMATHGoogle Scholar
  30. 30.
    Volterra, V., Variations and fluctuations of the number of individuals in animal species living together, Journal du Conseil permanent International pour l' Exploration de la Mer, 1928, vol. 3, no. 1, pp. 3–51.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • M. N. Gevorkyan
    • 1
  • A. V. Demidova
    • 1
  • T. R. Velieva
    • 1
  • A. V. Korol’kova
    • 1
  • D. S. Kulyabov
    • 1
    • 2
  • L. A. Sevast’yanov
    • 1
    • 3
  1. 1.Department of Applied Probability and InformaticsPeoples’ Friendship University of Russia (RUDN)MoscowRussia
  2. 2.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubna, Moscow oblastRussia
  3. 3.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubnaRussia

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