EDN: CHGKRS
Authors & Affiliations
Popykin A.I.
Scientific and Engineering Centre for Nuclear and Radiation Safety, Moscow, Russia
Popykin A.I. – Leading Researcher, Cand. Sci. (Phys.-Math.). Contacts: 2/8, Building 5, Malaya Krasnoselskaya st., Moscow, Russia, 107140. Tel.: +7 (499) 753-05-24, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it..
Abstract
S.B. Shikhov in his monograph “Problems of the Mathematical Theory of Reactors. Linear Analysis” established in the most general conditions that the solution of a linear non-stationary transport equation is represented by a semigroup of operators (hereinafter referred to as a semigroup) in a number of functional spaces and studied its properties. The structure of the spectrum of the (non-stationary) transport operator was established, including the condition for the presence of a leading eigenvalue in it. A leading eigenvalue is a simple real eigenvalue greater in modulus than all the others and which corresponds to a (unique) positive eigenfunction. One of the consequences of the established properties is the presence of an asymptotics of the semigroup, determined by a single exponential, to which a positive distribution in the phase space corresponds. This is a useful property of the non-stationary transport equation for substantiating perturbation theory and other methods for solving it. It is used in the qualitative analysis of pulse and some other experiments and has the following simple physical interpretation. When the initial condition is set, after some time the sensor placed at any point in the configuration space will first register a signal proportional to the neutron flux, which will then become exponentially dependent on time.
The article shows that, in certain cases, also quite general, the mentioned semigroup is Markov, i. e. it corresponds to a random (Markov) process. In this case, the transition probability of the Markov process has the same properties as the semigroup under consideration, i. e. it has exponential asymptotics, which is useful when considering this process.
The article uses minimal information from the theory of Markov processes and provides all the necessary definitions.
Keywords
reactor theory, non-stationary transport equation, smoothness properties of the solution of the transport equation, Cauchy problem, operator semigroups, spectrum of the transport operator, cone of non-negative functions, random modeling of particle transport, theory of random processes, Markov process, ergodic property
Article Text (PDF, in Russian)
References
- Shikhov S.B. Voprosy matematicheskoy teorii reaktorov. Lineynyy analiz [Problems of the Mathematical Theory of Reactors: Linear Analysis]. Moscow, Atomizdat Publ., 1973. 376 p.
- Kochubey Yu.K. Statisticheskoe modelirovanie kineticheskikh protsessov [Statistical Modeling of Kinetic Processes]. Sarov, RFYaTs-VNIIEF Publ., 2004. 246 p.
- Vladimirov V.S. Matematicheskie zadachi odnoskorostnoy teorii perenosa chastits [Mathematical Problems of the Single-Velocity Theory of Particle Transport]. Trudy MIAN im. V.A. Steklova [Proc. of the Steklov Mathematical Institute]. Moscow, Izd-vo AN SSSR Publ., 1961. Vol. 61.
- Venttsel' A.D. Kurs teorii sluchaynykh protsessov [Course in the Theory of Random Processes]. Moscow, Nauka Publ., 1996. 400 p.
- Germogenova T.A. Lokal'nye svoystva resheniya uravneniya perenosa [Local Properties of the Solution of the Transport Equation]. Moscow, Nauka, Fizmatlit Publ., 1986. 272 p.
- Kolmogorov A.N. Ob analiticheskikh metodakh v teorii veroyatnostey [On Analytical Methods in Probability Theory]. UMN – Russian Mathematical Surveys, 1938, issue 5, pp. 5–41.
- Feller W. An Introduction to Probability Theory and Its Applications. Vol. 1. New York London Sydney, John Wiley & Sons Inc., 1968. 527 p.
- Korolyuk V.S., Portenko N.N., Skorokhod A.V., Turbin A.F. Spravochnik po teorii veroyatnostey i matematicheskoy statistike [Handbook of Probability Theory and Mathematical Statistics]. Moscow, Nauka Publ., 1985. 640 p.
- Funktsional'nyy analiz. Pod obshchey redaktsiey S.G. Kreyna. Spravochnaya matematicheskaya biblioteka [Functional Analysis. Ed. S.G. Krein. Reference Mathematical Library]. Moscow, Nauka, Fizmatlit Publ., 1973. 540 p.
- Popykin A.I. O stokhasticheskoy formulirovke zadachi dlya nestatsionarnogo uravneniya perenosa neytronov [On a Stochastic Formulation of the Problem for the Non-Stationary Neutron Transport Equation]. Tezisy dokladov Nauchno-tekhnicheskoy konferentsii “Neytronno-fizicheskie problemy atomnoy energetiki” (Neytronika-2016) [Proc. of the Scientific and Technical Conference “Neutron-Physical Problems of Nuclear Power Engineering” (Neutronica-2016)]. Obninsk, November 23–25, 2016, p. 58.
- Popykin A.I. Reshenie sopryazhennogo nestatsionarnogo uravneniya perenosa i veroyatnost' [Solution of the Adjoint Non-Stationary Transport Equation and Probability]. Tezisy dokladov Nauchno-tekhnicheskoy konferentsii “Neytronno-fizicheskie problemy atomnoy energetiki” (Neytronika-2017). [Proc. of the Scientific and Technical Conference “Neutron-Physical Problems of Nuclear Power Engineering” (Neutronica-2017)]. Obninsk, November 28–30, 2017, pp. 44–45.
- Popykin A.I. Reshenie sopryazhennoy nestatsionarnoy sistemy uravneniy diffuzionnogo mnogogruppovogo priblizheniya i perekhodnaya veroyatnost' markovskogo protsessa [Solution of the Adjoint Non-Stationary System of Equations of the Diffusion Multigroup Approximation and the Transition Probability of a Markov Process]. Tezisy dokladov Nauchno-tekhnicheskoy konferentsii “Neytronno-fizicheskie problemy atomnoy energetiki” (Neytronika-2018) [Proc. of the Scientific and Technical Conference “Neutron-Physical Problems of Nuclear Power Engineering” (Neutronica-2018)]. Obninsk, November 28–30, 2018, pp. 47–48.
- Popykin A.I. Reshenie sopryazhennoy nestatsionarnoy zadachi dlya sopryazhennogo nestatsionarnogo uravneniya perenosa bez ucheta stolknoveniy i perekhodnaya veroyatnost' markovskogo protsessa [Solution of the adjoint non-stationary problem for the adjoint non-stationary transport equation without taking into account collisions and the transition probability of a Markov process]. Tezisy dokladov Nauchno-tekhnicheskoy konferentsii “Neytronno-fizicheskie problemy atomnoy energetiki” (Neytronika-2019) [Proc. of the Scientific and Technical Conference “Neutron-Physical Problems of Nuclear Power Engineering” (Neutronica-2019)]. Obninsk, November 27–29, 2019, pp. 68–70.
- Popykin A.I. Nestatsionarnoe uravnenie perenosa i sootvetstvuyushchiy emu markovskiy sluchaynyy protsess [Non-stationary transport equation and the corresponding Markov random process]. Tezisy dokladov Nauchno-tekhnicheskoy konferentsii “Neytronno-fizicheskie problemy atomnoy energetiki” (Neytronika-2022) [Proc. of the Scientific and Technical Conference “Neutron-Physical Problems of Nuclear Power Engineering” (Neutronica-2022)]. Obninsk, May 31 – June 02, 2022, pp. 40–41.
- Popykin A.I. Dopolnitel'noe svoystvo resheniya nestatsionarnogo uravneniya perenosa [Additional property of the solution of the non-stationary transport equation]. Tezisy dokladov Nauchno-tekhnicheskoy konferentsii “Neytronno-fizicheskie problemy atomnoy energetiki” (Neytronika-2024) [Proc. of the Scientific and Technical Conference “Neutron-Physical Problems of Nuclear Power Engineering” (Neutronica-2024). Obninsk, May 28–31, 2024, pp. 83–84.
UDC 621.039
Problems of Atomic Science and Technology. Series: Nuclear and Reactor Constants, 2025, no. 4, 4:5
