Atoms and Photons: Kinetic Equations with Delay
In his recent works, the author drew attention to the fact that the extraction of a part of a closed Hamiltonian system turns the original well-known Liouville differential equation into an integro-differential equation with a delayed time argument, which describes the dynamics of the selected subsy...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 260; no. 3; pp. 335 - 370 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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01.01.2022
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| Abstract | In his recent works, the author drew attention to the fact that the extraction of a part of a closed Hamiltonian system turns the original well-known Liouville differential equation into an integro-differential equation with a delayed time argument, which describes the dynamics of the selected subsystem as an open system. It was shown that the integral operator can be represented in the form of a fractional differential operator of distributed order. In this paper, we show how the kinetic theory of the system “atoms+photons” is transformed for the subsystem consisting of excited atoms. We present the derivation of the telegraph equation with delay, derive the Biberman–Holstein equation in the fractional differential form (with the fractional Laplace operator), and discuss boundary effects in the nonlocal transport model. The final section is devoted to laser technologies, which include free-electron lasers and laser cooling of atoms. |
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| AbstractList | In his recent works, the author drew attention to the fact that the extraction of a part of a closed Hamiltonian system turns the original well-known Liouville differential equation into an integro-differential equation with a delayed time argument, which describes the dynamics of the selected subsystem as an open system. It was shown that the integral operator can be represented in the form of a fractional differential operator of distributed order. In this paper, we show how the kinetic theory of the system “atoms+photons” is transformed for the subsystem consisting of excited atoms. We present the derivation of the telegraph equation with delay, derive the Biberman–Holstein equation in the fractional differential form (with the fractional Laplace operator), and discuss boundary effects in the nonlocal transport model. The final section is devoted to laser technologies, which include free-electron lasers and laser cooling of atoms. In his recent works, the author drew attention to the fact that the extraction of a part of a closed Hamiltonian system turns the original well-known Liouville differential equation into an integro-differential equation with a delayed time argument, which describes the dynamics of the selected subsystem as an open system. It was shown that the integral operator can be represented in the form of a fractional differential operator of distributed order. In this paper, we show how the kinetic theory of the system "atoms+photons" is transformed for the subsystem consisting of excited atoms. We present the derivation of the telegraph equation with delay, derive the Biberman--Holstein equation in the fractional differential form (with the fractional Laplace operator), and discuss boundary effects in the nonlocal transport model. The final section is devoted to laser technologies, which include free-electron lasers and laser cooling of atoms. Keywords and phrases: balance equation, radiation trapping, frequency redistribution, telegraph equation, laser beam, fractal medium. AMS Subject Classification: 65P40 |
| Audience | Academic |
| Author | Uchaikin, V. V. |
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Phys.2015318720010.4236/jamp.2015.32029 – reference: SchauflerSYakovlevVPSubrecoil laser cooling: Trapping versus diffusionLaser Phys.19946414419 – reference: HolsteinTImprisonment of resonance radiation in gasesPhys. Rev.1947721212123310.1103/PhysRev.72.1212 – reference: KondrashinMPSchauflerSSchleichWPYakovlevVPAnomalous kinetics of heavy particles in light mediaJ. Chem. Phys.20022841–2319330 – reference: GolubovskyYBKaganYMLyagushchenkoRIPopulation of resonance levels in a discharge of cylindrical configurationOpt. Spektroskop.19713112229 – reference: UchaikinVVSelf-similar anomalous diffusion and stable lawsUsp. Fiz. Nauk2003173884787610.3367/UFNr.0173.200308c.0847 – reference: KacMProbability and Related Topics if Physical Sciences1959London–New YorkInterscience – reference: NakhushevAMFractional Calculus and Its Applications2003MoscowFizmatlit1066.26005[in Russian] – reference: DattoliGRenieriATorreALectures in Free-Electron Laser Theory and Related Topics1995SingaporeWorld Scientific – reference: V. V. Uchaikin, “On the fractional differential Liouville equation as an equation of the dynamics of an open system,” Nauch. Ved. Belgorod. Univ. Ser. Mat. Fiz., 25 (196), No. 37, 58–67 (2014). – reference: F. Bardou, J.-P. Bouchaud, A. Aspect, and C. Cohen-Tannoudji, L´evy Statistics and Laser Cooling, Cambridge, Cambridge Univ. Press (2002). – reference: BibermanLMOn the theory of diffusion of resonant radiationZh. Eksp. Teor. Fiz.1947175416426 – reference: BoyadjievLDobnerHJOn a fractional integro-differential equation of Volterra typeIntegr. 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| SubjectTerms | Atoms & subatomic particles Differential equations Free electron lasers Hamiltonian functions Kinetic equations Kinetic theory Laser cooling Lasers Mathematics Mathematics and Statistics Open systems Operators (mathematics) Photons Subsystems |
| Title | Atoms and Photons: Kinetic Equations with Delay |
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