Atoms and Photons: Kinetic Equations with Delay

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 260; no. 3; pp. 335 - 370
• Main Author: Uchaikin, V. V.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2022; Springer; Springer Nature B.V
• Subjects: 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; frequency redistribution; telegraph equation; radiation trapping; fractal medium; balance equation; laser beam; 65P40
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-022-05698-y

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.