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Search Results (2)

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Keywords = linearly autonomous symmetry

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17 pages, 338 KiB  
Article
On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems
by Stanislav Yu. Lukashchuk
Mathematics 2022, 10(13), 2319; https://doi.org/10.3390/math10132319 - 2 Jul 2022
Cited by 2 | Viewed by 1350
Abstract
The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains [...] Read more.
The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point. Full article
(This article belongs to the Section E4: Mathematical Physics)
28 pages, 1148 KiB  
Article
Hidden Dynamical Symmetry and Quantum Thermodynamics from the First Principles: Quantized Small Environment
by Ashot S. Gevorkyan, Alexander V. Bogdanov and Vladimir V. Mareev
Symmetry 2021, 13(8), 1546; https://doi.org/10.3390/sym13081546 - 23 Aug 2021
Cited by 3 | Viewed by 2459
Abstract
Evolution of a self-consistent joint system (JS), i.e., a quantum system (QS) + thermal bath (TB), is considered within the framework of the Langevin–Schrödinger (L-Sch) type equation. As a tested QS, we considered two linearly coupled quantum oscillators that interact with TB. The [...] Read more.
Evolution of a self-consistent joint system (JS), i.e., a quantum system (QS) + thermal bath (TB), is considered within the framework of the Langevin–Schrödinger (L-Sch) type equation. As a tested QS, we considered two linearly coupled quantum oscillators that interact with TB. The influence of TB on QS is described by the white noise type autocorrelation function. Using the reference differential equation, the original L-Sch equation is reduced to an autonomous form on a random space–time continuum, which reflects the fact of the existence of a hidden symmetry of JS. It is proven that, as a result of JS relaxation, a two-dimensional quantized small environment is formed, which is an integral part of QS. The possibility of constructing quantum thermodynamics from the first principles of non-Hermitian quantum mechanics without using any additional axioms has been proven. A numerical algorithm has been developed for modeling various properties and parameters of the QS and its environment. Full article
(This article belongs to the Special Issue Symmetry in Particle Physics II)
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