General Non-Markovian Quantum Dynamics

Tarasov, Vasily E.

doi:10.3390/e23081006

Cited by 30 publications

(20 citation statements)

References 78 publications

(167 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the first time, the use of FC to take into account memory effects (non-Markovity) in open quantum systems was proposed in work [112] (see Chapter 20 in book [112, pp. 477-482] and [169,170]). The solutions of non-Markovian equations describing quantum systems with memory, which is based on FC, were proposed in these works.…”

Section: Non-markovian Dynamics Of Open Quantum Systemsmentioning

confidence: 99%

“…458-464, 468-477], [171]). Solutions of the proposed generalized Lindblad equations, which describe non-Markovian quantum dynamics, were derived in [112,157,170,171]. For construction of a theory of non-Markovian quantum dynamics, it is important to consider time-dependent parameters [172] and the relativistic quantum systems with memory.…”

Section: Non-markovian Dynamics Of Open Quantum Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Trends, directions for further research, and some open problems of fractional calculus

et al. 2022

Self Cite

View full text Add to dashboard Cite

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.

show abstract

Section: Non-markovian Dynamics Of Open Quantum Systemsmentioning

confidence: 99%

Section: Non-markovian Dynamics Of Open Quantum Systemsmentioning

confidence: 99%

Trends, directions for further research, and some open problems of fractional calculus

et al. 2022

Self Cite

View full text Add to dashboard Cite

show abstract

“…The proposed General FVC can be used as a mathematical tool in general fractional dynamics (GFDynamics) [44,45], in which non-locality in space is taken into account in general form.…”

Section: General Fractional Divergence In Occmentioning

confidence: 99%

“…It is important to have a general fractional calculus that allows us to describe nonlocality in the most general form. We proposed [44,45] to use general fractional calculus (GFC) to describe systems, media and fields with general form of nonlocality in space.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

General Fractional Vector Calculus

Tarasov

2021

Mathematics

Self Cite

View full text Add to dashboard Cite

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

show abstract

The 1st Level General Fractional Derivatives and Some of Their Properties

Luchko

2022

J Math Sci

View full text Add to dashboard Cite

In this paper, we first provide a short summary of the main properties of the so-called general fractional derivatives with the Sonin kernels introduced so far. These are integro-differential operators defined as compositions of the first order derivative and an integral operator of convolution type. Depending on succession of these operators, the general fractional derivatives of the Riemann-Liouville and of the Caputo types were defined and studied. The main objective of this paper is a construction of the 1st level general fractional derivatives that comprise both the general fractional derivative of the Riemann-Liouville type and the general fractional derivative of the Caputo type. We also provide some of their properties including the 1st and the 2nd fundamental theorems of Fractional Calculus for these derivatives and the suitably defined general fractional integrals.

show abstract

General Non-Markovian Quantum Dynamics

Cited by 30 publications

References 78 publications

Trends, directions for further research, and some open problems of fractional calculus

Trends, directions for further research, and some open problems of fractional calculus

General Fractional Vector Calculus

The 1st Level General Fractional Derivatives and Some of Their Properties

Contact Info

Product

Resources

About