Derivation of density operators for generalized entropies with quantum analysis

Ishihara, M.

doi:10.1016/j.physa.2019.123419

Cited by 7 publications

(3 citation statements)

References 25 publications

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“…For other applications of quantum analysis, see, for example, Refs. [11][12][13][14][15][16] Let us consider an operator f ( A) where f (x) is a smooth function of x. Then the Gâteaux differential of this operator is defined by [3] df…”

Section: Definition Of Operator Derivativementioning

confidence: 99%

Poisson bracket operator

Koide

2021

Preprint

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We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this operator, we derive the quantum canonical equation which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation.At the same time, the quantum canonical equation is applicable to c-number canonical variables and then coincides with the canonical equation in classical mechanics.Therefore the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to non-standard system where the Heisenberg is not applicable. As an example, we consider the application to the system where a c-number and a q-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.

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Section: Definition Of Operator Derivativementioning

confidence: 99%

Poisson bracket operator

Koide

2021

Preprint

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“…Let data matrix X= [xij], i=1,…, N, j=1, …m represent a stretch of m-dimensional chaotic [2,17] discrete time series at a relative scale, so xij is out of scale random values. Chaotic nature implies fast local changes; therefore, methods based on averaging over extended sliding windows such as parametric or non-parametric spectra analysis are hardly to be used.…”

Section: Introductionmentioning

confidence: 99%

A new approach to identifying the local structure of multidimensional chaotic time series

Makshanov

Zhuravlev

Tyndykar

2021

J. Phys.: Conf. Ser.

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The paper is devoted to the solution of the problems of mathematical supply of decision making during multichannel monitoring of large-scaled systems. The work also deals with space-time dynamics of multidimensional time series of different origins. Highly dynamical chaotic processes whose fine structure cannot be revealed by standard spectral methods are regarded. Technologies for dimension reduction based on data matrix representation on the first singular basis and multiple regression in projections’ space are developed.

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“…The quantum analysis was applied to the nonequilibrium response [14] and quantum correlation identities [15]. The density operators for some entropies were derived with the quantum analysis in the previous study [17]. It is expected that the quantum analysis works well in order to obtain the density operators with some constraints in the MEP.…”

Section: Introductionmentioning

confidence: 99%

Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics

Ishihara

2020

Preprint

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We derived the equation of the density operator for generalized entropy and generalized expectation value with quantum analysis when conserved quantities exist. The derived equation is simplified when the conventional expectation value is employed. The derived equation is also simplified when the commutation relations, [ρ, Ĥ] and [ρ, Q[a] ], are the functions of the density operator ρ, where Ĥ is the Hamiltonian, and Q[a] is the conserved quantity. We derived the density operators for the von Neumann entropy, the Tsallis entropy, and the Rényi entropy in the case of the conventional expectation value. We also derived the density operators for the Tsallis entropy and the Rényi entropy in the case of the escort average (the normalized q-expectation value), when the density operator commutes with the Hamiltonian and the conserved quantities. We found that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in the case of the conventional expectation value, even when conserved quantities do not commute. The simple extension of the argument is also shown in the case of the escort average, when the density operator ρ commutes with the Hamiltonian Ĥ and the conserved quantity Q[a] : [ρ, Ĥ] = [ρ, Q[a] ] = 0. These findings imply that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in some systems.

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Derivation of density operators for generalized entropies with quantum analysis

Cited by 7 publications

References 25 publications

Poisson bracket operator

Poisson bracket operator

A new approach to identifying the local structure of multidimensional chaotic time series

Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics

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