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Modeling the Linear Response from a Quantum Nonextensive System to a Dynamic External Disturbance

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Mathematical Models and Computer Simulations Aims and scope

Abstract

A dynamic theory of the linear reaction from nonextensive quasi-equilibrium multibody systems to an external time-dependent perturbation is developed in quantum statistical mechanics based on the Tsallis parametric nonadditive entropy associated with the density matrix. For nonextensive quantum systems, a modification of the Kubo theory developed in quantum mechanics is proposed. The linear reaction theory is constructed based on a generalized canonical form of the density matrix obtained by maximizing the Tsallis quantum entropy by averaging the observed values over the escort distribution. Generalized expressions for admittance and response functions are presented that describe the linear response of the system to a weak external mechanical impact. The paper discusses the symmetry property for the relaxation function under time reversal and the Onsager reciprocity relation for generalized susceptibility. It is shown that these properties known in classical quantum statistics remain valid for anomalous systems.

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Notes

  1. Further, the operators will be denoted by the caret superscript ^.

  2. The distribution \({{\hat {\rho }}_{{eq}}}\) can not only be a canonical distribution but any equilibrium distribution (in particular, a large canonical distribution—the most convenient one when calculating averages for Bose and Fermi systems). The canonical ensemble is more suitable for some other systems, for example, spin ones.

  3. The switching operator \({{\hat {a}}^{ \times }}\) conforms to the following rules [41]: \(\exp ({{\hat {a}}^{ \times }})\hat {b} = \sum {\tfrac{1}{{n!}}} {{({{\hat {a}}^{ \times }})}^{n}}\hat {b} = \sum {\tfrac{1}{{n!}}} [\hat {a}[\hat {a}..[\hat {a},\hat {b}]..] = \exp (\hat {a})\hat {b}\exp ( - \hat {a});\)\({{\hat {a}}^{ \times }}{{\hat {b}}^{ \times }} - {{\hat {b}}^{ \times }}{{\hat {a}}^{ \times }} = {{[\hat {a},\hat {b}]}^{ \times }}\).

  4. It should be recalled that operator \({{\hat {\mathcal{A}}}^{ + }}\) is called adjoint to operator \(\hat {\mathcal{A}}\) if for every pair of functions \({{\psi }_{1}}\) and \({{\psi }_{2}}\) the relation \(\left\langle {{{\psi }_{1}},\hat {\mathcal{A}}{{\psi }_{2}}} \right\rangle = \left\langle {{{{\hat {\mathcal{A}}}}^{ + }}{{\psi }_{1}},{{\psi }_{2}}} \right\rangle \) is valid. The Hermitian (or self-adjoint) operator \(\hat {\mathcal{A}}\) coincides with its adjoint \(\hat {\mathcal{A}} = {{\hat {\mathcal{A}}}^{ + }}\). The eigenvalues of the Hermitian operators are real numbers.

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  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    A. V. Kolesnichenko

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Translated by I. Pertsovskaya

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Kolesnichenko, A.V. Modeling the Linear Response from a Quantum Nonextensive System to a Dynamic External Disturbance. Math Models Comput Simul 12, 647–659 (2020). https://doi.org/10.1134/S2070048220050099

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