Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Mathematical Models and Computer Simulations

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

А. В. Колесниченко

Published Sep 1, 2020• 6 citations
Publisher

Abstract

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.

DOI
10.1134/s2070048220050099

Related papers

General linear response theory

No¨elle Pottier

Abstract Linear response theory is a general formalism applicable to any physical system slightly departing from equilibrium. It allows us to express the linear response functions and the generalized susceptibilities in terms of the equilibrium correlation functions of the relevant dynamical variables. For instance, in the case of the response to an applied external field, one of the two relevant dynamical variables represents the physical quantity conjugate to the field whereas the other represents the physical quantity whose out-of-equilibrium average is measured. The computation of the linear response functions relies on a first-order perturbation expansion of the density operator (or of the phase space distribution function) of the system coupled to the field. The expressions thus obtained constitute the Kubo formulas of linear response theory.

•Sep 17, 2009•0 citations

Quantum statistical mechanics for nonextensive systems

Ervin Kaminski Lenzi, R. S. Mendes, A. K. Rajagopal

The traditional basis of description of many-particle systems in terms of Green functions is here generalized to the case when the system is nonextensive, by incorporating the Tsallis form of the density matrix indexed by a nonextensive parameter q. This is accomplished by expressing the many-particle q Green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical Green function which now depends on this parameter. We study one- and two-particle Green functions in detail. From the one-particle Green function, we deduce some experimentally observable quantities such as the one-particle momentum distribution function and the one-particle energy distribution function. Special forms of the two-particle Green functions are related to physical dynamical structure factors, some of which are studied here. We deduce different forms of sum rules in the q formalism. A diagrammatic representation of the q Green functions similar to the traditional ones follows because the equations of motion for both of these are formally similar. Approximation schemes for one-particle q Green functions such as Hartree and Hartree-Fock schemes are given as examples. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the q averages.

arXiv:cond-mat/9904100•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 1, 1999•45 citations

Nonextensive statistical mechanics : implications to quantum information

A. K. Rajagopal, R. W. Rendell

The interactions and correlations among the constituents of manybody systems are manifested in characteristic physical properties such as ferromagnetism, superconductivity, etc. A list of some that have been studied in the last century is given in the Box (see below). A parallel development in quantum information was initially slow but over the past two decades progress has been very rapid. Fundamentally this is another aspect of quantum correlations in composite systems arising from the twin features of the superposition principle and the tensor product structure of state space. These features are not utilized in the same manner in quantum many-body physics. In the Box, a corresponding parallel list with properties of many-body systems is given because there has been an interplay between the two research efforts and their understanding. The basic quantum mechanical principles apply to both cases except different aspects are utilized because the goals are different in each. Both areas of investigation are based on a probabilistic foundation with a variational underpinning founded on an “entropy” maximization, which may be called Quantum Statistical Mechanics [1]. The specific form of the entropy as a functional of the density matrix will be made explicit presently. See Table I for definitions of density matrix and associated quantities. The equilibrium properties of the many-body systems are then given by a maximization of von Neumann entropy subject to certain constraints such as the average value of the Hamiltonian of the system “energy”. This leads to the familiar exponential probabilities of the Boltzmann-Gibbs (BG) form. Any nonequilibrium properties are studied by a quantum time evolution equation. Non-equilibrium properties such as the anomalous relaxation in time are often analyzed with the quantum version of the Tsallis entropy with an entropic index, q, which leads to powerlaw probabilities in contrast to the BG type [2]. (See the Box in the Introduction of Tsallis and Boon). In quantum information, the maximum entropy scheme is not of use because its origins are elsewhere as will be made clear presently. The evolution is replaced by processing of information

Europhysics news•Nov 1, 2005•9 citations

Linear response theory

Helmut Hofmann

Abstract This chapter presents a universal definition of linear response, which at first is studied for the example of the damped oscillator. From the quantal time evolution operator lowest order transition rates are deduced, together with microscopic response functions. Their fundamental properties are studied, in particular, the meaning of their reactive and dissipative parts for energy transfer. Correlation functions are introduced and related to response functions by way of the fluctuation dissipation theorem. For a random matrix model, the latter is shown to also hold true for microcanonical ensembles. Connections to thermal Green functions are established and an extension of response theory to unstable modes is discussed. With the help of the Mori product, relations among the static response, the adiabatic and the isothermal susceptibilities are given. Time dependent irreversible processes are studied by introducing relaxation functions and by following the variation of the density operator and of entropy. Onsager relations are presented and Kubo's expressions for transport coefficients are given.

•Apr 17, 2008•1 citation

Topics in non-equilibrium statistical mechanics

Michel Le Bellac, Fabrice Mortessagne, G. G. Batrouni

We have given in two previous chapters a first introduction to non-equilibrium phenomena. The present chapter is devoted to a presentation of more general approaches, in which time dependence will be made explicit, whereas in practice we had to limit ourselves to stationary situations in Chapters 6 and 8. In the first part of the chapter, we examine the relaxation toward equilibrium of a system that has been brought out of equilibrium by an external perturbation. The main result is that, for small deviations from equilibrium, this relaxation is described by equilibrium time correlation functions, called Kubo (or relaxation) functions: this result is also known as ‘Onsager's regression law’. The Kubo functions turn out to be basic objects of non-equilibrium statistical mechanics. First they allow one to compute the dynamical susceptibilities, which describe the response of the system to an external time dependent perturbation: the dynamical susceptibilities are, within a multiplicative constant, the time derivatives of Kubo functions. A second crucial property is that transport coefficients can be expressed in terms of time integrals of Kubo functions. As we limit ourselves to small deviations from equilibrium, our theory is restricted to a linear approximation and is known as linear response theory . The classical version of linear reponse is somewhat simpler than the quantum one, and will be described first in Section 9.1. We shall turn to the quantum theory in Section 9.2, where one of our main results will be the proof of the fluctuationdissipation theorem.

Cambridge University Press eBooks•Apr 8, 2004•5 citations

On Statistical Mechanics of Nonequilibrium Fluctuations

M. Ichiyanagi

Without assuming detailed balance, the linear response theory of systems, in which the notation of a quantum dynamical semigroup can be defined, is discussed. This theory is used to prove a fluctuation-dissipation theorem for nonstationary nonequilibrium states. We propose an extended version of the Liouville equation in which we incorporate the stochastic behavior of the subsystem of the macroscopic motions. For this we introduce a non-Hermitian Hamiltonian and set a Liouville equation for a density matrix which is not Hermitian. We show that a response to external perturbations reduces to a sum of two contributions, one for which an improved fluctuation-dissipation theorem is proved and the other due to stochastic excitations that does not satisfy a reciprocity relation of Onsager type.

Progress of Theoretical Physics•Jun 1, 1984•0 citations
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.
Abstract Linear response theory is a general formalism applicable to any physical system slightly departing from equilibrium. It allows us to express the linear response functions and the generalized susceptibilities in terms of the equilibrium correlation functions of the relevant dynamical variables. For instance, in the case of the response to an applied external field, one of the two relevant dynamical variables represents the physical quantity conjugate to the field whereas the other represents the physical quantity whose out-of-equilibrium average is measured. The computation of the linear response functions relies on a first-order perturbation expansion of the density operator (or of the phase space distribution function) of the system coupled to the field. The expressions thus obtained constitute the Kubo formulas of linear response theory.
The traditional basis of description of many-particle systems in terms of Green functions is here generalized to the case when the system is nonextensive, by incorporating the Tsallis form of the density matrix indexed by a nonextensive parameter q. This is accomplished by expressing the many-particle q Green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical Green function which now depends on this parameter. We study one- and two-particle Green functions in detail. From the one-particle Green function, we deduce some experimentally observable quantities such as the one-particle momentum distribution function and the one-particle energy distribution function. Special forms of the two-particle Green functions are related to physical dynamical structure factors, some of which are studied here. We deduce different forms of sum rules in the q formalism. A diagrammatic representation of the q Green functions similar to the traditional ones follows because the equations of motion for both of these are formally similar. Approximation schemes for one-particle q Green functions such as Hartree and Hartree-Fock schemes are given as examples. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the q averages.
The interactions and correlations among the constituents of manybody systems are manifested in characteristic physical properties such as ferromagnetism, superconductivity, etc. A list of some that have been studied in the last century is given in the Box (see below). A parallel development in quantum information was initially slow but over the past two decades progress has been very rapid. Fundamentally this is another aspect of quantum correlations in composite systems arising from the twin features of the superposition principle and the tensor product structure of state space. These features are not utilized in the same manner in quantum many-body physics. In the Box, a corresponding parallel list with properties of many-body systems is given because there has been an interplay between the two research efforts and their understanding. The basic quantum mechanical principles apply to both cases except different aspects are utilized because the goals are different in each. Both areas of investigation are based on a probabilistic foundation with a variational underpinning founded on an “entropy” maximization, which may be called Quantum Statistical Mechanics [1]. The specific form of the entropy as a functional of the density matrix will be made explicit presently. See Table I for definitions of density matrix and associated quantities. The equilibrium properties of the many-body systems are then given by a maximization of von Neumann entropy subject to certain constraints such as the average value of the Hamiltonian of the system “energy”. This leads to the familiar exponential probabilities of the Boltzmann-Gibbs (BG) form. Any nonequilibrium properties are studied by a quantum time evolution equation. Non-equilibrium properties such as the anomalous relaxation in time are often analyzed with the quantum version of the Tsallis entropy with an entropic index, q, which leads to powerlaw probabilities in contrast to the BG type [2]. (See the Box in the Introduction of Tsallis and Boon). In quantum information, the maximum entropy scheme is not of use because its origins are elsewhere as will be made clear presently. The evolution is replaced by processing of information
Abstract This chapter presents a universal definition of linear response, which at first is studied for the example of the damped oscillator. From the quantal time evolution operator lowest order transition rates are deduced, together with microscopic response functions. Their fundamental properties are studied, in particular, the meaning of their reactive and dissipative parts for energy transfer. Correlation functions are introduced and related to response functions by way of the fluctuation dissipation theorem. For a random matrix model, the latter is shown to also hold true for microcanonical ensembles. Connections to thermal Green functions are established and an extension of response theory to unstable modes is discussed. With the help of the Mori product, relations among the static response, the adiabatic and the isothermal susceptibilities are given. Time dependent irreversible processes are studied by introducing relaxation functions and by following the variation of the density operator and of entropy. Onsager relations are presented and Kubo's expressions for transport coefficients are given.
We have given in two previous chapters a first introduction to non-equilibrium phenomena. The present chapter is devoted to a presentation of more general approaches, in which time dependence will be made explicit, whereas in practice we had to limit ourselves to stationary situations in Chapters 6 and 8. In the first part of the chapter, we examine the relaxation toward equilibrium of a system that has been brought out of equilibrium by an external perturbation. The main result is that, for small deviations from equilibrium, this relaxation is described by equilibrium time correlation functions, called Kubo (or relaxation) functions: this result is also known as ‘Onsager's regression law’. The Kubo functions turn out to be basic objects of non-equilibrium statistical mechanics. First they allow one to compute the dynamical susceptibilities, which describe the response of the system to an external time dependent perturbation: the dynamical susceptibilities are, within a multiplicative constant, the time derivatives of Kubo functions. A second crucial property is that transport coefficients can be expressed in terms of time integrals of Kubo functions. As we limit ourselves to small deviations from equilibrium, our theory is restricted to a linear approximation and is known as linear response theory . The classical version of linear reponse is somewhat simpler than the quantum one, and will be described first in Section 9.1. We shall turn to the quantum theory in Section 9.2, where one of our main results will be the proof of the fluctuationdissipation theorem.
Without assuming detailed balance, the linear response theory of systems, in which the notation of a quantum dynamical semigroup can be defined, is discussed. This theory is used to prove a fluctuation-dissipation theorem for nonstationary nonequilibrium states. We propose an extended version of the Liouville equation in which we incorporate the stochastic behavior of the subsystem of the macroscopic motions. For this we introduce a non-Hermitian Hamiltonian and set a Liouville equation for a density matrix which is not Hermitian. We show that a response to external perturbations reduces to a sum of two contributions, one for which an improved fluctuation-dissipation theorem is proved and the other due to stochastic excitations that does not satisfy a reciprocity relation of Onsager type.