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

Progress of Theoretical Physics

Complex Spectral Representation of the Liouvillian and Kinetic Theory in Nonequilibrium Physics

Tomio Yamakoshi Petrosky

Published Mar 1, 2010• 19 citations
Publisher

Abstract

Kinetic theory in nonequilibrium statistical physics is discussed in terms of the complex spectral representation of the Liouvillian for quantum systems. We show that the collision operator appearing in the kinetic equation is the “self-energy part” of the Liouvillian. Solving the dispersion equation associated to the “self-energy part”, one can construct the resonance states of the Liouvillian which have complex eigenvalues. The imaginary part of the eigenvalue gives a decay rate in irreversible process in nonequilibrium statistical physics. As illustrations of the resonance states of the Liouvillian we consider two examples; one is a one-dimensional system in which a polaron is weakly interacting with a thermal reservoir consisting of an acoustic phonon field, and the other is a molecular chain that is described by the Davydov Hamiltonian, which has been introduced as a simple model that describes a protein chain. We show that the imaginary part of the spectrum of the Liouvillian in these systems has rich structures showing band structure, an accumulation point, and a fractal structure reminiscent of Hofstadter's butterfly.

DOI
10.1143/ptp.123.395

Related papers

Band structure and accumulation point in the spectrum of quantum collision operator in a one-dimensional molecular chain

Buang Ann Tay, Kazuki Kanki, Satoshi Tanaka +1 more

We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point, and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces.

arXiv:1101.3637•Journal of Mathematical Physics•Feb 1, 2011•4 citations

Liouvillian Green's functions and self-energies for energy-shift and decay phenomena

Marvin D. Girardeau

Green's functions and transition self-energies of dissipative quantum-mechanical many-body systems are formulated in terms of operator-basis expansions of Heisenberg equations of motion in Liouville space. Matrix elements of the Liouvillian with respect to arbitrary nonorthogonal operator bases play a primary role in the analysis. The self-energies are expressed as functionals of initial statistical ensemble averages which are presumed known and not restricted to systems at or near equilibrium; this provides a first-principles approach to use of spectral line shapes as a diagnostic of the nonequilibrium state of a dense gas, plasma, and/or radiation field. A perturbation expansion of the self-energy is derived and the general term is interpreted in terms of a subtraction of reducible parts. A separation into resonant and nonresonant contributions plays a crucial role in the analysis. The first-order resonant contribution yields a generalized statistical Hartree-Fock approximation, whereas the second-order contribution generalizes the Weisskopf-Wigner theory of decaying states to include thermal and, more generally, nonequilibrium environmental effects on transition-energy shifts and widths. Off-diagonal contributions to the self-energy are shown to be essential for inclusion of environmental contributions.

Physical Review A•Aug 1, 1983•19 citations

Microscopic description of irreversibility in quantum Lorentz gas by complex spectral analysis of the Liouvillian outside the Hilbert space

Tomio Yamakoshi Petrosky, Kazunari Hashimoto, Kazuki Kanki +1 more

Irreversible process of a weakly coupled one-dimensional quantum perfect Lorentz gas is studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouvillian. Without any phenomenological operations, such as a coarse-graining of space-time or a truncation of the higher order correlation, we obtained irreversible processes on a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free-length. The list of development of the complex spectral analysis of the Hamiltonian (instead of the Liouvillian) in quantum optical systems and in quantum nano-devices is also presented.

Chaos An Interdisciplinary Journal of Nonlinear Science•Sep 22, 2017•2 citations

Liouville extension of quantum mechanics: One-dimensional gas with δ-function interaction

Tomio Yamakoshi Petrosky, Gonzalo E. Ordonez

A one-dimensional quantum gas with \ensuremath{\delta}-function interaction is known to be integrable for any finite number of particles. We show that in the thermodynamic limit invariants of motion are destroyed in conjunction with a class of singular density matrices that appear commonly in both equilibrium and nonequilibrium statistical mechanics. The system becomes nonintegrable, due to Poincar\'e resonances. However, the eigenvalue problem of the Liouville--von Neumann operator ${L}_{H}$ may be solved in an extended functional space that contains the singular density matrices. We may in this way describe the approach to equilibrium. This is a simple example of the Liouville space extension of quantum mechanics, due to Prigogine and his co-workers. The fundamental quantity in this formulation is the probability (and not the wave function). We obtain indeed an irreducible complex spectral representation of ${L}_{H}.$ Irreducible means that the eigenstates of ${L}_{H}$ cannot be implemented by wave functions, and complex means that the eigenvalues are complex numbers, whose imaginary parts determine relaxation times of the system. The dynamical evolution is described by a semigroup. Due to dissipative effects the real part of the eigenvalues is no more a difference of energy levels, as is the case in standard quantum mechanics.

Physical Review A•Nov 1, 1997•22 citations

Accumulation Point in the Spectrum of the Collision Operator for a One-Dimensional Polaron System

Kazuki Kanki, Satoshi Tanaka, Buang Ann Tay +1 more

The eigenvalue problem of a collision operator in a quantum kinetic equation for a momentum relaxation process of a one-dimenstional polaron system is considered. There appear an accumulation point and band structures in the spectrum of the collision operator.

Progress of Theoretical Physics Supplement•Jan 1, 2010•6 citations

GENERALIZED KINETIC AND EVOLUTION EQUATIONS IN THE APPROACH OF THE NONEQUILIBRIUM STATISTICAL OPERATOR

A. L. KUZEMSKY

The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.

arXiv:cond-mat/0502194•International Journal of Modern Physics B•Mar 10, 2005•39 citations
Kinetic theory in nonequilibrium statistical physics is discussed in terms of the complex spectral representation of the Liouvillian for quantum systems. We show that the collision operator appearing in the kinetic equation is the “self-energy part” of the Liouvillian. Solving the dispersion equation associated to the “self-energy part”, one can construct the resonance states of the Liouvillian which have complex eigenvalues. The imaginary part of the eigenvalue gives a decay rate in irreversible process in nonequilibrium statistical physics. As illustrations of the resonance states of the Liouvillian we consider two examples; one is a one-dimensional system in which a polaron is weakly interacting with a thermal reservoir consisting of an acoustic phonon field, and the other is a molecular chain that is described by the Davydov Hamiltonian, which has been introduced as a simple model that describes a protein chain. We show that the imaginary part of the spectrum of the Liouvillian in these systems has rich structures showing band structure, an accumulation point, and a fractal structure reminiscent of Hofstadter's butterfly.
We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point, and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces.
Green's functions and transition self-energies of dissipative quantum-mechanical many-body systems are formulated in terms of operator-basis expansions of Heisenberg equations of motion in Liouville space. Matrix elements of the Liouvillian with respect to arbitrary nonorthogonal operator bases play a primary role in the analysis. The self-energies are expressed as functionals of initial statistical ensemble averages which are presumed known and not restricted to systems at or near equilibrium; this provides a first-principles approach to use of spectral line shapes as a diagnostic of the nonequilibrium state of a dense gas, plasma, and/or radiation field. A perturbation expansion of the self-energy is derived and the general term is interpreted in terms of a subtraction of reducible parts. A separation into resonant and nonresonant contributions plays a crucial role in the analysis. The first-order resonant contribution yields a generalized statistical Hartree-Fock approximation, whereas the second-order contribution generalizes the Weisskopf-Wigner theory of decaying states to include thermal and, more generally, nonequilibrium environmental effects on transition-energy shifts and widths. Off-diagonal contributions to the self-energy are shown to be essential for inclusion of environmental contributions.
Irreversible process of a weakly coupled one-dimensional quantum perfect Lorentz gas is studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouvillian. Without any phenomenological operations, such as a coarse-graining of space-time or a truncation of the higher order correlation, we obtained irreversible processes on a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free-length. The list of development of the complex spectral analysis of the Hamiltonian (instead of the Liouvillian) in quantum optical systems and in quantum nano-devices is also presented.
A one-dimensional quantum gas with \ensuremath{\delta}-function interaction is known to be integrable for any finite number of particles. We show that in the thermodynamic limit invariants of motion are destroyed in conjunction with a class of singular density matrices that appear commonly in both equilibrium and nonequilibrium statistical mechanics. The system becomes nonintegrable, due to Poincar\'e resonances. However, the eigenvalue problem of the Liouville--von Neumann operator LH{L}_{H}LH​ may be solved in an extended functional space that contains the singular density matrices. We may in this way describe the approach to equilibrium. This is a simple example of the Liouville space extension of quantum mechanics, due to Prigogine and his co-workers. The fundamental quantity in this formulation is the probability (and not the wave function). We obtain indeed an irreducible complex spectral representation of LH.{L}_{H}.LH​. Irreducible means that the eigenstates of LH{L}_{H}LH​ cannot be implemented by wave functions, and complex means that the eigenvalues are complex numbers, whose imaginary parts determine relaxation times of the system. The dynamical evolution is described by a semigroup. Due to dissipative effects the real part of the eigenvalues is no more a difference of energy levels, as is the case in standard quantum mechanics.
The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.