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Bulletin of the American Physical Society

One-dimensional atomic chains: rich physics in simple structures

Mingjie Liu, Vasilii I. Artyukhov, Boris I. Yakobson

Published Mar 4, 2015• 0 citations
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Complex Spectral Representation of the Liouvillian and Kinetic Theory in Nonequilibrium Physics

Tomio Yamakoshi Petrosky

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.

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One-Dimensional Chain of n -Level Atoms and Discrete Nonlinear Schrödinger Equation

Liu Xiao-juan, Yi Xiao, Wenhua Hai

The Hamiltonian of one-dimensional chain of n-level atoms is represented in terms of Boson operators by using the Dyson-Maleev transformation and it is shown that the finite-ladder effect disappears when n tends toward infinity. In this way, it is found that the Heisenberg equation of motion of this system is exactly described in the coherent state representation by the dark discrete nonlinear Schrödinger (DNLS) equation. It is also briefly shown that the DNLS equation has some general soliton solutions. This indicates that this simple system has richness of nonlinear waves.

Chinese Physics Letters•Apr 1, 1999•2 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.
The Hamiltonian of one-dimensional chain of n-level atoms is represented in terms of Boson operators by using the Dyson-Maleev transformation and it is shown that the finite-ladder effect disappears when n tends toward infinity. In this way, it is found that the Heisenberg equation of motion of this system is exactly described in the coherent state representation by the dark discrete nonlinear Schrödinger (DNLS) equation. It is also briefly shown that the DNLS equation has some general soliton solutions. This indicates that this simple system has richness of nonlinear waves.