論文の公開元へOne-Dimensional Quantum Transport Theory in Terms of the Complex Spectral Analysis of Liouvillian
概要
We can see irreversible phenomena everywhere in the world, and explaining them from the reversible microscopic dynamics is still a fundamental problem in physics. Conventional methods of dealing with irreversible phenomena include the analysis of kinetic equations, such as the Boltzmann equation. However, these kinetic equations are derived through phenomenological approximations such as coarsegraining. Therefore, it has been argued that irreversibility is based on the human inability to process macroscopic degrees of freedom and information. In contrast, Ilya Prigogine, in his theory of dissipative structure, states [1, 2] that the existence of the second law of thermodynamics, i.e., irreversibility, is a necessary condition for the spontaneous creation of sophisticated and complex systems while obeying the laws of physics. Thus it can be concluded that irreversibility is not caused by human inability to process macroscopic degrees of freedom and information.
Recently, as a justification from the reversible dynamical principle for the kinetic theory, a new formulation of nonequilibrium statistical mechanics based on the complex spectral representation of the Liouville-von Neumann operator (Liouvillian) was developed by Tomio Petrosky and Ilya Prigogine [3, 4]. According to this theorem, [5], the collision operator appearing in the kinetic equation is the “self-energy part” of the Liouvillian. By solving the dispersion equation associated with the“self-energy part,”one can construct the resonance states of the Liouvillian, which have complex eigenvalues. This representation gives us a microscopic foundation of the irreversible kinetic theory.
The theory of the complex spectral representation of the Liouvillian has been applied to a variety of physical systems with great success [6, 7, 8, 9, 10]. In our study, we have applied this theory to a one-dimensional (1D) quantum molecular chain model and analyzed non-equilibrium transport phenomena. The imaginary parts of the complex eigenvalues of the Liouvillian then give transport coefficients like the diffusion coefficient.
Due to mathematical simplicity and some anomalies resulting from extremely low dimensionality, many physicists are interested in the irreversible transport properties of one-dimensional (1D) system [11, 12, 13, 14, 6, 15, 16, 17, 18]. For example, the collision operator in the kinetic equation disappears for a 1D gas consisting of similar kinds of particles. This is because the momenta of the particles are simply exchanged during the collision process in the 1D system, and thus, the momentum distribution function cannot change in time. However, this is not the case for 1D quantum systems since quantum mechanics involves both forward and backward scattering [15]. Hence in this case, the irreversibility is purely a quantum effect. Even in the system researched for this paper, there are some interesting phenomena that occur due to this one-dimensionality which are discussed below.
Recent advances in experimental techniques have made it possible to realize low-dimensional systems . Thus, it has become increasingly important to analyze 1D transport phenomena, which used to be of mere mathematical interest. Furthermore, exciton propagation on proteins has been observed experimentally [19] , which served as motivation for choosing a model of the exciton propagation on a protein model.
In this study, we analyze Davydov’s one-dimensional protein molecular chain model as a working example of a 1D quantum system. Davydov model is introduced to explain energy transfer on a one-dimensional protein chain [17, 20, 18].
This model is a simple 1D quantum system composed of a tight-binding model with phonons as a thermal bath. Therefore, analyzing this model will reveal the universal nature of transport phenomena in several 1D quantum systems, which include not only in biosystems but also in condensed matter theory.
In our study, we restrict our focus to a situation in which a particle weakly couples to phonons. We should note that the present study is within the range of the Markov approximation. Nevertheless, we have discovered several unique properties of transport phenomena in the one-dimensional quantum system by analyzing them microscopically. This research will form the basis for future discussion of non-Markovian effects in one-dimensional quantum transport phenomena beyond the Markovian regime
The contents of this thesis are based on two papers [21, 22] that have already been published.
