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Chinese Journal of Physics

Effect of an Isotopic Impurity on the Energy Flow in a System of One-Dimensional Coupled Harmonic Oscillators

Keiko Kobayasi, Éi Iti Takizawa

Published Apr 1, 1964• 0 citations
OpenAlex

Abstract

linearSchrod- of coupled harmonic oscillators is investigated by means of the inger coordinates. Assuming such an initial canonical ensemble that half the system is at temperature T and other half at temperature zero, the correlation functions of particles in the system are calculated as functions of time. Thus, it is shown that the average kinetic and potential energies of each particle in the infinitely large system approach different stationary values in both sides of the impurity atom, and that the system approach a stationary state with a gap of energy distribution along the system at the impurity site. This result agrees with that obtained by Kashiwamura and Teramoto by means of trigonometric eigenfunctions of the dynamical system.The elements of position-velocity correlation matrix are also calculated. The energy flow in the large system is derived from these, and it is shown that even after a sufficiently long time the finite energy flow still exists at every point of the system though there is no temperature gradient at that point.It should be mentioned, however, that the method we used here is easier to calculate and simpler by far than that by means of the trigonometric eigenfunctions of the dynamical system.

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arXiv:cond-mat/0404711•Journal of Physics Condensed Matter•Jul 10, 2004•75 citations
linearSchrod- of coupled harmonic oscillators is investigated by means of the inger coordinates. Assuming such an initial canonical ensemble that half the system is at temperature T and other half at temperature zero, the correlation functions of particles in the system are calculated as functions of time. Thus, it is shown that the average kinetic and potential energies of each particle in the infinitely large system approach different stationary values in both sides of the impurity atom, and that the system approach a stationary state with a gap of energy distribution along the system at the impurity site. This result agrees with that obtained by Kashiwamura and Teramoto by means of trigonometric eigenfunctions of the dynamical system.The elements of position-velocity correlation matrix are also calculated. The energy flow in the large system is derived from these, and it is shown that even after a sufficiently long time the finite energy flow still exists at every point of the system though there is no temperature gradient at that point.It should be mentioned, however, that the method we used here is easier to calculate and simpler by far than that by means of the trigonometric eigenfunctions of the dynamical system.
Statistical dynamical feature of time development of one-dimensional harmonic lattice containing a few impurities is investigated. The fundamental solutions of equations of motion are obtained for a system of a finite number of particles as well as that of an infinite number of particles. The difference between those solutions is not appreciable at least within a range of time. Next, such an initial ensemble is introduced that a part of the lattice is canonically distributed with temperature T0 and the remaining part remains quite at rest. It is shown that the ensemble approaches a stationary state in a small sub-system for a more limited range of time. In the stationary state the mean energy of each particle is expressed as a function of characteristic parameters of the impurity particles.
The methods of information theory are applied to the problem of describing the transport of internal energy along a linear harmonic chain in a stationary state. It is shown that the average energy flux along the chain is a constant of the motion which can be directly incorporated into the canonical formalism. The classical probability distribution in phase space is constructed and used to evaluate the partition function in the thermodynamic limit. Thermodynamic relationships between the undetermined multipliers and the given information are explicitly determined, as are the singlet- and pair-distribution functions in configuration and momentum space and the correlation of the momentum of a specific particle in time. For a given system energy there is an upper bound for the mean rate of energy transport. As this limit is closely approached, both the range and duration of correlations become large. This behavior, as well as some other features of the model, is reminiscent of that near a critical point.
The nonequilibrium dynamics of an impurity immersed with a finite velocity into a one-dimensional system of weakly interacting bosons is studied within the framework of the time-dependent Gross-Pitaevskii equation. We uncover and characterize different regimes of relaxation dynamics. We find that the final impurity velocity remains constant in a large interval of sufficiently big and realistic initial velocities. The underlying physical mechanism is the emission of the dispersive density shock waves that carry away the excess of the initial impurity momentum, while locally the system remains in the same stationary state. In contrast, a heavy impurity with the same coupling constant relaxes differently and the regime of constant final velocity disappears. Furthermore, a fast heavy impurity exhibits damped velocity oscillations in time before reaching a stationary state. This process is accompanied by the oscillations of the local depletion of the boson density around the impurity, until their positions coincide and they continue the motion together. Decreasing the impurity-boson coupling or increasing the strength of repulsion between bosons, the oscillations get amplified. In the case of a heavy impurity with the mass bigger than the critical one, the ground-state energy as a function of momentum exhibits cusps and metastable branches. We show that they manifest themselves by a soliton emission, a considerable slowing down of the relaxation, and a change of the impurity direction of motion with respect to the initial one.
In this paper we analyze the properties of bound states of an atom interacting with a set of static impurities. We begin with the simplest system of a single atom interacting with two static impurities. We consider two types of atom-impurity interaction: (i) a zero-range potential represented by a regularized $\ensuremath{\delta}$ interaction and (ii) a more realistic polarization potential, representing the long-range part of the atom-ion interaction. For the former we obtain analytical results for energies of bound states. For the latter we perform numerical calculations based on the application of the finite-element method. We then discuss the case of a single atom interacting with a one-dimensional (1D) infinite chain of static ions. Such a setup resembles the Kronig-Penney model of a 1D crystalline solid, where the energy spectrum exhibits band-structure behavior. For this system we derive analytical results for the band structure of bound states, assuming a regularized $\ensuremath{\delta}$ interaction, and perform numerical calculations, considering the polarization potential to model an atom-impurity interaction. Both approaches agree quite well when the separation between impurities is much larger than the characteristic range of the interaction potential.
Abstract It is a fundamental problem to characterize nonequilibrium processes. For a moving one-dimensional potential, we explore the nonequilibrium dynamics of the initial energy eigenstates for a confined quantum system interacting with a large reservoir. For concreteness, we investigate a dragged harmonic oscillator linearly interacting with an assembly of harmonic oscillators, and explore the deviation from adiabatic processes by rigorously calculating the so-called persistent amplitude. In particular, we show that the phase of the persistent amplitude is considered to be common both for the ground and excited states. Also, we can define the quasi adiabatic processes in a well-defined double limit of small perturbation and a sufficiently long time in terms of the phase and absolute value of the persistent amplitude.
We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.