Coupled Anharmonic Oscillators Research Articles

Frequency–Amplitude Relationship of Coupled Anharmonic Oscillators

The frequency–amplitude relationship of coupled anharmonic oscillators is an important problem. Many powerful methods for solving this problem have been proposed. He’s parameter-expanding method is an important one. It holds the advantages of modified Lindstedt–Poincare parameter method and bookkeeping parameter method. The first iteration is enough. It is very effective and convenient and quite accurate to both linear and nonlinear problems. In this paper, He’s parameter-expanding method is applied to coupled anharmonic oscillators. The frequency-amplitude relationship and the first-order approximate solution of the oscillators are obtained respectively. The solution procedure shows that the method is very powerful and convenient to nonlinear oscillator. This method has great potential and can be applied to other types of nonlinear oscillator problems.

Reply to Comment on ‘Coupled anharmonic oscillators: the Rayleigh–Ritz approach versus the collocation approach’

The Comment by Amore et al (2011 Phys. Scr.83 047003) proves only that the results of the collocation approach become upper bounds for bound-state energies in the limit of an infinite number of grid points N. We point out that the results obtained at finite N may lie below the exact results and should be taken with care.

Comment on ‘Coupled anharmonic oscillators: the Raileigh–Ritz approach versus the collocation approach’

We show that some of the conclusions drawn recently about the performance and convergence of the little sinc functions collocation method are misleading. To this end, we prove the convergence of the method and recalculate the ground-state energy for the Pullen–Edmonds Hamiltonian. In addition, we show the results for another nonseparable Hamiltonian operator with a somewhat more complicated potential-energy function.

Polarization-Angle-Scanning 2DIR Spectroscopy of Coupled Anharmonic Oscillators: A Polarization Null Angle Method

Two-dimensional (2D) optical spectroscopy based on stimulated photon echo geometry requires multiple ultrashort pulses of which spatiotemporal properties and optical phases can be precisely controlled. Also, it is possible to change the incident beam polarization directions defined in a laboratory frame. Here, we introduce the polarization-angle-scanning (PAS) 2D spectroscopy and show that the diagonal and cross-peak amplitudes in the 2D spectrum can be arbitrarily modulated by spatially controlling the beam polarization directions. For a coupled anharmonic oscillator system, we specifically demonstrate that either diagonal or cross-peaks in the measured 2DIR spectra can be selectively eliminated and show that such polarization angles provide direct information on the relative angles between coupled transition dipoles and thereby on intricate details of molecular structures. We thus anticipate that the present PAS 2D optical spectroscopy can be a useful experimental method enabling us to probe structural evolutions of nonequilibrium state molecules by monitoring the time-dependent changes of the relative transition dipole directions.

Coupled anharmonic oscillators: the Rayleigh–Ritz approach versus the collocation approach

For a system of coupled anharmonic oscillators we compare the convergence rate of the variational collocation approach presented recently by Amore and Fernández (2010 Phys. Scr.81 045011) with the one obtained using the optimized Rayleigh–Ritz (RR) method. The monotonic convergence of the RR method allows us to obtain more accurate results at a lower computational cost.

Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators

We consider a Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice Z(2n) x Z(2), and subjected to stochastic forcing mimicking heat baths of temperatures T-1 and T-2 on the hyperplanes at O and N. We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for N large, is approximately a local equilibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile.

Landau Free Energy in the Mean-Field Approximation for Coupled Classical Anharmonic Oscillators

In the mean-field approximation, the expression for free energy contains the mean field which is to be determined by the self-consistency equation. When the mean field is taken as a free parameter equivalent to the order parameter in the Landau theory, the free energy constructed from the mean-field Hamiltonian possesses the same minimum property as the Landau free energy. Utility of such an ad hoc free energy is studied analytically and numerically in comparison with the Landau free energy, with particular attention to interacting classical anharmonic oscillators.

Overtone Spectroscopy of CHD3 in Liquid Argon Solutions

AbstractThe vibrational overtone spectrum of CHD3 has been obtained in liquid argon solutions at temperatures around 94 K. The spectra of C‐H and C‐D overtones below 10000 cm−1 were obtained with a Fourier transform spectrophotometer operating in the near‐infrared region and a low temperature cell. The C‐H spectrum around Δv = 5 was recorded using a resonant photoacoustic technique at cryogenic temperatures. The absorption bands were computer deconvoluted to obtain the peak absorption frequency, the band‐width, and the area of some important bands. Due to the small frequency shift of the absorption bands in solution with respect to the gas phase frequencies, the assignment of the transitions was made by comparison with experimental and calculated gas phase frequencies. The calculation of gas phase frequencies and vibrational states was made using the pure local mode model for the isolated C‐H bond and the Harmonically Coupled Anharmonic Oscillator (HCAO) model for the three equivalent C‐D bonds. In order to test the Fermi resonance interaction between C‐H stretching and bending vibrations, the absorption bands and relative intensities in solution were used to obtain the parameters of an effective Fermi resonance Hamiltonian. The values of the parameters obtained in solution and gas phase are very close in magnitude. The optimized parameters of the Fermi resonance Hamiltonian give a good description of the frequencies and relative intensities of the C‐H absorption bands in solution.

Linearly Coupled Anharmonic Oscillators and Integrability

The Painleve test for a linearly coupled anharmonic oscillator is performed. We show that this system does not pass the Painleve test. This suggests that this system is not integrable. Moreover, we apply Ziglin's (1983) theorem which provides a criterion for non-existence of first integrals besides the Hamiltonian. Calculating numerically the maximal one-dimensional Lyapunov exponent, we find regions with positive exponents. Thus, the system can show chaotic behaviour. Finally we compare our results with the quartic coupled anharmonic oscillator.

Coupled quartic anharmonic oscillators, Painlevé analysis, and integrability.

A detailed and systematic investigation of the Painlev\'e (P) properties of coupled quartic anharmonic oscillators is presented. Considering the two coupled oscillators we show that there exist four different parametric cases possessing P properties, two are identified with strong P property and the other two with weak P property. For each of these four cases explicit second integrals of motion can also be constructed directly. We then consider the three-coupled-oscillator system and identify three cases, one with strong P and the other two with weak P nature. We have explicitly derived the second and third integrals of motion for a special case of the strong P case, but for the remaining cases they have not yet been found. Finally, we extend the procedure to the N coupled oscillators and succeed to show that there exist three cases possessing P properties, which are the natural generalizations of the three-coupled-oscillator system.

Studies of Coupled Anharmonic Oscillator Problem Using Coherent States and Path Integral Approaches

The present review is devoted to the study of the problem of coupled anharmonic oscillators. A perturbative solution is obtained for the system of an undamped and the damped coupled anharmonic oscillators in the coherent state representation. The solution does not contain the vicious secular terms and shows, explicitly, the anharmonic effects of a coupled system. In order to derive the perturbative solution of a damped coupled system a new frame of time, called quasi time τ, has been exploited. The large quantum number behaviour of coupled anharmonic oscillators has been derived using the path integral method.

Coupled Anharmonic Oscillators. I. Equal-Mass Case

We develop a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators. Our technique is based on a generalization of the semiclassical approximation which was used to study equal-mass oscillators in the first paper of this series. The unequal-mass problem is much more difficult because the path which minimizes the classical action is not a straight line. Assuming that this tunneling path is known, we derive a general expression for the physical-optics approximation to the wave function of a tunneling particle. This derivation rests on the construction of a WKB approximation in curved space. We thus completely reduce the general quantum problem to the much simpler classical one of determining the path. Then we present a perturbation scheme for finding the classical path for systems of oscillators whose masses only differ by a small amount. Finally, we illustrate our techniques by solving a two-mode unequal-mass oscillator and comparing these results with a computer calculation. Our theoretical predictions and numerical calculations agree.

Computer Studies on the Approach to Thermal Equilibrium in Coupled Anharmonic Oscillators. II. One-Dimensional Case

One-dimensional anharmonic lattices with quadratic as well as quartic potentials between nearest neighbors are treated on computer to see the long-time behaviors of the vibration. As is the case of two-dimensional lattice, the induction period is found to exist and its length increases on decreasing the anharmonic coupling constant. The possible existence of the critical value of the anharmonic coupling constant is surmised, below which the system is almost periodic. Above this critical value, the system tends to reach a thermal equilibrium, which is confirmed by the long-time averages of the squares of the velocities and by the products of velocities of different particles. The significance of the errors introduced during computation for the ergodic property of the system is pointed out.

Computer Studies on the Approach to Thermal Equilibrium in Coupled Anharmonic Oscillators. I. Two Dimensional Case

Computer Studies on the Approach to Thermal Equilibrium in Coupled Anharmonic Oscillators. I. Two Dimensional Case