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## Fourier´s law from a chain of coupled anharmonic oscillators under energy conserving shot noise.

Landi, Gabriel Teixeira; Oliveira, Mario Jose de
Fonte: Águas de Lindóia Publicador: Águas de Lindóia
Tipo: Conferência ou Objeto de Conferência
ENG
Relevância na Pesquisa
56.73%
We analyze the transport of heat along a chain of particles interacting through anharmonic potentials consisting of quartic terms in addition to harmonic quadratic terms and subject to heat reservoirs at its ends. Each particle is also subject to an impulsive shot noise with exponentially distributed waiting times whose effect is to change the sign of its velocity, thus conserving the energy of the chain. We show that the introduction of this energy conserving stochastic noise leads to Fourier's law. That is for large system size L the heat current J behaves as J ‘approximately’ 1/L, which amounts to say that the conductivity k is constant. The conductivity is related to the current by J = kΔT/L, where ΔT is the difference in the temperatures of the reservoirs. The behavior of heat conductivity k for small intensities¸ of the shot noise and large system sizes L are obtained by assuming a scaling behavior of the type k = ‘L POT a Psi’(L’lambda POT a/b’) where a and b are scaling exponents. For the pure harmonic case a = b = 1, characterizing a ballistic conduction of heat when the shot noise is absent. For the anharmonic case we found values for the exponents a and b smaller then 1 and thus consistent with a superdiffusive conduction of heat without the shot noise. We also show that the heat conductivity is not constant but is an increasing function of temperature.

## Heat conduction in a chain of harmonic and anharmonic oscillators under the presence of an energy conserving noise

Oliveira, Mario Jose de; Landi, Gabriel Teixeira
Fonte: Águas de Lindóia Publicador: Águas de Lindóia
Tipo: Aula
ENG
Relevância na Pesquisa
56.89%
Reproducing Fourier's law of heat conduction from a microscopic stochastic model is a long standing challenge in statistical physics. As was shown by Rieder, Lebowitz and Lieb many years ago, a chain of harmonically coupled oscillators connected to two heat baths at different temperatures does not reproduce the diffusive behaviour of Fourier's law, but instead a ballistic one with an infinite thermal conductivity. Since then, there has been a substantial effort from the scientific community in identifying the key mechanism necessary to reproduce such diffusivity, which usually revolved around anharmonicity and the effect of impurities. Recently, it was shown by Dhar, Venkateshan and Lebowitz that Fourier's law can be recovered by introducing an energy conserving noise, whose role is to simulate the elastic collisions between the atoms and other microscopic degrees of freedom, which one would expect to be present in a real solid. For a one-dimensional chain this is accomplished numerically by randomly flipping - under the framework of a Poisson process with a variable “rate of collisions" - the sign of the velocity of an oscillator. In this poster we present Langevin simulations of a one-dimensional chain of oscillators coupled to two heat baths at different temperatures. We consider both harmonic and anharmonic (quartic) interactions...

## Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Bervillier, C.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.54%
The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential $V(x) =\beta x^{2}+x^{2m}$ ($m>0$). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE with two-point boundaries (one being located at infinity), the solution of which has singularities in the complex plane of the independent variable $x$. A conformal mapping of a suitably chosen angular sector of the complex plane of $x$ upon the unit disc centered at the origin makes convergent the transformed Taylor series of the generic solution so that the boundary condition at infinity can be easily imposed. In principle, this constraint, when applied on the logarithmic-derivative of the wave function, determines the eigenvalues to an arbitrary level of accuracy. In practice, for $\beta \geq 0$ or slightly negative, the accuracy of the results obtained is astonishingly large with regards to the modest computing power used. It is explained why the efficiency of the method decreases as $\beta$ is more and more negative. Various aspects of the method and comparisons with some seemingly similar methods, based also on expressing the solution as a Taylor series...

## Breather initial profiles in chains of weakly coupled anharmonic oscillators

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.54%
A systematic correlation between the initial profile of discrete breathers and their frequency is described. The context is that of a very weakly harmonically coupled chain of softly anharmonic oscillators. The results are structurally stable, that is, robust under changes of the on-site potential and are illustrated numerically for several standard choices. A precise genericity theorem for the results is proved.; Comment: 12 pages, 4 figures

## A new strategy to find bound states in anharmonic oscillators

Gomez, Francisco J.; Sesma, Javier
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.64%
A very simple procedure to calculate eigenenergies of quantum anharmonic oscillators is presented. The method, exact but for numerical computations, consists merely in requiring the vanishing of the Wronskian of two solutions which are regular respectively at the origin and at infinity. The first one can be represented by its series expansion; for the second one, an asymptotic expansion is available. The procedure is illustrated by application to quartic and sextic oscillators.; Comment: A cotribution to a book dedicated to Prof. Alberto Galindo, on occasion of his seventieth birthday

## Quantum entanglement of anharmonic oscillators

Joshi, Chaitanya; Jonson, Mats; Andersson, Erika; Ohberg, Patrik
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.86%
We investigate the quantum entanglement dynamics of undriven anharmonic (nonlinear) oscillators with quartic potentials. We first consider the indirect interaction between two such nonlinear oscillators mediated by a third, linear oscillator and show that it leads to a time-varying entanglement of the oscillators, the entanglement being strongly influenced by the nonlinear oscillator dynamics. In the presence of dissipation, the role of nonlinearity is strongly manifested in the steady state dynamics of the indirectly coupled anharmonic oscillators. We further illustrate the effect of nonlinearities by studying the coupling between an electromagnetic field in a cavity with one movable mirror which is modeled as a nonlinear oscillator. For this case we present a full analytical treatment, which is valid in a regime where both the nonlinearity and the coupling due to radiation pressure is weak. We show that, without the need of any conditional measurements on the cavity field, the state of the movable mirror is non-classical as a result of the combined effect of the intrinsic nonlinearity and the radiation pressure coupling. This interaction is also shown to be responsible for squeezing the movable mirror's position quadrature beyond the minimum uncertainty state even when the mirror is initially prepared in its ground state.; Comment: 1 column 18 pages

## The Bounded Anharmonic Oscillators: a simple approach

Maiz, F.; Nasr, M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.82%
We have developed a simple method to solve anharmonic oscillators equations. The idea of our method is mainly based on the partitioning of the potential curve into (n+1) small intervals, solving the Schr\"odinger equation in each subintervals, and writing the continuity conditions of each solutions at each subintervals boundary, which leads to the energy quantification condition, so to the energy levels, and finally, one can calculate the exact wave function associated to each energy level. Our method has been applied to three examples: the well-known square well, the bounded and unbounded harmonic oscillators, the Morse potential, and some anharmonic oscillators bounded by two infinite walls. Our method is more realistic, simpler, with high degree of accuracy, both satisfactory and not computationally complicated, and applicable for any forms of potential. Our results agree very well with the preceding ones.; Comment: 14 pages

## Coupled anharmonic oscillators: the Rayleigh-Ritz approach versus the collocation approach

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.54%
For a system of coupled anharmonic oscillators we compare the convergence rate of the variational collocation approach presented recently by Amore and Fernandez (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.; Comment: 7 pages, 1 figure

## Exact and approximate expressions for the period of anharmonic oscillators

Amore, Paolo; Fernandez, Francisco M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.54%
In this paper we present a straightforward systematic method for the exact and approximate calculation of integrals that appear in formulas for the period of anharmonic oscillators and other problems of interest in classical mechanics.; Comment: 7 pages

## Connection Between q-Deformed Anharmonic Oscillators and Quasi-Exactly Soluble Potentials

Bonatsos, Dennis; Daskaloyannis, C.; Mavromatis, Harry A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.54%
It is proved that quasi-exactly soluble potentials (QESPs) corresponding to an oscillator with harmonic, quartic and sextic terms, for which the $n+1$ lowest levels of a given parity can be determined exactly, may be approximated by WKB equivalent potentials corresponding to deformed anharmonic oscillators of SU$_q$(1,1) symmetry, which have been used for the description of vibrational spectra of diatomic molecules. This connection allows for the immediate approximate determination of all levels of the QESPs.; Comment: Plain TeX, 11 pages

## Unitary transformations of a family of two-dimensional anharmonic oscillators

Fernández, Francisco M.; Garcia, Javier
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.64%
In this paper we analyze a recent application of perturbation theory by the moment method to a family of two-dimensional anharmonic oscillators. By means of straightforward unitary transformations we show that two of the models studied by the authors are separable. Other is unbounded from below and therefore cannot be successfully treated by perturbation theory unless a complex harmonic frequency is introduced in the renormalization process. We calculate the lowest resonance by means of complex-coordinate rotation and compare its real part with the eigenvalue estimated by the authors. A pair of the remaining oscillators are equivalent as they can be transformed into one another by unitary transformations.

## Normal Heat Conductivity in a strongly pinned chain of anharmonic oscillators

Lefevere, R.; Schenkel, A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.65%
We consider a chain of coupled and strongly pinned anharmonic oscillators subject to a non-equilibrium random forcing. Assuming that the stationary state is approximately Gaussian, we first derive a stationary Boltzmann equation. By localizing the involved resonances, we next invert the linearized collision operator and compute the heat conductivity. In particular, we show that the Gaussian approximation yields a finite conductivity $\kappa\sim\frac{1}{\lambda^2T^2}$, for $\lambda$ the anharmonic coupling strength.; Comment: Introduction and conclusion modified

## A General Scheme for Construction of Coherent States of Anharmonic Oscillators

Molski, Marcin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.7%
A mixed supersymmetric-algebraic approach to construction of the minimum uncertainty coherent states of anharmonic oscillators is presented. It permits generating not only the well-known coherent states of the harmonic and Morse oscillators but also the so far unknown coherent states of the Wei Hua, Kratzer-Fues and generalized Morse and Kratzer-Fues oscillators. The method can be applied also to generate superpotentials indispensable for deriving the Schr\"odinger equation in the supersymmetric form amenable to direct solution in the SUSYQM scheme.; Comment: 5 pages. Submitted to Physical Review Letters

## Computing Energy Eigenvalues of Anharmonic Oscillators using the Double Exponential Sinc collocation Method

Gaudreau, Philippe; Slevinsky, Richard; Safouhi, Hassan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.65%
A quantum anharmonic oscillator is defined by the Hamiltonian ${\cal H}= -\frac{ {\rm d^{2}}}{{\rm d}x^{2}} + V(x)$, where the potential is given by $V(x) = \sum_{i=1}^{m} c_{i} x^{2i}$ with $c_{m}>0$. Using the Sinc collocation method combined with the double exponential transformation, we develop a method to efficiently compute highly accurate approximations of energy eigenvalues for anharmonic oscillators. Convergence properties of the proposed method are presented. Using the principle of minimal sensitivity, we introduce an alternate expression for the mesh size for the Sinc collocation method which improves considerably the accuracy in computing eigenvalues for potentials with multiple wells. We apply our method to a number of potentials including potentials with multiple wells. The numerical results section clearly illustrates the high efficiency and accuracy of the proposed method. All our codes are written using the programming language Julia and are available upon request.; Comment: 18 pages and 16 figures

## Spectral instability for even non-selfadjoint anharmonic oscillators

Henry, Raphaël
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.64%
We study the instability of the spectrum for a class of non-selfadjoint anharmonic oscillators, estimating the behavior of the instability indices (i. e. the norm of spectral projections) associated with the large eigenvalues of these oscillators. We get asymptotic expansions for the instability indices, extending the results of Davies and Davies-Kuijlaars.

## Euclidean Gibbs states of interacting quantum anharmonic oscillators

Kozitsky, Y.; Pasurek, T.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.76%
A rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting $\nu$-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set $\mathbb{L}\subset \mathbb{R}^d$, possibly irregular; the anharmonic potentials vary from site to site. The description is based on the representation of the Gibbs states in terms of path measures -- the so called Euclidean Gibbs measures. It is proven that: (a) the set of such measures $\mathcal{G}^{\rm t}$ is non-void and compact; (b) every $\mu \in \mathcal{G}^{\rm t}$ obeys an exponential integrability estimate, the same for the whole set $\mathcal{G}^{\rm t}$; (c) every $\mu \in \mathcal{G}^{\rm t}$ has a Lebowitz-Presutti type support; (d) $\mathcal{G}^{\rm t}$ is a singleton at high temperatures. In the case of attractive interaction and $\nu=1$ we prove that $|\mathcal{G}^{\rm t}|>1$ at low temperatures. The uniqueness of Gibbs measures due to quantum effects and at a nonzero external field are also proven in this case. Thereby, a qualitative theory of phase transitions and quantum effects, which interprets most important experimental data known for the corresponding physical objects, is developed. The mathematical result of the paper is a complete description of the set $\mathcal{G}^{\rm t}$...

## Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree

Jentschura, U. D.; Surzhykov, A.; Zinn-Justin, J.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.76%
We present a unified treatment, including higher-order corrections, of anharmonic oscillators of arbitrary even and odd degree. Our approach is based on a dispersion relation which takes advantage of the PT-symmetry of odd potentials for imaginary coupling parameter, and of generalized quantization conditions which take into account instanton contributions. We find a number of explicit new results, including the general behaviour of large-order perturbation theory for arbitrary levels of odd anharmonic oscillators, and subleading corrections to the decay width of excited states for odd potentials, which are numerically significant.; Comment: 5 pages, RevTeX

## Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators

Jentschura, U. D.; Surzhykov, A.; Zinn-Justin, J.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.93%
This is the third article in a series of three papers on the resonance energy levels of anharmonic oscillators. Whereas the first two papers mainly dealt with double-well potentials and modifications thereof [see J. Zinn-Justin and U. D. Jentschura, Ann. Phys. (N.Y.) 313 (2004), pp. 197 and 269], we here focus on simple even and odd anharmonic oscillators for arbitrary magnitude and complex phase of the coupling parameter. A unification is achieved by the use of PT-symmetry inspired dispersion relations and generalized quantization conditions that include instanton configurations. Higher-order formulas are provided for the oscillators of degrees 3 to 8, which lead to subleading corrections to the leading factorial growth of the perturbative coefficients describing the resonance energies. Numerical results are provided, and higher-order terms are found to be numerically significant. The resonances are described by generalized expansions involving intertwined non-analytic exponentials, logarithmic terms and power series. Finally, we summarize spectral properties and dispersion relations of anharmonic oscillators, and their interconnections. The purpose is to look at one of the classic problems of quantum theory from a new perspective...

## Calculation of the Characteristic Functions of Anharmonic Oscillators

Jentschura, Ulrich D.; Zinn-Justin, Jean
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.65%
The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrodinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is B_m(E,g) = n + 1/2, where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel-Kramers-Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function A_m(E,g). The evaluation of A_m(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.; Comment: 11 pages, LaTeX; three further typographical errors corrected

## Optimal linearization of anharmonic oscillators

Lee, Jungkun
Fonte: Rochester Instituto de Tecnologia Publicador: Rochester Instituto de Tecnologia
EN_US
Relevância na Pesquisa
56.64%
This investigation is based on the geometric analysis of phase trajectories and incurred vector fields associated with nonlinear oscillators. Optimal curve fitting techniques are applied in the phase plane, in an effort to generate a so-called "geometric averaging". The results are then compared with those generated by classical techniques such as harmonic balance and equivalent linearization, as well as by numerical integration. The investigation is extended to nonlinear mult iple-degree-of -freedom systems. Frequencies of oscillations and mode shapes are derived based on the optimal equivalent linearization process. The results are also compared with numerical integration for justification. It is shown that the proposed linearization methods are simple to implement and provide an efficient methodology for the analysis of nonlinear oscillations.