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

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.

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

Fonte: Águas de Lindóia
Publicador: Águas de Lindóia

Tipo: Aula

ENG

Relevância na Pesquisa

56.89%

#Heat conduction#Harmonic and anharmonic oscillators#Statistical physics#FÍSICA#MECÂNICA ESTATÍSTICA

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/12/2008

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/05/2002

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/11/2004

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

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## Quantum entanglement of anharmonic oscillators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## The Bounded Anharmonic Oscillators: a simple approach

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/12/2008

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/12/2010

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/09/2004

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/10/1995

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/09/2014

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/06/2007

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Spectral instability for even non-selfadjoint anharmonic oscillators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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.

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## Euclidean Gibbs states of interacting quantum anharmonic oscillators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/09/2006

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 $\mathcal{G}^{\rm t}$...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/01/2009

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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...

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## Calculation of the Characteristic Functions of Anharmonic Oscillators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Optimal linearization of anharmonic oscillators

Fonte: Rochester Instituto de Tecnologia
Publicador: Rochester Instituto de Tecnologia

Tipo: Tese de Doutorado

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.

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