We present numerical experiments on Hamiltonian nonlinear chains at fixed specific energy in the ... more We present numerical experiments on Hamiltonian nonlinear chains at fixed specific energy in the stochastic domain with a growing number of degrees of freedom, up to N = 2048. Previous results on the rates of changes of action variables, with reference to translational invariance, are confirmed and specified. Furthermore, for models with other conserved quantities beside energy, an increasing number
The asymptotic behavior of form factors for two-and thAee-particle bound states are investigated ... more The asymptotic behavior of form factors for two-and thAee-particle bound states are investigated in the case of spin-4 constituents in order to shed some light on the underlying J structure of the pi,on and nucleon. Here-the Blankenbecler-Sugar approach proves to be a powerful tool for studying dynamics at infinite momentum. For a two-body interaction which for large momentum transfer behaves as V(q,k) = [(q-k)2]-'-a we obtain for the twoand three-body form factors F2 = [Q2]-3/2-A and F3 = [Q2]-3-2A respectively in the case of scalar and pseudoscalar couplings and F2 = [Q']-1-A and F3 = [Q2]-2-2A for the vector coupling. The experimental pion and nucleon form factors are, e.g., consistently recovered assigning a quark-antiquark and three-quark structure to the pion and nucleon respectively and the quarks interacting via vector-gluon exchange (i.e., A-+ 0).
The asymptotic power behavior of the electromagnetic form factors are examined for two-and three-... more The asymptotic power behavior of the electromagnetic form factors are examined for two-and three-body s-w&e bound states both relativisitc and nonrelativistic. In the nonrelativistic case we consider local and separable two-body potentials and we make use of the Faddeev equations in order to define the three-body bound states. For local potentials which behave as G2,-e (0 < 0 5 1) for large momentum transfer, we obtain for the asymptotic power behavior of the form factors of the two-and three-body bound states F2(y2) = (ly*r)-3-e and F3(q2) = (13)-6-2e, respectively. For separable potentials V = g(IFI) g(Iktl) and g(IFI) = (l~l)-'-e we find F,(2) = (lTl)-2-e
The threshold behavior of the structure function vW2 is I_ evaluated for twoand three-body bound ... more The threshold behavior of the structure function vW2 is I_ evaluated for twoand three-body bound states described by the Comparison with previous results on form factors show*that the Drell-Yan-West relation is satisfied in all the models considered here.
ABSTRACT We evaluate the power spectra of the time series for the following simple observables in... more ABSTRACT We evaluate the power spectra of the time series for the following simple observables in the Fermi-Pasta-Ulam model: harmonic energy, kinetic energy, microcanonical density, Frenet-Serret curvature and the Lyapunov variable. For some of these observables, also in the stochastic regime, the spectra show a well defined quasi-harmonic structure, with harmonic frequencies shifted with a single rescaling factor, as calculated in a previous paper. Even higher frequencies are excited: as replicas of the harmonic window at low energy, to end up with a smooth distribution at high energy, showing a power law behaviour (flicker noise). In the intermediate region the shape depends on the observables, but in all cases the crossover is the maximum of the shifted harmonic spectrum. This establishes an intrinsic short-time scale depending only on the energy density, as does the frequency rescaling factor. For the curvature, we also evaluate the standard deviation: above threshold, at increasing energy, it decreases exactly as the inverse of the rescaling factor. This can be interpreted as a focalization around `effective tori&#39; of a harmonic-like regime which apparently coexist with the chaotic motion.
We study the relevance of several physical parameters in regulating the rate of energy exchanges ... more We study the relevance of several physical parameters in regulating the rate of energy exchanges for nonlinear systems in the stochastic domain. In particular, exploring the structural features which differentiate models of radiant cavities from nonlinear chains, we prove that the "superfreezing", i.e. the quasi harmonical behaviour of the action variables for high frequencies, essentially depends on the presence of an external potential acting on the centre of mass, while it seems roughly independent of the shape of the spectrum.
Results of calculations on a model of a radiant cavity, performed in order to explore the relatio... more Results of calculations on a model of a radiant cavity, performed in order to explore the relation between stochasticity and geometrical structure of phase space, are presented. The rate of energy exchanges, as indicator of stochasticity, is found to be quite effective. Furthermore, a trend to equipartition for such a quantity is observed at increasing energy, and this implies an increasing rigidity of high harmonic modes also in the stochastic regime of motion. Such a feature may be correlated to the shape of the spectrum which characterizes the radiant cavity with respect to nonlinear chains.
In the context of Pad~ approximation theory some methods are studied for the calculation of the r... more In the context of Pad~ approximation theory some methods are studied for the calculation of the resonances and the phase shifts in potential scattering. They are extensions of a method already used for the determination of the bound states and are based on the variational content of the Padd approximation. Essentially these methods amount to work with off-shell amplitudes taking the off-shell momenta as variational parameters. Numerical examples are reported which always show a definite improvement over the results of the conventional on-shell Padd approximation.
In anharmonic chains with even potentials, including classical Fermi- Pasta-Ulam model, we show h... more In anharmonic chains with even potentials, including classical Fermi- Pasta-Ulam model, we show how ordered structures can coexist with high degree stochasticity.
Some difficulties have been lately pointed out in the application of the Pad6 approximant (P.A.) ... more Some difficulties have been lately pointed out in the application of the Pad6 approximant (P.A.) technique to the calculation of the bound states (1,2). These difficulties arise from the fact that, in this context, the bound states are usually studied as poles of the P.A.'s to the on-shell scattering amplitude. Ill fact, although it has been shown (3) that, for a large class of potentials, the normalized denominators of the diagonal P.A.'s converge to the Jost function, at any finite order these denominators have not only the right-hand cut in the complex energy plane, but also a left-hand one. The latter is characteristic of the on-shell perturbative terms, but, of course, it does not appear in the exact Jost function. An indirect way to circumvent~ueh difficulties has been proposed by TURCHETT1 and VILLA~I (a). It seems, however, that the most natural way consists in applying the P.A. technique either to the diagonal elements of the resolvent operator B(z)= 1~(H-z)= 1/(Ho+gV-z) of the Hamiltonian, or to those of the related operator T(z) = gV-g2 VR(z) V.
A model ofnonlinear oscillators related to classical electrodynamics is studied. Using an indicat... more A model ofnonlinear oscillators related to classical electrodynamics is studied. Using an indicator based on the energy exchange of harmonic modes, it is possible to establish that equipartition of energy and equipartition of exchanges go hand in hand: this entails a phenomenon of quasi-harmonic behavior ofhigh modes which we call "superfreezing". Comparisons with similar experiments on other models, where superfreezing does not occur, seem to indicate the kind ofnonlinear coupling and the shape of the spectrum as responsible for this phenomenon.
We study the relevance of several physical parameters in regulating the rate of energy exchanges ... more We study the relevance of several physical parameters in regulating the rate of energy exchanges for nonlinear systems in the stochastic domain. In particular, exploring the structural features which differentiate models of radiant cavities from nonlinear chains, we prove that the "superfreezing", i.e. the quasi harmonical behaviour of the action variables for high frequencies, essentially depends on the presence of an external potential acting on the centre of mass, while it seems roughly independent of the shape of the spectrum.
We present numerical experiments on Hamiltonian nonlinear chains at fixed specific energy in the ... more We present numerical experiments on Hamiltonian nonlinear chains at fixed specific energy in the stochastic domain with a growing number of degrees of freedom, up to N = 2048. Previous results on the rates of changes of action variables, with reference to translational invariance, are confirmed and specified. Furthermore, for models with other conserved quantities beside energy, an increasing number
The asymptotic behavior of form factors for two-and thAee-particle bound states are investigated ... more The asymptotic behavior of form factors for two-and thAee-particle bound states are investigated in the case of spin-4 constituents in order to shed some light on the underlying J structure of the pi,on and nucleon. Here-the Blankenbecler-Sugar approach proves to be a powerful tool for studying dynamics at infinite momentum. For a two-body interaction which for large momentum transfer behaves as V(q,k) = [(q-k)2]-'-a we obtain for the twoand three-body form factors F2 = [Q2]-3/2-A and F3 = [Q2]-3-2A respectively in the case of scalar and pseudoscalar couplings and F2 = [Q']-1-A and F3 = [Q2]-2-2A for the vector coupling. The experimental pion and nucleon form factors are, e.g., consistently recovered assigning a quark-antiquark and three-quark structure to the pion and nucleon respectively and the quarks interacting via vector-gluon exchange (i.e., A-+ 0).
The asymptotic power behavior of the electromagnetic form factors are examined for two-and three-... more The asymptotic power behavior of the electromagnetic form factors are examined for two-and three-body s-w&e bound states both relativisitc and nonrelativistic. In the nonrelativistic case we consider local and separable two-body potentials and we make use of the Faddeev equations in order to define the three-body bound states. For local potentials which behave as G2,-e (0 < 0 5 1) for large momentum transfer, we obtain for the asymptotic power behavior of the form factors of the two-and three-body bound states F2(y2) = (ly*r)-3-e and F3(q2) = (13)-6-2e, respectively. For separable potentials V = g(IFI) g(Iktl) and g(IFI) = (l~l)-'-e we find F,(2) = (lTl)-2-e
The threshold behavior of the structure function vW2 is I_ evaluated for twoand three-body bound ... more The threshold behavior of the structure function vW2 is I_ evaluated for twoand three-body bound states described by the Comparison with previous results on form factors show*that the Drell-Yan-West relation is satisfied in all the models considered here.
ABSTRACT We evaluate the power spectra of the time series for the following simple observables in... more ABSTRACT We evaluate the power spectra of the time series for the following simple observables in the Fermi-Pasta-Ulam model: harmonic energy, kinetic energy, microcanonical density, Frenet-Serret curvature and the Lyapunov variable. For some of these observables, also in the stochastic regime, the spectra show a well defined quasi-harmonic structure, with harmonic frequencies shifted with a single rescaling factor, as calculated in a previous paper. Even higher frequencies are excited: as replicas of the harmonic window at low energy, to end up with a smooth distribution at high energy, showing a power law behaviour (flicker noise). In the intermediate region the shape depends on the observables, but in all cases the crossover is the maximum of the shifted harmonic spectrum. This establishes an intrinsic short-time scale depending only on the energy density, as does the frequency rescaling factor. For the curvature, we also evaluate the standard deviation: above threshold, at increasing energy, it decreases exactly as the inverse of the rescaling factor. This can be interpreted as a focalization around `effective tori&#39; of a harmonic-like regime which apparently coexist with the chaotic motion.
We study the relevance of several physical parameters in regulating the rate of energy exchanges ... more We study the relevance of several physical parameters in regulating the rate of energy exchanges for nonlinear systems in the stochastic domain. In particular, exploring the structural features which differentiate models of radiant cavities from nonlinear chains, we prove that the "superfreezing", i.e. the quasi harmonical behaviour of the action variables for high frequencies, essentially depends on the presence of an external potential acting on the centre of mass, while it seems roughly independent of the shape of the spectrum.
Results of calculations on a model of a radiant cavity, performed in order to explore the relatio... more Results of calculations on a model of a radiant cavity, performed in order to explore the relation between stochasticity and geometrical structure of phase space, are presented. The rate of energy exchanges, as indicator of stochasticity, is found to be quite effective. Furthermore, a trend to equipartition for such a quantity is observed at increasing energy, and this implies an increasing rigidity of high harmonic modes also in the stochastic regime of motion. Such a feature may be correlated to the shape of the spectrum which characterizes the radiant cavity with respect to nonlinear chains.
In the context of Pad~ approximation theory some methods are studied for the calculation of the r... more In the context of Pad~ approximation theory some methods are studied for the calculation of the resonances and the phase shifts in potential scattering. They are extensions of a method already used for the determination of the bound states and are based on the variational content of the Padd approximation. Essentially these methods amount to work with off-shell amplitudes taking the off-shell momenta as variational parameters. Numerical examples are reported which always show a definite improvement over the results of the conventional on-shell Padd approximation.
In anharmonic chains with even potentials, including classical Fermi- Pasta-Ulam model, we show h... more In anharmonic chains with even potentials, including classical Fermi- Pasta-Ulam model, we show how ordered structures can coexist with high degree stochasticity.
Some difficulties have been lately pointed out in the application of the Pad6 approximant (P.A.) ... more Some difficulties have been lately pointed out in the application of the Pad6 approximant (P.A.) technique to the calculation of the bound states (1,2). These difficulties arise from the fact that, in this context, the bound states are usually studied as poles of the P.A.'s to the on-shell scattering amplitude. Ill fact, although it has been shown (3) that, for a large class of potentials, the normalized denominators of the diagonal P.A.'s converge to the Jost function, at any finite order these denominators have not only the right-hand cut in the complex energy plane, but also a left-hand one. The latter is characteristic of the on-shell perturbative terms, but, of course, it does not appear in the exact Jost function. An indirect way to circumvent~ueh difficulties has been proposed by TURCHETT1 and VILLA~I (a). It seems, however, that the most natural way consists in applying the P.A. technique either to the diagonal elements of the resolvent operator B(z)= 1~(H-z)= 1/(Ho+gV-z) of the Hamiltonian, or to those of the related operator T(z) = gV-g2 VR(z) V.
A model ofnonlinear oscillators related to classical electrodynamics is studied. Using an indicat... more A model ofnonlinear oscillators related to classical electrodynamics is studied. Using an indicator based on the energy exchange of harmonic modes, it is possible to establish that equipartition of energy and equipartition of exchanges go hand in hand: this entails a phenomenon of quasi-harmonic behavior ofhigh modes which we call "superfreezing". Comparisons with similar experiments on other models, where superfreezing does not occur, seem to indicate the kind ofnonlinear coupling and the shape of the spectrum as responsible for this phenomenon.
We study the relevance of several physical parameters in regulating the rate of energy exchanges ... more We study the relevance of several physical parameters in regulating the rate of energy exchanges for nonlinear systems in the stochastic domain. In particular, exploring the structural features which differentiate models of radiant cavities from nonlinear chains, we prove that the "superfreezing", i.e. the quasi harmonical behaviour of the action variables for high frequencies, essentially depends on the presence of an external potential acting on the centre of mass, while it seems roughly independent of the shape of the spectrum.
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Papers by Carlo Alabiso