Electronic trajectories in atomic physics: The chemical bond in the H2 + ion.

The H2 + ion is the simplest example in which a chemical bond exists, created by one electron between two protons. As all chemical bonds, it is usually considered inexplicable in a classical frame. Here, in view of the extremely large velocities attained by the electron near the protons, we consider a relativistic extension of the standard classical three-body model. This has a great impact since the reference unperturbed system (clamped protons) is no more integrable, and indeed by molecular dynamics simulations, we find that the modification entails the existence of a large region of strongly chaotic motions for the unperturbed system, which lead, for the full system, to a collapse of the molecule. For motions of generic type, with the electron bouncing between the protons, there exists an open region of motions regular enough for producing a bond. Such a region is characterized by the property that the electron's trajectories have an angular momentum pφ along the inter-nuclear axis of the order of the reduced Planck's constant ℏ. Moreover, special initial data exist for which the experimental bond length and oscillation frequency of the protons (but not the dissociation energy) are well reproduced. Also, well reproduced is the quantum potential, albeit only in an extended interval about the minimum.

Chaoticity threshold in magnetized plasmas: numerical results in the weak coupling regime.

The present paper is a numerical counterpart to the theoretical work [Carati et al., Chaos 22, 033124 (2012)]. We are concerned with the transition from order to chaos in a one-component plasma (a system of point electrons with mutual Coulomb interactions, in a uniform neutralizing background), the plasma being immersed in a uniform stationary magnetic field. In the paper [Carati et al., Chaos 22, 033124 (2012)], it was predicted that a transition should take place when the electron density is increased or the field decreased in such a way that the ratio ωp/ωc between plasma and cyclotron frequencies becomes of order 1, irrespective of the value of the so-called Coulomb coupling parameter Γ. Here, we perform numerical computations for a first principles model of N point electrons in a periodic box, with mutual Coulomb interactions, using as a probe for chaoticity the time-autocorrelation function of magnetization. We consider two values of Γ (0.04 and 0.016) in the weak coupling regime Γ â‰ª 1, with N up to 512. A transition is found to occur for ωp/ωc in the range between 0.25 and 2, in fairly good agreement with the theoretical prediction. These results might be of interest for the problem of the breakdown of plasma confinement in fusion machines.

Transition from order to chaos, and density limit, in magnetized plasmas.

It is known that a plasma in a magnetic field, conceived microscopically as a system of point charges, can exist in a magnetized state, and thus remain confined, inasmuch as it is in an ordered state of motion, with the charged particles performing gyrational motions transverse to the field. Here, we give an estimate of a threshold, beyond which transverse motions become chaotic, the electrons being unable to perform even one gyration, so that a breakdown should occur, with complete loss of confinement. The estimate is obtained by the methods of perturbation theory, taking as perturbing force acting on each electron that due to the so-called microfield, i.e., the electric field produced by all the other charges. We first obtain a general relation for the threshold, which involves the fluctuations of the microfield. Then, taking for such fluctuations, the formula given by Iglesias, Lebowitz, and MacGowan for the model of a one component plasma with neutralizing background, we obtain a definite formula for the threshold, which corresponds to a density limit increasing as the square of the imposed magnetic field. Such a theoretical density limit is found to fit pretty well the empirical data for collapses of fusion machines.

Classical helium atom with radiation reaction.

We study a classical model of helium atom in which, in addition to the Coulomb forces, the radiation reaction forces are taken into account. This modification brings in the model a new qualitative feature of a global character. Indeed, as pointed out by Dirac, in any model of classical electrodynamics of point particles involving radiation reaction one has to eliminate, from the a priori conceivable solutions of the problem, those corresponding to the emission of an infinite amount of energy. We show that the Dirac prescription solves a problem of inconsistency plaguing all available models which neglect radiation reaction, namely, the fact that in all such models, most initial data lead to a spontaneous breakdown of the atom. A further modification is that the system thus acquires a peculiar form of dissipation. In particular, this makes attractive an invariant manifold of special physical interest, the zero-dipole manifold that corresponds to motions in which no energy is radiated away (in the dipole approximation). We finally study numerically the invariant measure naturally induced by the time-evolution on such a manifold, and this corresponds to studying the formation process of the atom. Indications are given that such a measure may be singular with respect to that of Lebesgue.

Relaxation properties in classical diamagnetism.

It is an old result of Bohr that, according to classical statistical mechanics, at equilibrium a system of electrons in a static magnetic field presents no magnetization. Thus a magnetization can occur only in an out of equilibrium state, such as that produced through the Foucault currents when a magnetic field is switched on. It was suggested by Bohr that, after the establishment of such a nonequilibrium state, the system of electrons would quickly relax back to equilibrium. In the present paper, we study numerically the relaxation to equilibrium in a modified Bohr model, which is mathematically equivalent to a billiard with obstacles, immersed in a magnetic field that is adiabatically switched on. We show that it is not guaranteed that equilibrium is attained within the typical time scales of microscopic dynamics. Depending on the values of the parameters, one has a relaxation either to equilibrium or to a diamagnetic (presumably metastable) state. The analogy with the relaxation properties in the Fermi Pasta Ulam problem is also pointed out.

On the energy transfer to small scales in a discrete model of one-dimensional turbulence.

We consider a discrete one-dimensional model that was investigated numerically by Daumont and Peyrard [Chaos 13, 624 (2003)] as a model for turbulence in fluids, i.e., for the energy transfer from large to small scales. They found numerically that there exist two different regimes for the energy spectrum at high energies and low energies, respectively, and gave an analytical explanation for the high-energy spectrum. An analytical explanation is given here for the low-energy spectrum, which corresponds to the laminar regime.

Fermi-Pasta-Ulam phenomenon for generic initial data.

The well-known Fermi-Pasta-Ulam (FPU) phenomenon (lack of attainment of equipartition of the mode energies at low energies for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initial data of full measure, sampled out from the microcanonical ensemble by standard Monte Carlo techniques. Mixing is tested by looking at the decay of the autocorrelations of the mode energies, and it is found that the high-frequency modes have autocorrelations that tend instead to positive values. Indications are given that such a nonmixing property survives in the thermodynamic limit. It is left as an open problem whether mixing actually occurs, i.e., whether the autocorrelations vanish as time tends to infinity.

The Fermi-Pasta-Ulam problem as a challenge for the foundations of physics.

The Fermi-Pasta-Ulam (FPU) problem is discussed in connection with its physical relevance, and it is shown how apparently there exist only two possibilities: either the FPU problem is just a curiosity, or it has a fundamental role for the foundations of physics, casting a new light on the relations between classical and quantum mechanics. To this end, a short review is given of the main conceptual proposals that have been advanced. Particular emphasis is given to the perspective of a metaequilibrium scenario, which appears to be the only possible one for the FPU paradox to survive in the physically relevant case of infinitely many particles.

Analytical estimate of stochasticity thresholds in Fermi-Pasta-Ulam and straight phi(4) models.

We consider an infinitely extended Fermi-Pasta-Ulam model. We show that the slowly modulating amplitude of a narrow wave packet asymptotically satisfies the nonlinear Schrödinger equation (NLS) on the real axis. Using well known results from inverse scattering theory, we then show that there exists a threshold of the energy of the central normal mode of the packet, with the following properties. Below threshold the NLS equation presents a quasilinear regime with no solitons in the solution of the equation, and the wave packet width remains narrow. Above threshold generation of solitons is possible instead and the packet of normal modes can spread out. Analogous results are obtained for the straight phi(4) model. We also give an analytical estimate for such thresholds. Finally, we make a comparison with the numerical results known to us and show that, they are in remarkable agreement with our estimates.

Analog of Planck's formula and effective temperature in classical statistical mechanics far from equilibrium

We study the statistical mechanics very far from equilibrium for a classical system of harmonic oscillators colliding with point particles (mimicking a heat reservoir), for negligible initial energies of the oscillators. It is known that for high frequencies the times of relaxation to equilibrium are extremely long, so that one meets with situations of quasiequilibrium very far from equilibrium similar to those of glassy systems. Using recent results from the theory of dynamical systems, we deduce a functional relation between energy variance and mean energy that was introduced by Einstein phenomenologically in connection with Planck's formula. It is then discussed how this leads to an analog of Planck's formula. This requires using Einstein's relation between specific heat and energy variance to define an effective temperature in a context of quasiequilibrium far from equilibrium, as is familiar for glassy systems.