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The randomness of some irreversible quantum phenomena is a central question because irreversible phenomena break quantum coherence and thus yield an irreversible loss of information. The case of quantum jumps observed in the fluorescence of a single two-level atom illuminated by a quasi-resonant laser beam is a worked example where statistical interpretations of quantum mechanics still meet some difficulties because the basic equations are fully deterministic and unitary. In such a problem with two different time scales, the atom makes coherent optical Rabi oscillations between the two states, interrupted by random emissions (quasi-instantaneous) of photons where coherence is lost. To describe this system, we already proposed a novel approach, which is completed here. It amounts to putting a probability on the density matrix of the atom and deducing a general "kinetic Kolmogorov-like" equation for the evolution of the probability. In the simple case considered here, the probability only depends on a single variable θ describing the state of the atom, and p(θ,t) yields the statistical properties of the atom under the joint effects of coherent pumping and random emission of photons. We emphasize that p(θ,t) allows the description of all possible histories of the atom, as in Everett's many-worlds interpretation of quantum mechanics. This yields solvable equations in the two-level atom case.
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Following the idea that dissipation in turbulence at high Reynolds number is dominated by singular events in space-time and described by solutions of the inviscid Euler equations, we draw the conclusion that in such flows, scaling laws should depend only on quantities appearing in the Euler equations. This excludes viscosity or a turbulent length as scaling parameters and constrains drastically possible analytical pictures of this limit. We focus on the drag law deduced by Newton for a projectile moving quickly in a fluid at rest. Inspired by this Newton's drag force law (proportional to the square of the speed of the moving object in the limit of large Reynolds numbers), which is well verified in experiments when the location of the detachment of the boundary layer is defined, we propose an explicit relationship between the Reynolds stress in the turbulent wake and quantities depending on the velocity field (averaged in time but depending on space). This model takes the form of an integrodifferential equation for the velocity which is eventually solved for a Poiseuille flow in a circular pipe.
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Because the collapse of massive stars occurs in a few seconds, while the stars evolve on billions of years, the supernovae are typical complex phenomena in fluid mechanics with multiple time scales. We describe them in the light of catastrophe theory, assuming that successive equilibria between pressure and gravity present a saddle-center bifurcation. In the early stage we show that the loss of equilibrium may be described by a generic equation of the Painlevé I form. This is confirmed by two approaches, first by the full numerical solutions of the Euler-Poisson equations for a particular pressure-density relation, secondly by a derivation of the normal form of the solutions close to the saddle-center. In the final stage of the collapse, just before the divergence of the central density, we show that the existence of a self-similar collapsing solution compatible with the numerical observations imposes that the gravity forces are stronger than the pressure ones. This situation differs drastically in its principle from the one generally admitted where pressure and gravity forces are assumed to be of the same order. Moreover it leads to different scaling laws for the density and the velocity of the collapsing material. The new self-similar solution (based on the hypothesis of dominant gravity forces) which matches the smooth solution of the outer core solution, agrees globally well with our numerical results, except a delay in the very central part of the star, as discussed. Whereas some differences with the earlier self-similar solutions are minor, others are very important. For example, we find that the velocity field becomes singular at the collapse time, diverging at the center, and decreasing slowly outside the core, whereas previous works described a finite velocity field in the core which tends to a supersonic constant value at large distances. This discrepancy should be important for explaining the emission of remnants in the post-collapse regime. Finally we describe the post-collapse dynamics, when mass begins to accumulate in the center, also within the hypothesis that gravity forces are dominant.
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Catastrophes of all kinds can be roughly defined as short-duration, large-amplitude events following and followed by long periods of "ripening." Major earthquakes surely belong to the class of "catastrophic" events. Because of the space-time scales involved, an experimental approach is often difficult, not to say impossible, however desirable it could be. Described in this article is a "laboratory" setup that yields data of a type that is amenable to theoretical methods of prediction. Observations are made of a critical slowing down in the noisy signal of a solder wire creeping under constant stress. This effect is shown to be a fair signal of the forthcoming catastrophe in two separate dynamical models. The first is an "abstract" model in which a time-dependent quantity drifts slowly but makes quick jumps from time to time. The second is a realistic physical model for the collective motion of dislocations (the Ananthakrishna set of equations for unstable creep). Hope thus exists that similar changes in the response to noise could forewarn catastrophes in other situations, where such precursor effects should manifest early enough.
Asunto(s)
Desastres , Algoritmos , Terremotos , Elasticidad , Diseño de Equipo , Geografía/métodos , Plomo/química , Modelos Estadísticos , Modelos Teóricos , Movimiento (Física) , Oscilometría/métodos , Física/métodos , Presión , Factores de TiempoRESUMEN
The intensity of classical bright solitons propagating in linearly coupled identical fibers can be distributed either in a stable symmetric state at strong coupling or in a stable asymmetric state if the coupling is small enough. In the first case, if the initial state is not the equilibrium state, the intensity may switch periodically from fiber to fiber, while in the second case the asymmetrical state remains forever, with most of its energy in either fiber. The latter situation makes a state of propagation with two exactly reciprocal realizations. In the quantum case, such a situation does not exist as an eigenstate because of the quantum tunneling between the two fibers. Such a tunneling is a purely quantum phenomenon without counterpart in the classical theory. We estimate the rate of tunneling by quantizing a simplified dynamics derived from the original Lagrangian equations with test functions. This tunneling could be within reach of the experiments, particularly if the quantum coherence of the soliton can be maintained over a sufficient amount of time.
RESUMEN
Above the first lasing threshold the degenerate optical parametric oscillator with saturable absorber displays successive Hopf and Turing instabilities. Various spiral patterns and defect turbulent patterns are numerically observed on the light intensity profiles. Close to the Hopf threshold, a normal form is derived which leads to a complex Ginzburg-Landau equation where a bi-Laplacian instead of a Laplacian drives the formation of spirals. At resonance the predictions of the normal form are compared with the numerical observations of the full equations. Above the Hopf threshold, the spirals destabilize, breaking into slowly evolving patterns with small spirals and filaments. Further above the threshold, when both the Turing and Hopf bifurcations interplay, a new spiral pattern emerges, with large notched arms.