Quasi Stationary States

The recurrence coe#cients of generalized Charlier polynomials satisfy a system of nonlinear recurrence relations. We simplify the recurrence relations, show that they are related to certain discrete Painleve equations, and analyze the asymptotic behaviour.

For systems with long-range interactions, the two-body potential decays at large distances as V (r) ∼ 1/r α , with α ≤ d, where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameters space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.

A generic feature of systems with long-range interactions is the presence of quasistationary states with non-Gaussian single particle velocity distributions. For the case of the Hamiltonian mean-field model, we demonstrate that a maximum entropy principle applied to the associated Vlasov equation explains known features of such states for a wide range of initial conditions. We are able to reproduce velocity distribution functions with an analytic expression which is derived from the theory with no adjustable parameters. A normal diffusion of angles is detected, which is consistent with Gaussian tails of velocity distributions. A dynamical effect, two oscillating clusters surrounded by a halo, is also found and theoretically justified.

The Vlasov equation is well known to provide a good description of the dynamics of mean-field systems in the N → ∞ limit. This equation has an infinity of stationary states and the case of homogeneous states, for which the single-particle distribution function is independent of the spatial variable, is well characterized analytically. On the other hand, the inhomogeneous case often requires some approximations for an analytical treatment: the dynamics is then best treated in action-angle variables, and the potential generating inhomogeneity is generally very complex in these new variables. We here treat analytically the linear stability of toy-models where the inhomogeneity is created by an external field. Transforming the Vlasov equation into action-angle variables, we derive a dispersion relation that we accomplish to solve for both the growth rate of the instability and the stability threshold for two specific models: the Hamiltonian Mean-Field model with additional asymmetry and the mean-field φ 4 model. The results are compared with numerical simulations of the N-body dynamics. When the inhomogeneous state is stationary stable, we expect to observe in the N-body dynamics Quasi-Stationary-States (QSS), whose lifetime diverges algebraically with N.

We numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions in the quasi-stationary states of the Hamiltonian mean field model. This is an example, in an N -body Hamiltonian system, of anomalous transport properties characterized by nonexponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. It is noteworthy that, at statistical equilibrium also, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties.

In a granular gas of rough particles the spin of a grain is correlated with its linear velocity. We develop an analytical theory to account for these correlations and compare its predictions to numerical simulations, using Direct Simulation Monte Carlo as well as Molecular Dynamics. The system is shown to relax from an arbitrary initial state to a quasi-stationary state, which is characterized by time-independent, finite correlations of spin and linear velocity. The latter are analysed systematically for a wide range of system parameters, including the coefficients of tangential and normal restitution as well as the moment of inertia of the particles. For most parameter values the axis of rotation and the direction of linear momentum are perpendicular like in a sliced tennis ball, while parallel orientation, like in a rifled bullet, occurs only for a small range of parameters. The limit of smooth spheres is singular: any arbitrarily small roughness unavoidably causes significant translation-rotation correlations, whereas for perfectly smooth spheres the rotational degrees of freedom are completely decoupled from the dynamic evolution of the gas.

The interaction of an atomic gas confined inside a cavity with a strong electromagnetic field is numerically and theoretically investigated in a regime where recoil effects are not negligible. The spontaneous appearance of a density grating (atomic bunching) accompanied by the onset of coherent backpropagation is found to be ruled by a continuous phase-transition. Numerical tests allow us to convincingly prove that the transition is steered by the appearence of a periodic atomic density modulation. Consideration of different experimental relaxation mechanisms induces us to analyze the problem in nearly analytic form, in the large detuning limit, using both a Vlasov approach and a Fokker-Planck description. The application of our predictions to recent experimental findings, reported in [Phys. Rev. Lett. 91, 183601 (2003)], yields a semi-quantitative agreement with the observations.

We present a new physical model resolving a long-standing mystery of the power-law distributions of the blinking times in single colloidal quantum dot fluorescence. The model considers the non-radiative relaxation of the exciton through multiple recombination centers. Each center is allowed to switch between two quasi-stationary states. We point out that the conventional threshold analysis method used to extract the

We consider a π-mode solution of the Fermi-Pasta-Ulam β system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime, where is unstable, first weakly and then strongly chaotic. We introduce, as indicator of stochasticity, the ratio ρ (when is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is a numerical evidence that the thermostatistic is governed by the Tsallis distribution.

Non-spherical precipitates are the main strengthening source of the age-hardenable aluminum alloys. In the majority of the precipitation hardening models presented so far, the simple spherical shape has been assumed. Moreover, the models which considered the actual shape of the precipitates, are derived based on the simple assumption of steady-state diffusion problem solutions. In the present study, the classical Kampmann and Wagner numerical model of spherical precipitates is extended to model the kinetics of spheroidal shape precipitates evolution during the aging treatment of Al alloys. To do so, a new rate law is proposed using a similarity solution of the transient diffusion problem around the spheroidal precipitates, with different aspect ratios, and moving boundaries. Moreover, a modified age-hardening model which considered the effects of precipitate shape, size and volume fraction, is used to predict the variation of the alloy hardness during the aging process. The accuracy of the proposed model is shown by comparing the predicted features of precipitates, and hardness evolution with the published experimental data. Also, the validity of the existing approximate solutions for the aging problem is discussed.

We study the original α-Fermi-Pasta-Ulam (FPU) system with N = 16, 32 and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory, i.e. we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first non trivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that for small amplitude random waves the time scale of such interactions is extremely large and it is of the order of 1/ǫ 8 , where ǫ is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

THEORETICAL AND NUMERICAL 1. The macroscopic dispersive flux of solutes undergoing aqueous phase bimolecular reactions can be described in the same manner as that of a non-reactive solute. In a linear gradient dependent relationship for the macrodispersive flux the macrodispersion coefficient remains unchanged on account of transformation reactions. 2. The concentration variance and the cross-covariance between the reactant concentrations also undergo a macrodispersive flux that can be described using a linear gradient relationship with the macrodispersion coefficient being the same as that for the mean dispersive flux of a non-reactive solute. 3. The dominant balance in the cross-covariance budget is between its rates of production and dissipation of concentration cross-covariance, with, and without reactions. The same balance holds for the concentration variances. 4. The characteristic time-scale over which small-scale mixing (local dispersion) dissipates the concentration variance and the cross-covariance, which we refer to as the odissipation time scale,6 and has been previously referred to as the ovariance residence time,6 is unaffected by reaction. 5. The macrodispersion coefficient and the dissipation time scale appear to approach constant asymptotic values, although the dissipation time scale approaches that asymptotic constant level much slower than the macrodispersion coefficient. 6. Exploiting the dominant production dissipation balance in the concentration cross-covariance budget yields a simple formula for the concentration cross covariance. The macrodispersion coefficients, the dissipation time-scale, and the product of the mean reactant concentration gradients control the reactant concentration cross-covariance. The influence of the flow microstructure and small-scale mixing are present in this model through the macrodispersion coefficient and the dissipation time-scale. This formula for the concentration cross-covariance was shown to apply under reactive and non-reactive conditions. 7. The simple result for the concentration cross-covariance that reflects the production-dissipation balance that characterizes second order fluctuation budgets can be applied to upscale bimolecular reactions and results in a macroscopic gradient dependent effective rate parameter. The effective rate parameter can be larger than its intrinsic value (inferred under well-mixed conditions) if the macroscopic gradients have identical signs (initially overlapping case) or smaller than the intrinsic value if the macroscopic gradients have opposite signs (initially non-overlapping reactants). The effective reaction rate parameter for the bimolecular reaction derived here provides a specific example of how mixing in a heterogeneous environment controls the reaction rates. The macro-kinetics themselves are dependent on the macroscopic concentration gradient, because these gradients control the small-scale concentration fluctuations of the reactants. It is easy to incorporate the gradient dependent reaction macro-kinetics in models that routinely evaluate the mean concentration field and its gradients. 8. The upscaling of dual-Monod kinetics expression -that is widely used to model biodegradation kinetics -is illustrated based on the expressions for concentration covariance and variances developed in this work. It is shown how reactant segregation can appreciably slow down the attenuation rate of an aerobically biodegradable contaminant. 9. An analytical dissolution rate coefficient that is independent of scale was developed in order to yield more realistic models for mass transfer in rock fractures. The analytical solution yields the mean velocity, dispersion coefficient and dissolution rate coefficient, which are independent of scale and may be applied to NAPL and solid dissolution. Research Training: D. Scott Felton finished his MS thesis in December 2000. Jagadeesh Anmala finished his PhD dissertation in August 2000. Outreach Activities: CHAPMAN CONFERENCE PROPOSAL Based on the positive feedback we have received from a broad range of researchers Kapoor have initiated the process for organizing a Chapman Conference on the topic of this proposal. CONFERENCE PRESENTATIONS Anmala, J., V. Kapoor, Mixing and bimolecular reaction kinetics in stratified flows, EOS Trans. AGU, 80(46), F414-F415, 1999.

A dilute Al-Sc alloy (Al-0.12 Sc, at.%, Al-Sc), its counterpart with a Li addition (Al-2.9 Li-0.11 Sc, at.%, Al-Li-Sc), as well as a quaternary alloy (Al-5.53 Li-0.048 Sc-0.009 Yb, at.%, Al-Li-Sc-Yb) were isothermally aged at 325 • C, and in some cases isochronally aged to 450 • C. As the ␣ -Al 3 (Li,Sc) and Al 3 (Li,Sc,Yb) precipitates, with L1 2 structure, coarsen in the two Li-containing alloys, their Li and Yb concentrations decrease and their Sc concentration increases. A significant interfacial excess of Li also segregates at the ␣-Al matrix/␣ -Al 3 Sc(Li,Sc,Yb) precipitate interface: 5.99 ± 0.05 atoms nm -2 in Al-Li-Sc and 13.2 ± 0.4 atoms nm -2 in Al-Li-Sc-Yb after aging isochronally to 450 • C. During compression creep at 300 • C, the aged alloys exhibit threshold stresses between 8 and 22 MPa. A recent threshold stress model based on elastic interactions between dislocations and precipitates predicts correctly that Li additions in the Al-Li-Sc alloy reduce the threshold stress, while Yb in the Al-Li-Sc-Yb alloy increases it. The model is also in agreement with the threshold stresses of all Al-Sc-X alloys published to date.

In a granular gas of rough particles the spin of a grain is correlated with its linear velocity. We develop an analytical theory to account for these correlations and compare its predictions to numerical simulations, using Direct Simulation Monte Carlo as well as Molecular Dynamics. The system is shown to relax from an arbitrary initial state to a quasi-stationary state, which is characterized by time-independent, finite correlations of spin and linear velocity. The latter are analysed systematically for a wide range of system parameters, including the coefficients of tangential and normal restitution as well as the moment of inertia of the particles. For most parameter values the axis of rotation and the direction of linear momentum are perpendicular like in a sliced tennis ball, while parallel orientation, like in a rifled bullet, occurs only for a small range of parameters. The limit of smooth spheres is singular: any arbitrarily small roughness unavoidably causes significant translation-rotation correlations, whereas for perfectly smooth spheres the rotational degrees of freedom are completely decoupled from the dynamic evolution of the gas.

The behaviour of sulfonated polyethersulfone membranes in the reverse osmosis treatment of aqueous solutions containing ammonium, cyanide and sulphate ions is described in this paper. Experimental tests were performed in an INDEVEN planar membrane test module, which provides data concerning the behaviour of the membrane in cross flow conditions with a minimal surface area, low feed and short times. Aqueous solutions of different concentrations of ammonium, cyanide and sulphate were treated in the test module, with both concentrate and permeate being recycled in order to keep the feed concentration practically constant, and so simulate a continuous process in a quasi-stationary state. The results show that this kind of membrane can be used satisfactorily to reduce concentrations of these ions in water, with rejection coefficients equal or higher to ninety percent. The water permeability coefficient and the solute permeability coefficients, of ammonium, cyanide and sulphate, have been obtained according the solution-diffusion model.

Non-equilibrium quasi-stationary states resulting from curvature driven interchange instabilities and driftwave instabilities in a low beta, weakly ionized, magnetized plasma are investigated in the context of laboratory experiments in a toroidal configuration. Analytic modelling, numerical simulations and experimental results are discussed with emphasis on identifying the unstable modes and understanding the physics of anomalous particle and energy fluxes and their linkage to self-organized pressure profiles.

Palabras clave: mesofílico, simulación matemática, metanogénico, bioproceso RESUMEN En los últimos años, el modelado matemático ha sido empleado para tratar de representar los cambios químicos que ocurren en el ambiente. Tradicionalmente los modelos se fundamentan en los balances de materia y la cinética química de los bioprocesos. En este estudio se aplican y simulan las ecuaciones obtenidas para un reactor biológico isotérmico tipo tanque agitado mediante ecuaciones lineales adimensionales en estado semi-estacionario. Para darle validez a los resultados derivados de la solución analítica, estos se comparan con los resultados experimentalmente obtenidos a nivel laboratorio para un digestor metanogénico en el régimen mesofílico de temperatura y en donde se trataron los vertidos residuales provenientes de una industria alcoholera bajo diferentes condiciones de operación, empleando como consorcio microbiano fluido ruminal vacuno. Los resultados del modelo concuerdan favorablemente bien con la producción de biogás y la disminución de la carga orgánica (expresada como DQO), teniendo un factor de ajuste promedio para ambos casos de R 2 =0.9960 y una variación del 0.01601, cuando se propone una cinética de reacción de primer orden y empleando en el medio células libres. Además que dicho modelo es capaz de predecir cuándo el sistema alcanza el estado estacionario.

The Smaller Alignment Index (SALI) is a new mathematical tool for chaos detection in the phase space of Hamiltonian Dynamical Systems. With temporal behavior very specific to movements ordered or chaotic, the SALI method is very efficient in distinguishing between chaotic and regular movements. In this work, this method will be applied in the study of stellar orbits immersed in a gravitational potential of barred galaxies, once the motion of a test particle, in a rotating barred galaxy model is given by a Hamiltonian function. Using an analytical potential representative of a galaxy with bar (two degrees of freedom), we integrate some orbits and apply SALI in order to verify their stabilities. In this paper, we will discuss a few cases illustrating the trajectories of chaotic and regular orbits accompanied by the graph containing the behavior of SALI. All calculations and integrations were performed with the LP-VIcode program. 1 Introduction One of the schemes more used to classify ...

General circulation models (GCMs) can be used to develop diagnostics for identifying weather regimes. The author has looked for three-dimensional (3D) weather regimes associated with a 10-yr run of the U.K. UGAMP GCM with perpetual January boundary conditions; 3D low-pass empirical orthogonal functions (EOFs), using both the 500-and 250-mb streamfunctions () have been computed. These EOFs provide a low-order phase space in which weather regimes are studied. The technique here is an extension to 3D of that of Haines and Hannachi. They found, within the 500-mb EOF phase space, two local minima of area-averaged-tendency (based on barotropic vorticity dynamics), which were identified as ϮPacific-North America (PNA). In this work, the author demands that both the flow and its tendency be within the phase space spanned by the 3D EOFs. The streamfunction tendency is computed from the two-level quasigeostrophic potential vorticity equation and projected onto the EOF phase space. This projection produces a finite dynamical system whose singular points are identified as the quasi-stationary states. Two blocking solutions and one zonal solution are found over the Pacific. The first blocking solution is closer to the west coast of North America than the other blocking, which is shifted slightly westward and has a larger scale, rather similar to the ϩPNA pattern, indicating that blocking over the Pacific may have two phases in the model. Further investigation of the GCM trajectory within the EOF phase space using a mixture analysis shows the existence of realistic three-dimensional weather regimes similar to the singular points. The same solutions were found when the transient eddy contributions to the climatological quasigeostrophic potential vorticity budget were included. It is also shown that this extended technique allows a direct study of the stability of these quasistationary states and helps in drawing transition pictures and determining the transition times between them.

In this paper, we present an investigation of the timerelaxation of the electron energy distribution function (EEDF) in the nitrogen afterglow of an 2 = 433 MHz flowing discharge at = 3 3 torr, in a tube with inner radius = 1 9 cm. We solve the time-dependent Boltzmann equation, including the term for creation of new electrons in associative/Penning reactions, coupled to a system of rate balance equations for the heavy-particles. The EEDFs are also obtained experimentally, from second derivatives of digitized probe characteristics measured using a triple probe technique, and compared with the calculations. It is shown that an equilibrium between the vibrational distribution function of ground-state molecules N 2 (1 6 +) and low-energy electrons is rapidly established, in times 10 7 s. In these early instants of the postdischarge, a dip is formed in the EEDF around 4 eV. The EEDF finally reaches a quasi-stationary state for 10 6 s, although the electron density still continues to decrease beyond this instant. Collisions of highly excited N 2 (1 6 + 35) molecules with N(4) atoms are in the origin of a maximum in the electron density occurring downstream from the discharge at 2 10 2 s. These reactions create locally the metastable states N 2 (3 6 +) and N 2 (1 6), which in turn ionize the gas in associative/Penning processes. Slow electrons remain for very long times in the postdischarge and can be involved in electron stepwise processes with energy thresholds smaller than 2-3 eV.

To control a heat source easily in the forming process of steel plate with heating, the electromagnetic induction process has been used as a substitute of the flame heating process. However, only few studies have analyzed the deformation of a workpiece in the induction heating process by using a mathematical model. This is mainly due to the difficulty of modeling the heat flux from the inductor traveling on the conductive plate during the induction process. In this study, the heat flux distribution over a steel plate during the induction process is first analyzed by a numerical method with the assumption that the process is in a quasi-stationary state around the inductor and also that the heat flux itself greatly depends on the temperature of the workpiece. With the heat flux, heat flow and thermo-mechanical analyses on the plate to obtain deformations during the heating process are then performed with a commercial FEM program for 34 combinations of heating parameters. An artificial neural network is proposed to build a simplified relationship between deformations and heating parameters that can be easily utilized to predict deformations of steel plate with a wide range of heating parameters in the heating process. After its architecture is optimized, the artificial neural network is trained with the deformations obtained from the FEM analyses as outputs and the related heating parameters as inputs. The predicted outputs from the neural network are compared with those of the experiments and the numerical results. They are in good agreement.

present a new physical model resolving a long-standing mystery of the power-law distributions of the blinking times in single colloidal quantum dot fluorescence. The model considers the non-radiative relaxation of the exciton through multiple recombination centers. Each center is allowed to switch between two quasi-stationary states. We point out that the conventional threshold analysis method used to extract the exponents of the distributions for the on-times and off-times has a serious flaw: The qualitative properties of the distributions strongly depend on the threshold value chosen for separating the on and off states. Our new model explains naturally this threshold dependence, as well as other key experimental features of the single quantum dot fluorescence trajectories, such as the power-law power spectrum (1/f noise).

We present a new physical model resolving a long-standing mystery of the power-law distributions of the blinking times in single colloidal quantum dot fluorescence. The model considers the non-radiative relaxation of the exciton through multiple recombination centers. Each center is allowed to switch between two quasi-stationary states. We point out that the conventional threshold analysis method used to extract the

The problem of nonlinear adjustment of localized front-like perturbations to a state of geostrophic equilibrium (balanced state) is studied in the framework of rotating shallow-water equations with no dependence on the along-front coordinate. We work in Lagrangian coordinates, which turns out to be conceptually and technically advantageous. First, a perturbation approach in the cross-front Rossby number is developed and splitting of the motion into slow and fast components is demonstrated for non-negative potential vorticities. We then give a non-perturbative proof of existence and uniqueness of the adjusted state, again for configurations with nonnegative initial potential vorticities. We prove that wave trapping is impossible within this adjusted state and, hence, adjustment is always complete for small enough departures from balance. However, we show that retarded adjustment occurs if the adjusted state admits quasi-stationary states decaying via tunnelling across a potential barrier. A description of finite-amplitude periodic nonlinear waves known to exist in configurations with constant potential vorticity in this model is given in terms of Lagrangian variables. Finally, shock formation is analysed and semi-quantitative criteria based on the values of initial gradients and the relative vorticity of initial states are established for wave breaking showing, again, essential differences between the regions of positive and negative vorticity.

We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is derived starting from the Levy-Levy-Solomon microscopic model using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model, under a suitable scaling, we derive a Fokker-Planck equation and show that the equation admits a self-similar lognormal behavior. Several numerical examples are also reported to validate our analysis.

Recent studies on the Fermi-Pasta-Ulam (FPU) paradox, like the theory of q-breathers and the metastability scenario, dealing mostly with the energy localization properties in the FPU space of normal modes (q-space), motivated our first work on q-tori in the FPU problem [8]. The q-tori are lowdimensional invariant tori hosting trajectories that present features relevant to the interpretation of FPU recurrences as well as the energy localization in q-space. The present paper is a continuation of our work in [8]. Our new results are: We extend a method of analytical computation of q-tori, using Poincaré-Lindstedt series, from the β to the α-FPU and we reach significantly higher expansion orders using an improved computer-algebraic program. We probe numerically the convergence properties as well as the level of precision of our computed series. We develop an additional algorithm in order to systematically locate values of the incommensurable frequencies used as an input in the PL series construction of q-tori corresponding to progressively higher values of the energy. We generalize a proposition proved in [8] regarding the so-called 'sequence of propagation' of an initial excitation in the PL series. We show by concrete examples how the latter interprets the localization patterns found in numerical simulations. We focus, in particular, on various types of extensive initial excitations that lead to q-tori solutions with exponentially localized profiles. Finally, we discuss the relation between q-tori, q-breathers (viewed as one-dimensional q-tori), and the so-called 'FPU-trajectories' invoked in the original study of the FPU problem.

Within the framework of the quasi-geostrophic approximation, the interactions of two identical initially circular vortex patches are studied using the contour dynamics/surgery method. The cases of barotropic vortices and of vortices in the upper layer of a two-layer fluid are considered. Diagrams showing the end states of vortex interactions and, in particular, the new regime of vortex triplet formation are constructed for a wide range of external parameters. This paper shows that, in the nonlinear evolution of two such (like-signed) vortices, the filaments and vorticity fragments surrounding the merged vortex often collapse into satellite vortices. Therefore, the conditions for the formation and the quasi-steady motions of a new type of triplet-shaped vortex structure are obtained.

We investigate the evolution of phase space close to complex unstable periodic orbits in two galactic type potentials. They represent characteristic morphological types of disc galaxies, namely barred and normal (non-barred) spiral galaxies. These potentials are known for providing building blocks to support observed features such as the peanut, or X-shaped bulge, in the former case and the spiral arms in the latter. We investigate the possibility that these structures are reinforced, apart by regular orbits, also by orbits in the vicinity of complex unstable periodic orbits. We examine the evolution of the phase space structure in the immediate neighbourhood of the periodic orbits in cases where the stability of a family presents a successive transition from stability to complex instability and then to stability again, as energy increases. We find that we have a gradual reshaping of invariant structures close to the transition points and we trace this evolution in both models. We conclude that for time scales significant for the dynamics of galaxies, there are weakly chaotic orbits associated with complex unstable periodic orbits, which should be considered as structure-supporting, since they reinforce the morphological features we study.

We investigate equilibrium solutions for tripolar vortices in a two-layer quasi-geostrophic flow. Two of the vortices are like-signed and lie in one layer. An opposite-signed vortex lies in the other layer. The families of equilibria can be spanned by the distance (called separation) between the two like-signed vortices. Two equilibrium configurations are possible when the opposite-signed vortex lies between the two other vortices. In the first configuration (called ordinary roundabout), the opposite signed vortex is equidistant to the two other vortices. In the second configuration (eccentric roundabouts), the distances are unequal. We determine the equilibria numerically and describe their characteristics for various internal deformation radii. The two branches of equilibria can co-exist and intersect for small deformation radii. Then, the eccentric roundabouts are stable while unstable ordinary roundabouts can be found. Indeed, ordinary roundabouts exist at smaller separations th...

"Quasistationary" states are approximately time-independent out of equilibrium states which have been observed in a variety of systems of particles interacting by longrange interactions. We investigate here the conditions of their occurrence for a generic pair interaction V (r → ∞) ∼ 1/r γ with γ > 0, in d > 1 dimensions. We generalize analytic calculations known for gravity in d = 3 to determine the scaling parametric dependences of their relaxation rates due to two body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for γ < d − 1, the existence of quasi-stationary states is ensured by the large distance behavior of the interaction alone, while for γ > d − 1 it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft-core in the interaction potential.

Out of equilibrium magnetised solutions of the XY-Hamiltonian Mean Field (XY-HMF) model are build using an ensemble of integrable uncoupled pendula. Using these solutions we display an out-of equilibrium phase transition using a specific reduced set of the magnetised solutions.

In this paper the lifetime of quasi-stationary states (QSS) in the α−HMF model are investigated at the long range threshold (α = 1). It is found that QSS exist and have a diverging lifetime τ (N) with system size which scales as τ (N)∼ log N , which contrast to the exhibited power law for α < 1 and the observed finite lifetime for α > 1. Another feature of the long range nature of the system beyond the threshold (α > 1) namely a phase transition is displayed for α = 1.5. The definition of a long range system is as well discussed.

Each oscillator in a linear chain (a string) interacts with a local Ising spin in contact with a thermal bath. These spins evolve according to Glauber dynamics. Below a critical temperature, there appears an equilibrium, time-independent, rippled state in the string that is accompanied by a nonzero spin polarization. On the other hand, the system is shown to form "metastable", nonequilibrium but long-lived ripples in the string for slow spin relaxation. The system vibrates rapidly about these quasi-stationary states which can be described as snapshots of a coarse-grained stroboscopic map. For moderate observation times, ripples are observed irrespective of the final thermodynamically stable state (rippled or not). Interestingly, the system can be considered as a "minimal" model to understand rippling in clamped graphene sheets.

We investigate the probability density of rescaled sum of iterates of sine-circle map within quasiperiodic route to chaos. When the dynamical system is strongly mixing (i.e., ergodic), standard Central Limit Theorem (CLT) is expected to be valid, but at the edge of chaos where iterates have strong correlations, the standard CLT is not necessarily to be valid anymore. We discuss here the main characteristics of the probability densities for the sums of iterates of deterministic dynamical systems which exhibit quasi-periodic route to chaos. At the golden-mean onset of chaos for the sinecircle map, we numerically verify that the probability density appears to converge to a q-Gaussian with q < 1 as the golden mean value is approached.

We consider a simple unstructured individual based stochastic epidemic model with contact tracing. Even in the onset of the epidemic, contact tracing implies that infected individuals do not act independent of each other. Nevertheless, it is possible to analyze the embedded non-stationary Galton±Watson process. Based upon this analysis, threshold theorems and also the probability for major outbreaks can be derived. Furthermore, it is possible to obtain a deterministic model that approximates the stochastic process, and in this way, to determine the prevalence of disease in the quasi-stationary state and to investigate the dynamics of the epidemic.

Nonstationary regimes of the wave turbulence evolution are considered in the framework of isotropic kinetic equation. It is predicted analytically and confirmed by numerical experiment that there is a class of wave systems in which any initial distribution of the turbulence energy in k-space comes into a universal, Kolmogorovtype spectrum in a finite time. Before and after the formation of the Kolmogorov spectrum, two different self-similar regimes of evolution occur: the first one is responsible for explosively forming the universal spectrum and the second one determines energy dissipation.

We investigate numerically and analytically size-polydisperse granular mixtures immersed into a molecular gas. We show that the equipartition of granular temperatures of particles of different sizes is established; however, the granular temperatures significantly differ from the temperature of the molecular gas. This result is surprising since, generally, the energy equipartition is strongly violated in driven granular mixtures. Qualitatively, the obtained results do not depend on the collision model, being valid for a constant restitution coefficient ε, as well as for the ε for viscoelastic particles. Our findings may be important for astrophysical applications, such as protoplanetary disks, interstellar dust clouds, and comets.

The kinetics of coarsening of ?' precipitates in binary Ni AI alloys containing nominally 5.72, 5.74 and 5.78 wt% A1 and aged at 630 C were investigated by transmission electron microscopy and magnetic analysis. The alloys were each pre-aged at a temperature just below its solvus prior to re-aging at 630uC. The volume fractions of 7' were low, between 0.02 and 0.03. The pre-aging treatment produced microstructures containing precipitates that nucleated heterogeneously on dislocations and grain boundaries, leaving empty matrix areas in which coherent 7' precipitates subsequently nucleated and coarsened. Measurements were made only on these precipitates. The coarsening kinetics are anomalous in that the rate constants decrease with increasing volume fraction, in contradiction with all the theories of this process. Furthermore, the increase is quite large, exceeding a factor of four over a range of volume fractions that increased by less than 40%, Additionally, the rate constants exceed the values expected from the literature by factors of 10~40. The distributions of particle sizes were measured, and with few exceptions were found to agree with those of earlier investigations. It is suggested that elastic interactions among the 7,' precipitates play a pivotal role in decelerating the kinetics of coarsening, overpowering the expected accelerating effect of increasing volume fraction.

By using the Enskog-Boltzmann approach, we study the steady-state dynamics of a granular discorectangle placed in a two-dimensional bath of thermalized hard disks. Hard core collisions are assumed elastic between disks and inelastic between the discorectangle and the disks, with a normal restitution coefficient α < 1. Assuming a Gaussian ansatz for the probability distribution functions, we obtain analytical expressions for the granular temperatures. We show the absence of equipartition and investigate both the role of the anisotropy of the discorectangle and of the relative ratio of the bath particles to the linear sizes of the discorectangle. In addition, we investigate a model of a discorectangle with two normal restitution coefficients for collisions along the straight and curved surfaces of the discorectangle. In this case one observes equipartition for a non trivial ratio of normal restitution coefficients.

The possibility of observing phenomena peculiar to long-range interactions, and more specifically in the so-called Quasi-Stationary State (QSS) regime is investigated within the framework of two devices, namely the Free-Electron Laser (FEL) and the Collective Atomic Recoil Laser (CARL). The QSS dynamics has been mostly studied using the Hamiltonian Mean-Field (HMF) toy model, demonstrating in particular the presence of first versus second order out-ofequilibrium phase transitions from magnetized to unmagnetized regimes. Here, we give evidence of the strong connections between the HMF model and the dynamics of the two mentioned devices, and we discuss the perspectives to observe some specific QSS features experimentally. In particular, a dynamical analog of the phase transition is present in the FEL and in the CARL in its conservative regime. Regarding the dissipative CARL, a formal link is established with the HMF model. For both FEL and CARL, calculations are performed with reference to existing experimental devices, namely the FERMI@Elettra FEL under construction at Sincrotrone Trieste (Italy) and the CARL system at LENS in Florence (Italy).

This paper presents symbolic analysis of time series data for estimation of multiple faults in permanent magnet synchronous motors (P M SM). The analysis is based on an experimentally validated dynamic model, where the flux linkage of the permanent magnet and friction in the motor bearings are varied in the simulation model to represent different stages of degradation. The fault magnitudes are estimated from the time series of the instantaneous line current. The behavior patterns of the P M SM are compactly generated as quasi-stationary state probability histograms associated with the finite state automata of its symbolic dynamic representation. The proposed fault estimation method is suitable for real-time execution on a limited-memory platforms, such as those used in sensor network nodes.

We investigate in detail a recent model of colliding mobile agents [M.C. González, P.G. Lind, H.J. Herrmann, Phys. Rev. Lett. 96 (2006) 088702. cond-mat/0602091], used as an alternative approach for constructing evolving networks of interactions formed by collisions governed by suitable dynamical rules. The system of mobile agents evolves towards a quasi-stationary state which is, apart from small fluctuations, well characterized by the density of the system and the residence time of the agents. The residence time defines a collision rate, and by varying this collision rate, the system percolates at a critical value, with the emergence of a giant cluster whose critical exponents are the ones of twodimensional percolation. Further, the degree and clustering coefficient distributions, and the average path length, show that the network associated with such a system presents non-trivial features which, depending on the collision rules, enables one not only to recover the main properties of standard networks, such as exponential, random and scale-free networks, but also to obtain other topological structures. To illustrate, we show a specific example where the obtained structure has topological features which characterize the structure and evolution of social networks accurately in different contexts, ranging from networks of acquaintances to networks of sexual contacts.

Collisions between granular particles are irreversible processes which cause dissipation of mechanical energy by fragmentation or heating of the colliders. The knowledge of these phenomena is essential for the understanding of the behaviour of complex systems of granular particles. We have developed a model for inelastic collisions of granular particles and calculated the velocity restitution coefficients, which describe all possible collisions in the system. The knowledge of these coefficients allows for event-driven many-particle simulations which cannot be performed in the frame of molecular dynamics. This approach has the advantage that very large particle numbers can be treated which are necessary for the understanding of intrinsic large-scale phenomena in granular systems.

The elastic energy of ordered arrays of ledges on the coherent planar face of a y' precipitate in nickel alloys was calculated as a function of the lateral growth of the ledges. This energy is separated into three components: the self energy of the ledges, the interaction among ledges and the interaction of the ledges with the underlying cube-shaped precipitate. The interaction between the cube and the ledges is always repulsive. The self energy of the ledges and the interaction among ledges decreases with a decrease in their height to width ratio. In the interplay between these two energies, the interaction energies may greatly reduce the elastic energy of the system, overwhelming the increase in the self energy. Hence the interaction among ledges generates a tendency for ordering of the ledges, favoring the formation of high and densely spaced ledges. Too dense arrays are prohibited, however, by the self energy of the ledges and an optimum spacing between ledges exists. 0 1997 Acta Metallurgica Inc.

Nonstationary regimes of the wave turbulence evolution are considered in the framework of isotropic kinetic equation. It is predicted analytically and confirmed by numerical experiment that there is a class of wave systems in which any initial distribution of the turbulence energy in k-space comes into a universal, Kolmogorovtype spectrum in a finite time. Before and after the formation of the Kolmogorov spectrum, two different self-similar regimes of evolution occur: the first one is responsible for explosively forming the universal spectrum and the second one determines energy dissipation.