Narrow Results

We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.

A simple spin model is studied, motivated by the dynamics of traders in a market, where expectation bubbles and crashes occur. The dynamics is governed by interactions, which are frustrated across different scales: while ferromagnetic couplings connect each spin to its local neighborhood, an additional coupling relates each spin to the global magnetization. This new coupling is allowed to be anti-ferromagnetic. The resulting frustration causes a metastable dynamics with intermittency and phases of chaotic dynamics. The model reproduces main observations of real economic markets as power-law distributed returns and clustered volatility.

We study the extreme long-time behavior of the metastable phase of the three-dimensional Ising model with Glauber dynamics in an applied magnetic field and at a temperature below the critical temperature. For these simulations, we use the advanced simulation method of projective dynamics. The algorithm is described in detail, together with its application to the escape from the metastable state. Our results for the field dependence of the metastable lifetime are in good agreement with theoretical expectations and span more than 50 decades in time.

The synchronization of asynchronous signals can lead to metastable behavior and malfunction of digital circuits. It is believed — but not proved — that metastability principally cannot be avoided. Confusion exists about its practical importance. This paper shows that metastable behavior can be avoided by usage of quantum synchronizers in principle, but not in practice, and that conventional synchronizers unavoidably show metastable behavior in principle, but not in practice, if properly designed.

Problems associated with analyticity of thermodynamic functions close to first-order phase transitions are briefly reviewed. The bubble model for correlation functions is then applied to planar Ising-like models at subcritical temperatures (T<T_c) with a bulk magnetic field h. The fluctuation sum is used to calculate the susceptibility χ(h) from the bubble correlation function. We show that χ(h), calculated this way, must contain an essential singularity at h=0 i.e. at the first-order phase boundary. This has important implications to metastability, where we demonstrate that if the ensemble is restricted such that the magnetization stays positive when h goes negative, χ(h) has an infinite number of poles along the negative real axis with a limit-point at h=0. For an unrestricted ensemble, a Yang-Lee circle theorem is derived.

An overview of advanced dynamical algorithms capable of spanning the widely disparate time scales that govern the decay of metastable phases in discrete spin models is presented. The algorithms discussed include constrained transfer-matrix, Monte Carlo with Absorbing Markov Chains (MCAMC), and projective dynamics (PD) methods. The strengths and weaknesses of each of these algorithms are discussed, with particular emphasis on identifying the parameter regimes (system size, temperature, and field) in which each algorithm works best.

The Projected Dynamics method was originally developed to study metastable decay in ferromagnetic discrete spin models. Here, we apply it to a classical, continuous Heisenberg model with anisotropic ferromagnetic interactions, which evolves under a Monte Carlo dynamic. The anisotropy is sufficiently large to allow comparison with the Ising model. We describe the Projected Dynamics method and how to apply it to this continuous-spin system. We also discuss how to extract metastable lifetimes and how to extrapolate from small systems to larger systems.

The type IIA nonsupersymmetric meta-stable brane configuration consisting of three NS5-branes, D4-branes and anti-D4-branes where the electric gauge theory superpotential has a quartic term for the bifundamentals besides a mass term is constructed. By adding the orientifold 4-plane and 6-plane to this brane configuration, we also describe the intersecting brane configurations of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of corresponding gauge theories.

From an supersymmetric electric gauge theory with the gauge group SU(N_c) × SU(N′_c) with fundamentals for each gauge group, the bifundamentals and a symmetric flavor and a conjugate symmetric flavor for SU(N_c), we apply Seiberg dual to each gauge group independently and obtain two supersymmetric dual magnetic gauge theories with dual matters including the gauge singlets. By analyzing the F-term equations of the dual magnetic superpotentials, we describe the intersecting brane configurations of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of these gauge theories. The case where the above symmetric flavor is replaced by an antisymmetric flavor is also discussed.

Starting from an supersymmetric electric gauge theory with the multiple product gauge group and the bifundamentals, we apply Seiberg dual to each gauge group, obtain the supersymmetric dual magnetic gauge theories with dual matters including the gauge singlets. Then we describe the intersecting brane configurations, where there are NS-branes and D4-branes (and anti-D4-branes), of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of this gauge theory.

We also discuss the case where the orientifold 4-planes are added into the above brane configuration. Next, by adding an orientifold 6-plane, we apply to an supersymmetric electric gauge theory with the multiple product gauge group (where a single symplectic or orthogonal gauge group is present) and the bifundamentals. Finally, we describe the other cases where the orientifold 6-plane intersects with NS-brane.

From an supersymmetric electric gauge theory with the gauge group with fundamentals for each gauge group and the bifundamentals, we apply Seiberg dual to each gauge group and obtain the supersymmetric dual magnetic gauge theories with dual matters including the additional gauge singlets. By analyzing the F-term equations of the dual magnetic superpotentials, we describe the intersecting brane configurations of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of this gauge theory. We apply also to the case for supersymmetric electric gauge theory with the gauge group with flavors for each gauge group and the bifundamentals. Finally, we describe the meta-stable brane configurations of multiple product gauge groups.

We construct the type IIA nonsupersymmetric meta-stable brane configurations corresponding to the various supersymmetric gauge theories. The D6-branes are both displaced and rotated where these deformations are described as the mass term and quartic term for the fundamental flavors, respectively. The multiplicity of the NS5-branes occurs in the superpotential order for adjoint, symmetric, or bifundamental matters. A rich pattern of nonsupersymmetric meta-stable states as well as the supersymmetric stable states is found.

Starting from an supersymmetric electric gauge theory with the gauge group with fundamentals for each gauge group, the bifundamentals, a symmetric flavor and a conjugate symmetric flavor for SU(N_c), we apply Seiberg dual to each gauge group, obtain the supersymmetric dual magnetic gauge theories with dual matters including the gauge singlets, and describe the intersecting brane configurations of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of this gauge theory. We also discuss the case where a symmetric flavor is replaced by an antisymmetric flavor. Next we apply to the case for supersymmetric electric gauge theory with the gauge group with flavors for each gauge group and the bifundamentals. Finally, we describe the case where the orientifold 6-plane charge is reversed.

In this review, we present a theory of cosmological constant and dark energy (DE), based on the topological structure of the vacuum. The multiple point principle (MPP) is reviewed. It demonstrates the existence of the two vacua into the SM. The Froggatt–Nielsen’s prediction of the top-quark and Higgs masses is given in the assumption that there exist two degenerate vacua in the SM. This prediction was improved by the next-order calculations. We also considered Sidharth’s theory of cosmological constant based on the noncommutative geometry of the Planck scale space–time, what gives an extremely small DE density providing the accelerating expansion of the Universe. Theory of two degenerate vacua — the Planck scale phase and electroweak (EW) phase — is also reviewed, topological defects in these vacua are investigated, also the Compton wavelength phase suggested by Sidharth is discussed. A general theory of the phase transition and the problem of the vacuum stability in the SM is reviewed. Assuming the existence of a new scalar $S$ bound state $6t+6\bar{t}$, earlier predicted by Froggatt, Nielsen and Laperashvili, we try to provide the vacuum stability in the SM and exact accuracy of the MPP.

In the present paper we argue that the correction to the Higgs mass coming from the bound state of 6 top and 6 antitop quarks, predicted early by Froggatt and ourselves, leads to the Standard Model vacuum stability and confirms the accuracy of the multiple point principle (principle of degenerate vacua) for all experimentally valued parameters (Higgs mass, top-quark mass, etc.). Fitting to get the vacuum degeneracy requires a mass of the bound state, just in the region of the new two photon state in Large Hadron Collider, 750–760 GeV.

The modern equilibrium statistical mechanics, intended to link microscopic dynamics to thermodynamic laws for macroscopic quantities, is faced with significant difficulties, as applied to the description of the macroscopic properties of real condensed media within wide thermodynamic ranges, including the vicinities of the phase transition points. A particular problem is the absence of metastable states in the Gibbs statistical mechanics of the systems composed of finite number of particles. Nevertheless, accordance between equilibrium statistical mechanics and thermodynamics of condensed media is achievable if we take account of the mutual correlation (the feedback) between the microscopic properties of molecules and the macrostate of the corresponding medium. This can be done via usage of the "enhanced" Hamilton operator of the considered many-particle system, which contains some temperature-dependent term(s), and the following introduction of the generalized equilibrium distribution over microstates. For illustration of the reasonableness of the proposed approach and of its practical availability in the applications to computing the macroscopic characteristics of condensed media, a cell model of melting/crystallization and metastable supercooled liquid for a water-like medium is presented.

The significant process, voltage and temperature (PVT) variations seen with modern technologies make strictly synchronous design inefficient. Asynchronous design with its flexible timing is a promising alternative, but prototyping is difficult on the available FPGA platforms which are clock centric and do not provide the required functional primitives like mutual exclusion or Muller C-elements. The solutions proposed in the literature so far work nicely in principle but cannot safely handle metastability issues that are inevitable even at some interfaces in asynchronous designs. In this paper, we propose reliable implementations of the fundamental function blocks required to safely convert potential intermediate voltage levels that result from metastability into late transitions that can be reliably handled in the asynchronous domain. These are high- and low-threshold buffers as well as a Schmitt-trigger. We give elaborate background analysis for the proposed circuits and also present the associated routing constraints to make the Schmitt-trigger circuit work properly in spite of the uncertain routing within FPGAs. Furthermore, we propose a procedure for an “in situ reliability assessment” of the specific Schmitt-trigger element under consideration, which also applies to metastability containment with high- or low-threshold buffers only. Our proof of concept is based on experimental results for both Xilinx and Altera FPGA platforms.

The increasing number of clock domain crossings in modern systems-on-chip makes the careful consideration of metastability paramount. However, the manifestation of metastability at a flip-flop output is often unduly reduced to late transitions only, while glitches are hardly ever accounted for. In this paper we study the occurrence of glitches resulting from metastability in detail. To this end we propose a measurement circuit whose principle substantially differs from the conventional approach, and by that allows to reliably detect glitches. By means of experimental measurements on an FPGA target we can clearly identify late transitions, single glitches and double glitches as possible manifestations of metastability. Some of these behaviors are unexpected as they do not follow from the traditional modeling theory. We also study the dependence of metastable behavior on supply voltage. Beyond confirming that, as reported in previous literature, the metastable decay constant $\tau$ is voltage-dependent, we also produce strong evidence that the relative occurrence of glitches is not voltage-dependent.

We present the analysis of the mean switching time and its standard deviation of short overdamped Josephson junctions, driven by a direct current and a periodic signal. The effect of noise enhanced stability is investigated. It is shown that fluctuations may both decrease and increase the switching time.

The study of the noise induced effects on the dynamics of a chain molecule crossing a potential barrier, in the presence of a metastable state, is presented. A two-dimensional stochastic version of the Rouse model for a flexible polymer has been adopted to mimic the molecular dynamics and to take into account the interactions between adjacent monomers. We obtain a nonmonotonic behavior of the mean first passage time and its standard deviation, of the polymer center of inertia, with the noise intensity. These findings reveal a noise induced effect on the mean crossing time. The role of the polymer length is also investigated.