The metastable behavior of the three-dimensional stochastic Ising model I

The metastable behavior of the stochastic Ising model in a finite three-dimensional torus is studied in the limit as the temperature goes to zero. All metastable states are characterized and a hierarchic structure is found. For a large class of initial states, the logarithmic asymptotics of the hitting time of the states are studied with all spins +l or − 1. Project supported in part by the State Education Commission of China, the National Natural Science Foundation of China, the Tianyuan Foundation and the National 863 Project.     

Noncommutative central limit theorem for Gibbs temperature states

For Gibbs temperature states, the scheme of the proof of the noncommutative central limit theorem is given by using the commutative central limit theorem for corresponding Euclidean measures. Applications are constructed for the model of a temperature-anharmonic crystal and the generalized Ising model with compact continuous configuration space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 299–306, March, 1995.This work was partially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

Modèles de verre de spins à champ moyen et à interactions multiples : résultats rigoureux à basse température

We study the Ising case of the p-spin interaction mean field model for spin glasses. For large p, below the critical temperature, the configuration space naturally decomposes in a sequence of “states”. In the thermodynamic limit, the overlaps between two generic elements of any two different “states” are zero.We also construct systems that provably exhibit several phase transitions between what one might in replica theory call different levels of symmetry breaking.     

Hydrodynamic limit for the Ginzburg–Landau ∇ϕ interface model with non-convex potential

The hydrodynamic limit of the Ginzburg–Landau $??$ interface model was derived in Funaki and Spohn (1997) and Nishikawa (2003) for strictly convex potentials. This paper deals with non-convex potentials under suitable assumptions on the free energy and identification of the extremal Gibbs measures which have been recently established at sufficiently high temperature in Cotar and Deuschel (2012). Because of the non-convexity, many difficulties arise, especially, on the identification of equilibrium states. We show the equivalence between the stationarity and the Gibbs property under quite general settings, and we complete the identification of equilibrium states. We also establish some uniform estimates for variances of extremal Gibbs measures.     

Diffusive wave in the low Mach limit for non-viscous and heat-conductive gas

The low Mach number limit for one-dimensional non-isentropic compressible Navier–Stokes system without viscosity is investigated, where the density and temperature have different asymptotic states at far fields. It is proved that the solution of the system converges to a nonlinear diffusion wave globally in time as Mach number goes to zero. It is remarked that the velocity of diffusion wave is proportional with the variation of temperature. Furthermore, it is shown that the solution of compressible Navier–Stokes system also has the same phenomenon when Mach number is suitably small.     

Spectrum and States of the BCS Hamiltonian in a Finite Domain. III. BCS Hamiltonian with Mean-Field Interaction

We investigate the spectrum of a model Hamiltonian with BCS and mean-field interaction in a finite domain under periodic boundary conditions. The model Hamiltonian is considered on the states of pairs and waves of density charges and their excitations. It is represented as the sum of three operators that describe noninteracting pairs, the interaction between pairs, and the interaction between pairs and waves of density charges. The last two operators tend to zero in the thermodynamic limit, and the spectrum of the model Hamiltonian coincides with the spectrum of noninteracting pairs with chemical potential shifted by mean-field interaction. It is shown that the model and approximating Hamiltonians coincide in the thermodynamic limit on their ground and excited states and both have two branches of eigenvalues and eigenvectors.     

Algebraic Quantum Theory of the Josephson Microwave Radiator

A closed, entirely quantum mechanical Josephson oscillator model is considered, consisting of two superconductors in tunneling contact, which weakly interact with the photon field. For each superconductor we use, for notational simplicity, the BCS model in the strong coupling approximation and restrict ourselves to Andersons quasi-spin formulation. The thermodynamic limit of the global (non-equilibrium) dynamics is formulated for a large variety of states. It arises a generalization of previously developed cocycle equations, connecting the collective behaviour of the two superconductors with the photon field dynamics. In the physically most interesting situations, where the two superconductors are in thermal equilibrium (below the critical temperature) at voltage difference V, we show, that for arbitrary initial states the outgoing multi-photon states are quantum optically all-order coherent and constitute an almost monochromatic radiation of frequency $ 2eV/\hbar $. Furthermore, we deduce in detail, how the collective behaviour of the superconductors influences the collective (that are the optical) features of the emitted microwave photons. Communicated by Joel Feldman submitted 14/02/03, accepted: 04/03/03     

DNS Curves in a Production/Inventory Model

In this paper, we investigate the bifurcation behavior of an inventory/production model close to a Hamilton-Hopf bifurcation. We show numerically that two different types of DNS curves occur: If the initial states are far from the bifurcating limit cycle, the limit cycle can be approached along different trajectories with the same cost. For a subcritical bifurcation scenario, the hyperbolic equilibrium state and the hyperbolic limit cycle coexist for some parameter range. When both the long term states yield approximately the same cost, a second DNS curve separates their domains of attraction. At the intersection of these two DNS curves, a threefold Skiba point in the state space is found. The acronym DNS stands for Dechert, Nishimura, and Skiba (Ref. 1)     

A model of nonautonomous dynamics driven by repeated harmonic interaction

We consider an exactly solvable model of nonautonomous W*-dynamics driven by repeated harmonic interaction. The dynamics is Hamiltonian and quasifree. Because of inelastic interaction in the large-time limit, it leads to relaxation of initial states to steady states. We derive the explicit entropy production rate accompanying this relaxation. We also study the evolution of different subsystems to elucidate their eventual correlations and convergence to equilibriums. In conclusion, we prove that the W*-dynamics manifests a universal stationary behavior in a short-time interaction limit.     

Stress and deformation states in the annulus made from shape memory material

The contribution deals with the stress and deformation states in an annulus made of shape memory material (SMM). The loading‐unloading process is going on at constant temperature, which is below temperature M_f (martensite finish temperature). The research is focused on the determination of the stress and deformation states in the annulus after loading and unloading process. These results are necessary for treating process of constrained recovery in shape memory annulus. The real loading‐unloading function in the stress‐strain coordinate system of shape memory material (Ni–Ti–Cu) is included in the mathematical model.     

Ground States for a Stationary Mean-Field Model for a Nucleon

In this paper we consider a variational problem related to a model for a nucleon interacting with the ω and σ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit, which is of a very different nature than the nonrelativistic limit in the atomic physics. Ground states are shown to exist for a large class of values for the parameters of the problem, which are determined by the values of some physical constants.     

Limit Distributions for Conflict Dynamical System with Point Spectra

Ukrainian Mathematical Journal - We construct a model of conflict dynamical system whose limit states are associated with singular distributions. It is proved that a criterion for the appearance of...     

Controlled Markov Chains with Weak and Strong Interactions: Asymptotic Optimality and Applications to Manufacturing

This paper deals with the asymptotic optimality of a stochastic dynamic system driven by a singularly perturbed Markov chain with finite state space. The states of the Markov chain belong to several groups such that transitions among the states within each group occur much more frequently than transitions among the states in different groups. Aggregating the states of the Markov chain leads to a limit control problem, which is obtained by replacing the states in each group by the corresponding average distribution. The limit control problem is simpler to solve as compared with the original one. A nearly-optimal solution for the original problem is constructed by using the optimal solution to the limit problem. To demonstrate, the suggested approach of asymptotic optimal control is applied to examples of manufacturing systems of production planning.     

Transient Metastability and Selective Decay for the Coherent Zonal Structures in Plasma Drift Wave Turbulence

The emergence of persistent zonal structures is studied in freely decaying plasma flows. The plasma turbulence with drift waves can be described qualitatively by the modified Hasegawa–Mima (MHM) model, which is shown to create enhanced zonal jets and more physically relevant features compared with the original Charney–Hasegawa–Mima model. We analyze the generation and stability of the zonal state in the MHM model following the strategy of the selective decay principle. The selective decay and metastable states are defined as critical points of the enstrophy at constant energy. The critical points are first shown to be invariant solutions to the MHM equation with a special emphasis on the zonal modes, but the metastable states consist of a zonal state plus drift waves with a specific smaller wavenumber. Further, it is found with full mathematical rigor that any initial state will converge to some critical point solution at the long-time limit under proper dissipation forms, while the zonal states are the only stable ones. The selective decay process of the solutions can be characterized by the transient visits to several metastable states, then the final convergence to a purely zonal state. The selective decay and metastability properties are confirmed by numerical simulations with distinct initial structures. One highlight in both theory and numerics is the tendency of Landau damping to destabilize the selective decay process.

     

Generalized Ferroelectric Model on a Square Lattice

A completely general square-lattice ferroelectric model is considered, in which the configuration energies have arbitrary values at each vertex. It is found that the partition function can be evaluated by straightforward extensions of the method used for the regular ice, F- and KDP-models, provided three sets of conditions are satisfied. These ensure that the two ordered F-model states are degenerate (in the limit of a large lattice), so we conclude that the application of any staggered field which breaks this degeneracy also renders the problem insoluble by the present methods. The partition function for these ordered states is obtained explicitly and found to be given quite simply as a product of certain parameters associated with each vertex. Finally some conjectures are made concerning the spontaneous staggered polarization.     

Time dependent critical fluctuations of a one dimensional local mean field model

Summary One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.     

Asymptotic properties of excited states in the Thomas-Fermi limit

Excited states are stationary localized solutions of the Gross-Pitaevskii equation with a harmonic potential and a repulsive nonlinear term that have zeros on a real axis. The existence and the asymptotic properties of excited states are considered in the semi-classical (Thomas-Fermi) limit. Using the method of Lyapunov-Schmidt reductions and the known properties of the ground state in the Thomas-Fermi limit, we show that the excited states can be approximated by a product of dark solitons (localized waves of the defocusing nonlinear Schrödinger equation with nonzero boundary conditions) and the ground state. The dark solitons are centered at the equilibrium points where a balance between the actions of the harmonic potential and the tail-to-tail interaction potential is achieved.     

Study of temperature Green’s functions of graphene-like systems in a half-space

We consider the formalism of temperature Green’s functions to study the electronic properties of a semiinfinite two-dimensional graphene lattice at a given temperature. Under most general assumptions about the graphene boundary structure, we calculate the propagator in the corresponding diagram technique. The obtained propagator survives limit transitions between physically different states of the system boundary, i.e., a zig-zag edge and a boundary condition in the “infinite mass” approximation, and also correctly describes the problem where the electron–hole symmetry is violated because of the presence of an external potential applied to the graphene boundary. We illustrate the use of the propagator, its analytic properties, and specific features of calculating with it in the example of determining the dependence of the electron density on the distance to the system boundary.     

On a phase transition model

An Allen–Cahn phase transition model with a periodic nonautonomous term is presented for which an infinite number of transition states is shown to exist. A constrained minimization argument and the analysis of a limit problem are employed to get states having a finite number of transitions. A priori bounds and an approximation procedure give the general case. Decay properties are also studied and a sharp transition result with an arbitrary interface is proved.     

Unstable limit cycles in an electric power system and basin boundary of voltage collapse

It has been reported that a saddle node bifurcation or a blue sky bifurcation causes voltage collapse in an electric power system. In these references, computer simulations are carried out with the voltage magnitude of the generator bus terminal held constant. The generator model described by differential equations of internal flux linkages allows the voltage magnitude of the generator bus terminal to change. By using this model, we have carried out computer simulations of the power system to determine the cause of voltage collapse. It is a cyclic fold bifurcation of the stable limit cycle caused by an unstable limit cycle that leads to the voltage collapse. The involvement of complicated sequences of unstable limit cycles with cyclic fold bifurcations is confirmed, and the voltage collapse which is caused by perturbation for steady states is related to these unstable limit cycles. This is very interesting from the point of view of a nonlinear problem. From the point of view of a power system, the power system will fluctuate in practice even in normal operation, and may sometimes operate beyond the limit of its stability in recent year. It is very important in this situation that we clarify bifurcations of limit cycles on the power system.