Some Rigorous Results on a Stochastic GOY Model

A stochastic infinite dimensional version of the GOY model is rigorously investigated. Well posedness of strong solutions, existence and p-integrability of invariant measures is proved. Existence of solutions to the zero viscosity equation is also proved. With these preliminary results, the asymptotic exponents ζ_p of the structure function are investigated. Necessary and sufficient conditions for ζ₂≥ 2/3 and ζ₂=2/3 are given and discussed on the basis of numerical simulations.     

Resonance in Defect Turbulence under Periodic External Force

The periodically forced spatially extended Brusselator is investigated in the chaotic regime. We explore resonant or non-resonant patterns generated under various forcing frequencies and forcing amplitudes. Resonant spatially uniform oscillation and irregular structures are found. Furthermore two types of regular spatial patterns are generated under appropriate parameters. Our results of numerical simulations demonstrate that periodic force can give rise to resonant patterns in forced systems of spatiotemporal chaos similar to the situation of forced systems of regular oscillations.     

An Alternative Model of the Damped Harmonic Oscillator Under the Influence of External Force

In this paper we introduce the modified time-dependent damped harmonic oscillator. An exact solution of the wave function for both Schrödinger picture and coherent state representation are given. The linear and quadratic invariants are also discussed and the corresponding eigenvalues and eigenfunctions are calculated. The Hamiltonian is transformed to SU(1,1) Lie algebra and an application to the generalized coherent state is discussed. It has been shown that when the system is under critical damping case the maximum squeezing is observed in the first quadrature F _x . However, for the overcritical damping case the maximum squeezing occurs in the second quadrature F _y . Also it has been shown that the system for both cases is sensitive to the variation in the coherent state phase.     

Equilibrium Phase Transitions in Coupled Map Lattices: A Pedestrian Approach

A class of piecewise linear coupled map lattices with simple symbolic dynamics is constructed. It can be solved analytically in terms of the statistical mechanics of spin lattices. The corresponding Hamiltonian is written down explicitly in terms of the parameters of the map. The approach follows the line of recent mathematical investigations. But the presentation is kept elementary so that phase transitions in the dynamical model can be studied in detail. Although the method works only for map lattices with repelling invariant sets some of the conclusions, i.e., the role of local curvature of the single site map and properties of the nearest neighbour coupling might play an important role for phase transitions in general dynamical systems.     

A Numerical Investigation of the Pinning Phenomenon in Quasi-Periodic Frenkel Kontrova Model Under an External Force

We derive the relativistic Vlasov equation from quantum Hartree dynamics for fermions with relativistic dispersion in the mean-field scaling, which is naturally linked with an effective semiclassic limit. Similar results in the non-relativistic setting have been recently obtained in Benedikter et al. (Arch Rat Mech Anal 221(1): 273–334, 2016). The new challenge that we have to face here, in the relativistic setting, consists in controlling the difference between the quantum kinetic energy and the relativistic transport term appearing in the Vlasov equation.     

The Convection Instability of Bhnard-Rayleigh Fluid Under Varied External Force

A new model of Bknard-Rayleigh convection has been put forward from mantle convection and liquid crystal, where the external force deviates from constant to a lineal function of vertical dimension z. The dimensionless parameter ε is induced to describe the scale of this kind of deviations. Through linear stability analyses, we find that even-odd symmetry broken appears. By numerical calculation under rigid boundary conditions, we find that α_c is almost α constant but R_c decreases nonlinearly as ε increases, so the increase of ε benefits convection. When ε=0, all results reduce to the former works automatically.     

On the Ising Model with Strongly Anisotropic External Field

In this paper we analyze the equilibrium phase diagram of the two-dimensional ferromagnetic n.n. Ising model when the external field takes alternating signs on different rows. We show that some of the zero-temperature coexistence lines disappear at every positive sufficiently small temperature, whereas one (and only one) of them persists for sufficiently low temperature.     

Transitions in a Logistic Growth Model Induced by Noise Coupling and Noise Color

With unified colored noise approximation, the logistic growth model is used to analyze cancer cell population when colored noise is included. It is found that both the coupling between noise terms and the noise color can induce continuous first-order-like and re-entrance-like phase transitions in the system. The coupling and the noise color can also increase tumor cell growth for small number of cell mass and repress tumor cell growth for large number of cell mass. It is shown that the approximate analytic expressions are consistent with the numerical simulations.     

Transitions in a Logistic Growth Model Induced by Noise Coupling and Noise Color

With unified colored noise approximation, the logistic growth model is used to analyze cancer cell population when colored noise is included. It is found that both the coupling between noise terms and the noise color can induce continuous first-order-like and re-entrance-like phase transitions in the system. The coupling and the noise color can also increase tumor cell growth for small number of cell mass and repress tumor cell growth for large number of cell mass. It is shown that the approximate analytic expressions are consistent with the numerical simulations.     

Non-Equilibrium Behaviour in Unidirectionally Coupled Map Lattices

A spatially one dimensional coupled map lattice with a local and unidirectional coupling is introduced. This model is studied analytically by a perturbation theory that is valid for small coupling strength. In parameter space three phases with different ergodic behaviour are observed. Via coarse graining the deterministic model is mapped to a stochastic spin model that can be described by a master equation. Because of the anisotropic coupling non-equilibrium behaviour is found on the coarse grained level. However, the stationary statistical properties of the spin dynamics can still be described with a nearest neighbour Ising model whereby the ordering is predominantly antiferromagnetic.     

Path Integral Approach of Quantum Heisenberg Planar Spin Glass Model in the Presencesof Random Field

A quantum version of the site-random Heisenberg planar XY model in a random field is presented in the boson representation. Like classical spherical model in the spin space, the model can be solved exactly within the coherent state path integral-representation. The phase diagram is obtained, and the effects of the randomness and quantum fluctuations on the onset of a spin glass phase are discussed.     

On forcing functions in Kauffman's random Boolean networks

The phase transition between frozen and chaotic behavior in Kauffman's cellular automata on a nearest neighbor square lattice does not agree with the percolation threshold of the forcing functions.     

Kinetic Gaussian Model with Long-Range Interactions

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/rd δ (δ＞ 0) on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z = 2 if δ＞ 2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0 ＜δ＜ 2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.     

Existence and Order of the Phase Transition of the Ising Model with Fixed Magnetization

Properties of the two dimensional Ising model with fixed magnetization are deduced from known exact results on the two dimensional Ising model. The existence of a continuous phase transition is shown for arbitrary values of the fixed magnetization when crossing the boundary of the coexistence region. Modifications of this result for systems of spatial dimension greater than two are discussed.     

Peculiar Quantum Phase Transitions and Hidden Supersymmetry in a Lipkin-Meshkov-Glick Model

In this paper we theoretically report an unconventional quantum phase transition of a simple Lipkin- Meshkow-Glick model： an interacting collective spin system without external magnetic field. It is shown that this model with integer-spin can exhibit a flrst-order quantum phase transition between different disordered phases, and more intriguingly, possesses a hidden supersymmetry at the critical point. However, for half-integer spin we predict another flrst-order quantum phase transition between two different long-range-ordered phases with a vanishing energy gap, which is induced by the destructive topological quantum interference between the intanton and anti-instanton tunneling paths and accompanies spontaneously breaking of supersymmetry at the same critical point. We also show that, when the total spin-value varies from half-integer to integer this model can exhibit an abrupt variation of Berry phase from π to zero.     

Kinetic Gaussian Model with Long-Range Interactions

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/r^d+δ (δ>0) on d-dimensional hypercubic lattices (d=1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z=2 if δ>2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0<δ<2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.     

Stability of incoherence in a population of coupled oscillators

We analyze a mean-field model of coupled oscillators with randomly distributed frequencies. This system is known to exhibit a transition to collective oscillations: for small coupling, the system is incoherent, with all the oscillators running at their natural frequencies, but when the coupling exceeds a certain threshold, the system spontaneously synchronizes. We obtain the first rigorous stability results for this model by linearizing the Fokker-Planck equation about the incoherent state. An unexpected result is that the system has pathological stability properties: the incoherent state is unstable above threshold, butneutrally stable below threshold. We also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise.     

Interference in the Quantum Ising Model

Using the measure of interference defined in this paper, we investigate the quantum phase transition of one-dimensional Ising chains. We find that thermal fluctuations affect the interference more strongly at the critical point. We also show that the derivative of the interference with respect to the coupling parameter, A, can be depressed by the thermal fluctuation. Finally, we find that this suppression is due to multi-particle excitations.     

Effective-Field Theory for Kinetic Ising Model on Honeycomb Lattice

As an analytical method, the effective-field theory （EFT） is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the honeycomb lattice （Z = 3）. The Liapunov exponent A is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. In the field amplitude ho / ZJ-temperature T/ZJ plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn. In contrast to previous analytical results that predicted a tricritical point separating a dynamic phase boundary line of continuous and discontinuous transitions, we find that the transition is always continuous. There is inconsistency between our results and previous analytical results, because they do not introduce sufficiently strong fluctuations.     

Tricritical Points and Reentry in the Quantum Hopfield Neural-Network Model

We examine a quantum Hopfield neural-network model in the presence of trimodal random transverse fields and random neuronal thresholds within the method of statistical physics. We use the Trotter decomposition to map the problem into an equivalent classical random Hopfield-type Ising model and obtain phase transitions between the ferromagnetic retrieval and the paramagnetic phases. The influence of competition between the diluted random transverse fields and the diluted random thresholds on the system is discussed, and some interesting results such as tricritical points and reentrance are analyzed.