Attractors and characteristic exponents

A natural definition of an attractor as an invariant measure is given (based on the ergodic theory of axiom A diffeomorphisms) and some results are proved which support this definition. It is also proved that if an attractor has every characteristic exponent less than zero in a set of nonzero measure, then the support set of the attractor is an asymptotic stable periodic orbit.     

On characteristic exponents in turbulence

Ruelle has found upper bounds to the magnitude and to the number of non-negative characteristic exponents for the Navier-Stokes flow of an incompressible fluid in a domain . The latter is particularly important because it yields an upper bound to the Hausdorff dimension of attracting sets. However, Ruelle's bound on the number has three deficiences: (i) it relies on some unproved conjectures about certain constants; (ii) it is valid only in dimensions 3 and not 2; (iii) it is valid only in the limit . In this paper these deficiences are remedied and, in addition, the final constants in the inequality are improved.Work partially supported by U.S. National Science Foundation grant No. PHY-8116101-A01     

The critical exponents of the matrix-valued Gross-Neveu model

We study the large-N limit of the matrix-valued Gross-Neveu model in d > 2 dimensions. The method employed is a combination of the approximate recursion formula of Polyakov and Wilson with the solution to the zero-dimensional large-N counting problem of Makeenko and Zarembo. The model is found to have a phase transition at a finite value for the critical temperature and the critical exponents are approximated by ν = 1/(2(d − 2)) and η = d − 2. We test the validity of the approximation by applying it to the usual vector models where it is found to yield exact results to leading order in 1/N.     

落球法测液体的粘性系数的研究

导出并讨论小球达到终极速度时，下落的距离与小球直径之间的关系，根据斯托克斯定律和雷诺数，导出满足实验条件的小球直径的范围。     

The neutrino ball in the standard model

The model of the quasars as a neutrino ball based on the standard model is presented. A realistic quasar with the typical massM=8.42 10⁹ M , the radius of the neutrino ballR=1.66 a.u., and the neutrino numberN=1.07 10⁷² is obtained when the electroweak phase transition is first-order.     

The relevance of connectance on the Lyapunov characteristic exponents of products of symplectic random matrices

We study the effect of connectance on the Lyapunov characteristic exponents of products of symplectic random matrices, which mimic the chaotic behaviour of a large class of hamiltonian systems. It is shown that no significative modifications appear in the spectrum of the Lyapunov characteristic exponents when the number of interacting neighbours is increased.     

Large volume limit of the distribution of characteristic exponents in turbulence

For spatially extended conservative or dissipative physical systems, it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity. In the case of a turbulent viscous fluid, however, this simple idea is complicated by the phenomenon of intermittency. In the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation. These bounds have a reasonable large volume behavior. For two-dimensional fluids a particularly striking result is obtained: the total information creation is bounded above by a fixed multiple of the total energy dissipation (at fixed viscosity). The distribution of characteristic exponents is estimated in an intermittent model of turbulence (see [7]), and it is found that a change of behavior occurs at the valueD=2.6 of the self-similarity dimension.     

Critical exponents of the three-state potts model

The Fernandez-Pacheco duality invariant renormalization group is applied to the hamiltonian version of the two-dimensional three-state Potts model. The fixed point is located at exactly the self-dual critical point K¹ = 1. The thermal exponent is calculated to be y_T=1.1814. This value is in excellent agreement with the low temperature series expansion result of Zwanzig and Ranshaw (y_T = 1.174) and the strong coupling expansion result of Elitzur, Pearson and Shigemitsu (y_T = 1.190). It also seems to lend strong support to den Nijs' recent conjecture that the exact value should be y_T = 6/5.     

Finite-size scaling exponents of the Lipkin-Meshkov-Glick model

We study the ground state properties of the critical Lipkin-Meshkov-Glick model. Using the Holstein-Primakoff boson representation, and the continuous unitary transformation technique, we compute explicitly the finite-size scaling exponents for the energy gap, the ground state energy, the magnetization, and the spin-spin correlation functions. Finally, we discuss the behavior of the two-spin entanglement in the vicinity of the phase transition.     

落球法测液体粘度实验的改进

采用光电计时装置,对落球法测定液体粘度实验作了改进,减少了由手控计时引起的误差,提高了实验的准确度,扩大了测量范围．     

Corner exponents in the two-dimensional Potts model

The critical behavior at a corner in two-dimensional Ising and three-state Potts models is studied numerically on the square lattice using transfer operator techniques. The local critical exponents for the magnetization and the energy density for various opening angles are deduced from finite-size scaling results at the critical point for isotropic or anisotropic couplings. The scaling dimensions compare quite well with the values expected from conformal invariance, provided the opening angle is replaced by an effective one in anisotropic systems.     

Gauge-invariant critical exponents for the Ginzburg-Landau model

The critical behavior of the Ginzburg-Landau model is described in a manifestly gauge-invariant manner. The gauge-invariant correlation-function exponent is computed to first order in the 4-d and 1/n expansion, and found to agree with the ordinary exponent obtained in the covariant gauge, with the parameter alpha=1-d in the gauge-fixing term ( partial differential (mu)A(mu))(2)/2alpha.     

Anomalous critical exponents in the anisotropic Ashkin-Teller model

We perform a rigorous computation of the specific heat of the Ashkin-Teller model in the case of small interaction and we explain how the universality-nonuniversality crossover is realized when the isotropic limit is reached. We prove that, even in the region where universality for the specific heat holds, anomalous critical exponents appear: for instance, we predict the existence of a previously unknown anomalous exponent, continuously varying with the strength of the interaction, describing how the difference between the critical temperatures rescales with the anisotropy parameter.     

Critical exponents for the Reggeon quantum spin model

The Reggeon quantum spin (RQS) model on the transverse lattice in D dimensional impact parameter space has been conjectured to have the same critical behaviour as the Reggeon field theory (RFT). Thus from a high “temperature” series of ten (D = 2) and twenty (D = 1) terms for the RQS model we extrapolate to the critical temperature T = T_c by Padé approximants to obtain the exponents η=0.238±0.008, z=1.16±0.01, v=1.271±0.007 for D=2 and η=0.317±0.002, z=1.272±0.007, v=1.736±0.001, λ=0.57±0.03 for D=1. These exponents naturally interpolate between the D=0 and D=4?ε results for RFT as expected on the basis of the universality conjecture.     

Critical exponents of the Ising model on low-dimensional fractal media

The critical behavior of the Ising model on fractal substrates with noninteger Hausdorff dimension d_H<2 and infinite ramification order is studied by means of the short-time critical dynamic scaling approach. Our determinations of the critical temperatures and critical exponents β, γ, and ν are compared to the predictions of the Wilson-Fisher expansion, the Wallace-Zia expansion, the transfer matrix method, and more recent Monte Carlo simulations using finite-size scaling analysis. We also determined the effective dimension (d_ef), which plays the role of the Euclidean dimension in the formulation of the dynamic scaling and in the hyperscaling relationship d_ef=2β/ν+γ/ν. Furthermore, we obtained the dynamic exponent z of the nonequilibrium correlation length and the exponent θ that governs the initial increase of the magnetization. Our results are consistent with the convergence of the lower-critical dimension towards d=1 for fractal substrates and suggest that the Hausdorff dimension may be different from the effective dimension.     

Critical exponents of the random-field O(N) model

The critical behavior of the random-field Ising model has long been a puzzle. Different methods predict that its critical exponents in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components contrary to the experiments and simulations. We calculate the exponents of the random-field O(N) model with the (4+epsilon)-expansion and obtain values different from the exponents of the pure ferromagnet in 2+epsilon dimensions. An infinite set of relevant operators missed in previous studies leads to a breakdown of the (6-epsilon)-expansion.     

Critical exponents of the Gross-Neveu model from the effective average action

The phase transition of the Gross-Neveu model with N fermions is investigated by means of a nonperturbative evolution equation for the scale dependence of the effective average action. The critical exponents and scaling amplitudes are calculated for various values of N in d = 3. It is also explicitly verified that the Neveu-Yukawa model belongs to the same universality class as the Gross-Neveu model.     

Rigorous bounds of the Lyapunov exponents of the one-dimensional random Ising model

We find analytic upper and lower bounds of the Lyapunov exponents of the product of random matrices related to the one-dimensional disordered Ising model, using a deterministic map which transforms the original system into a new one with smaller average couplings and magnetic fields. The iteration of the map gives bounds which estimate the Lyapunov exponents with increasing accuracy. We prove, in fact, that both the upper and the lower bounds converge to the Lyapunov exponents in the limit of infinite iterations of the map. A formal expression of the Lyapunov exponents is thus obtained in terms of the limit of a sequence. Our results allow us to introduce a new numerical procedure for the computation of the Lyapunov exponents which has a precision higher than Monte Carlo simulations.