Field-theoretical approach to static critical phenomena in semi-infinite systems

The critical behaviour of a semi-infinite system withO(n) spin symmetry is studied in 4- dimensions near the ordinary transition using renormalization-group methods of field theory and -expansion techniques. It is found that, to all orders in , all surface exponents can be expressed in terms of two bulk exponents and a single surface exponent which follows from the anomalous dimension of the derivative (x ,0) of The order parameter (x,x ) at the surface (x =0). As a byproduct, Barber's scaling law 2₁ – ₁₁ = + is obtained. The surface exponents are calculated to second order in . Our results show that the scaling relation = ^–1 proposed by Bray and Moore is incorrect. The behaviour of various scaling functions close to the surface (i.e. forx correlation length) is determined with the help of short-distance expansions. We also treat corrections to scaling and logarithmic corrections in four dimensions. Our results for the logarithmic corrections of the layer and local susceptibilities disagree with those obtained by Guttmann and Reeve.A brief account of some of the results presented here was given in [1]. The surface exponents were independently calculated to order ² by Reeve and Guttmann [2] using an alternative method     

Crisis-induced unstable dimension variability in a dynamical system

Unstable dimension variability is an extreme form of non-hyperbolic behavior in chaotic systems whose attractors have periodic orbits with a different number of unstable directions. We propose a new mechanism for the onset of unstable dimension variability based on an interior crisis, or a collision between a chaotic attractor and an unstable periodic orbit. We give a physical example by considering a high-dimensional dissipative physical system driven by impulsive periodic forcing.     

A rigorous replica trick approach to Anderson localization in one dimension

LetH=?Δ+V onl ²(?), whereV(x),x∈?, are i.i.d.r.v.'s, and letG _L (x,y;E+iη)= 〈x|(H _L?(E+iη))^?1|y〉, whereH _L denotes the operatorH restricted to {?L, ?L+1,...,L} with Dirichlet boundary conditions. We use a supersymmetric replica trick to prove that $$E(|G_L (0,x; E + i\eta )|^2 ) \leqq K\eta ^{ - 2} \exp \{ - m|\log \eta |^{ - \sigma } |x|\} $$ for somem>0, σ>0,K<∞, uniformly inL andE. This estimate, together with the usual necessary estimate on the density of states, implies zero conductivity and gives exponential localization by the Fröhlich, Martinelli, Scoppola, and Spencer method.     

Measure and dimension of solenoidal attractors of one dimensional dynamical systems

Letf:MM be aC -map of the interval or the circle with non-flat critical points. A closed invariant subsetAM is called a solenoidal attractor off if it has the following structure: , where{I _k ⁽ⁿ⁾ is the cycle of intervals of periodp _n. We prove that the Lebesgue measure ofA is equal to zero and if sup(p _n+1/p_n)< then the Hausdorff dimension ofA is strictly less than 1.     

Field-theoretical Renormalization-Group approach to critical dynamics of crosslinked polymer blends

We consider a crosslinked polymer blend that may undergo a microphase separation. When the temperature is changed from an initial value towards a final one very close to the spinodal point, the mixture is out equilibrium. The aim is the study of dynamics at a given time t, before the system reaches its final equilibrium state. The dynamics is investigated through the structure factor, S(q, t), which is a function of the wave vector q, temperature T, time t, and reticulation dose D. To determine the phase behavior of this dynamic structure factor, we start from a generalized Langevin equation (model C) solved by the time composition fluctuation. Beside the standard de Gennes Hamiltonian, this equation incorporates a Gaussian local noise, ζ. First, by averaging over ζ, we get an effective Hamiltonian. Second, we renormalize this dynamic field theory and write a Renormalization-Group equation for the dynamic structure factor. Third, solving this equation yields the behavior of S(q, t), in space of relevant parameters. As result, S(q, t) depends on three kinds of lengths, which are the wavelength q ⁻¹, a time length scale R(t) ∼ t ^1/z , and the mesh size ξ ^*. The scale R(t) is interpreted as the size of growing microdomains at time t. When R(t) becomes of the order of ξ ^*, the dynamics is stopped. The final time, t ^*, then scales as t ^* ∼ ξ ^{* z}, with the dynamic exponent z = 6−η. Here, η is the usual Ising critical exponent. Since the final size of microdomains ξ ^* is very small (few nanometers), the dynamics is of short time. Finally, all these results we obtained from renormalization theory are compared to those we stated in some recent work using a scaling argument.     

混沌动力系统小波变换模数的关联维数

关联维数是描述动力系统混沌的重要参数.在仿真计算的基础上,发现尽管小尺度小波变换时,奇异吸引子的整体形状发生了改变,但小波变换模数的关联维数与动力系统本身是一致的.同时仿真计算还表明,随尺度的增加,关联积分与距离的标度关系逐渐变差,但在一定范围内仍可较准确地计算出关联维数. 关键词： 混沌 关联维数 小波分析 尺度 相空间     

Bubbling transition to spatial mode excitation in an extended dynamical system

We investigated the transition to spatio-temporal chaos in spatially extended nonlinear dynamical systems possessing an invariant subspace with a low-dimensional attractor. When the latter is chaotic and the subspace is transversely stable we have a spatially homogeneous state only. The onset of spatio-temporal chaos, i.e. the excitation of spatially inhomogeneous modes, occur through the loss of transversal stability of some unstable periodic orbit embedded in the chaotic attractor lying in the invariant subspace. This is a bubbling transition, since there is a switching between spatially homogeneous and nonhomogeneous states with statistical properties of on-off intermittency. Hence the onset of spatio-temporal chaos depends critically both on the existence of a chaotic attractor in the invariant subspace and its being transversely stable or unstable.     

Classical simulation of infinite-size quantum lattice systems in one spatial dimension

Invariance under translation is exploited to efficiently simulate one-dimensional quantum lattice systems in the limit of an infinite lattice. Both the computation of the ground state and the simulation of time evolution are considered.     

Correlation dimension of the flow and spatial development of dynamical chaos in a boundary layer

The relation between the spatial development of turbulence in a real hydrodynamical system (a boundary layer on a plate in a wind tunnel) and the increase of dimension downstream is established experimentally.     

Dynamical system related to quasiperiodic Schrödinger equations in one dimension

A dynamical map is obtained from a class of quasiperiodic discrete Schrödinger equations in one dimension which include the Fibonacci system. The potentials are constant except for steps at special points.     

Mass extinction in a dynamical system of evolution with variable dimension

Introducing the effect of extinction into the so-called replicator equations in mathematical biology, we construct a general model where the diversity of species, i.e., the dimension of the equation, is a time-dependent variable. The system shows very different behavior from the original replicator equation, and leads to mass extinction when the system initially has high diversity. The present theory can serve as a mathematical foundation for the paleontologic theory for mass extinction. This extinction dynamics is a prototype of dynamical systems where the variable dimension is inevitable.     

A dynamical system approach to SOC models of Zhang's type

We discuss Zhang's model of SOC in the framework of hyperbolic dynamical systems with singularities. The fractal structure of the invariant energy distribution, correlation decay-like phenomena, and symbolic coding are discussed.