Repeller structure in a hierarchical model. II. Metric properties

We study the renormalization dynamics deriving from a hierarchical tight-binding Schrödinger equation. In the Part I of this work we analyzed the topological structure of the recurrent set-a chaotic repeller- and its relation with the spectral problem. In this part we turn our attention to the metric properties of the repeller. We first study periodic orbits and their bifurcation unfolding, and we organize them on a binary tree. We then apply a thermodynamic formalism which provides a complete characterization of the scaling properties of the energy spectrum. The distributionf() of local dimensions is determined by computing both a generalized -function through the periodic orbits and the bandwidths of periodic approximants of the Schrödinger operator. When the growth rateR of the potential is smaller than 1, we find evidence of a phase transition, implying that two different classes of states coexist in the spectrum. The asymptotic behavior of the Lebesgue measure of the spectrum is also studied. A linear scaling of to 0 is observed forR 1_–, while forR > 1, the measure of the periodic approximants goes to 0 as R^–h with the hierarchical orderh. Finally, we show that the localized state, present for R<1, is characterized by a superexponential scaling of the bandwidth.     

Turbulence spectra

The scaling invariance of the Navier-Stokes equations in the limit of infinite Reynolds number is used to derive laws for the inertial range of the turbulence spectrum. Whether the flow is homogeneous or not, the spectrum is chosen to be that given by a well-chosen biorthogonal decomposition. If the flow is hoogeneous, this spectrum coincides with the classical Fourier (energy) spectrum which exhibits Kolmogorov's k^–5/3 power law if the scaling exponent is assumed to be 1/3. In the more general case where the homogeneity assumption is relaxed, the spectrum is discrete and decays exponentially fast under the assumption that the flow is invariant (in a deterministic or statistical sense) under only one subgroup of the scaling coefficient of one scaling group of the equations (corresponding to one value of the scaling exponent). If the flow is invariant under two subgroups of scaling coefficients and, the spectrum becomes maximal, equal toR ₊. Finally, when a full symmetry, namely an invariance under a whole group, is assumed and the spectrum becomes continuous, the decaying law for the spectral density is derived and found to be independent of the specific value ofh These ideas are then applied to locally self-similar flows with multiple dilation centers (localized in space and time) and multiple scaling exponents, extending the concept of multifractals to space and time.     

Cantor spectrum and singular continuity for a hierarchical Hamiltonian

We study the spectrum of the HamiltonianH onl ₂() given by (H)(n)=(n+1)+(n–1)+V(n)(n) with the hierarchical (ultrametric) potentialV(2 ^m (2l+1))=(1–R ^m )/(1–R), corresponding to 1-, 2-, and 3-dimensional Coulomb potentials for 0<R<1,R=1 andR>1, respectively, in a suitably chosen valuation metric. We prove that the spectrum is a Cantor set and gaps open at the eigenvaluese _n (1)<e _n (2)<...<e _n (2 ⁿ –1) of the Dirichlet problemH=E, (0)=(2 ⁿ )=0,n1. In the gap opening ate _n (k) the integrated density of states takes on the valuek/2 ⁿ . The spectrum is purely singular continuous forR1 when the potential is unbounded, and the Lyapunov exponent vanishes in the spectrum. The spectrum is purely continuous forR<1 in (H)[–2, 2] and =0 here, but one cannot exclude the presence of eigenvalues near the border of the spectrum. We also propose an explicit formula for the Green's function.Work supported by the Fonds National Suisse de la Recherche Scientifique, Grant No. 2.042-0.86 (H.K. and R.L.) and 2.483-0.87 (A.S.)On leave from the Dipartimento di Fisica, Università degli Studi di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy     

Reflection and Absorption Spectra of Fluorite in the UV Spectral Region

We have calculated the R(E) and ₁(E) spectra from the theoretical ₂(E) spectra of five models in the region 8–27 eV and the ₂(E) and ₁(E) spectra from the experimental R(E) spectrum in the region 6–35 eV. The results are compared with the known theoretical ₂(E) spectra of five models. The basic features of all of the R(E), ₂(E), and ₁(E) spectra have been revealed. It is established that the experimental R(E) spectrum and the ₂(E) and ₁(E) spectra calculated with the use of experimental data are in good agreement with the results of theoretical calculations for the models of ₂(E). On the basis of the known theoretical calculations of the fluorite zones, a scheme of the nature of the principal maxima of R(E) and ₂(E) is suggested.     

Dimensional crossover in the large-N limit

We consider dimensional crossover for anO(N) Landau-Ginzburg-Wilson model on ad-dimensional film geometry of thicknessL in the large-N limit. We calculate the full universal crossover scaling forms for the free energy and the equation of state. We compare the results obtained using environmentally friendly renormalization with those found using a direct, non-renormalization-group approach. A set of effective critical exponents are calculated and scaling laws for these exponents are shown to hold exactly, thereby yielding nontrivial relations between the various thermodynamic scaling functions.     

Self-avoiding walks and trees in spread-out lattices

LetG _R be the graph obtained by joining all sites ofZ ^d which are separated by a distance of at mostR. Let (G _R ) denote the connective constant for counting the self-avoiding walks in this graph. Let (G _R ) denote the coprresponding constant for counting the trees embedded inG _R . Then asR, (G _R ) is asymptotic to the coordination numberk _R ofG _R , while (G _R ) is asymptotic toek _R. However, ifd is 1 or 2, then (G _R )-k _R diverges to –.Dedicated to Oliver Penrose on this occasion of his 65th birthday.     

Time Correlations in Fluid Transport Obtained by Sequential Rephasing Gradient Pulses

We present a basic experiment by which the evolution of the displacement probability density (propagator) of static or flowing fluid inNsuccessive time intervals is obtained by single labeling, coupled with multiple rephasing events during the course of a pulsed field-gradient sequence. We term this type of sequence SERPENT: SEquential Rephasing by Pulsed field-gradients Encoding N Time-intervals. Realizations of the SERPENT experiment for the caseN= 2 which include spin echo, stimulated echo, and Carr–Purcell pulse sequences are suggested. They have in common a spatial spin-labeling of the initial magnetization by a gradient of area q₀, followed by successive rephasing via gradients q₁and q₂at timest= Δ₁andt= Δ₂, respectively, where q₀+ q₁+ q₂= 0. A two-dimensional Fourier transform with respect to q₁and q₂gives directly the joint probability densityW₂(R₁, Δ₁; R₂, Δ₂) for displacements R₁and R₂in times Δ₁and Δ₂, respectively. q₁and q₂may be in arbitrary directions. Assuming R₁R₂, the correlation coefficient ρ_R1,R2then reflects the time-history of the fluctuating velocities. The behavior of the cross moment R₁(Δ₁) · R₂(Δ₂) can be obtained from either a full two-dimensional or a set of one-dimensional SERPENT measurements. Experimental results are presented for water flowing through a bed of packed glass beads. While Δ₁is appropriately chosen to sample the short-time velocity field within the system, increasing Δ₂clearly shows the loss of correlation when the average fluid element displacement exceeds the bead diameter.     

Effect of electron-hole scattering on the resistivity and Hall coefficient in YBaCuO

Coupled electron-hole (e-h) Boltzmann equations are applied to evaluate the resistivity and Hall coefficientR _H for a two-band model system with e-h, impurity, and phonon scatterings. We show that the anomalous temperature dependences =T andR _H ^–1 =T observed on YBaCuO compounds can be obtained by assuming a special two-band model in which the e-h scattering is responsible for the resistivity and the chemical potential varies linearly with temperature.     

Critical exponents,hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2 ₄ – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2 ₄ –. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R _g ² >/<R _e ² >=0.14026±0.00007, <R _m ² >/<R _e ² >=0.43961±0.00034, and ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent ₁=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ₁. Earlier Monte Carlo estimates ofv, which were 0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R _g ² >/<R _e ² >=0.1599±0.0002 and ^*=0.2471±0.0003; since ^*>0, hyperscaling holds. The approach to ^* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2 ₄ – for two-dimensional SAWs.     

Semi-infinite Potts model and percolation at surfaces

The effects of surfaces on percolation are investigated near the bulk percolation threshold ind=6– dimensions. Using field-theoretic methods, this is done within the framework of a semi-infinite continuousq-state Potts model withq1. Renormalization-group equations are obtained which imply that the usual scaling laws for surface and bulk exponents are valid to all orders in , and the surface exponents at the ordinary and special transition are computed to order . Our result for ₁ ^ord is in conformity with the one by Carton.     

Analytical properties of mean two-particle density matrix

The analytical properties of the mean two-particle density matrixZ(t)=d R ₁ d R ₂ <_c(R ₁:R ₂;t)(R ₁,R ₂;t)> in the right-hand halfplane of the complex variable t is considered; herec,v(R₁,R ₂; t) are single-particle density matrices. It is proven that, in the case of a Gaussian field, the function Z(t) is analytical in the region Ret > 0. It is shown that the frequency dependence of the light-absorption coefficient in disordered semiconductors is determined by the asymptote of the function Z(t) as t.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 46–50, June, 1987.     

Space-Charge Field Induced in a Photorefractive Crystal with an Applied Square-Wave Electric Field for Large Contrast of an Interference Pattern

In the quasi-static approximation, the nonlinear equation for the space-charge field E _sc formed in a photorefractive crystal with an applied square-wave voltage under the action of light with an inhomogeneous transverse intensity distribution is derived. The spatial spectrum of the field E _sc induced in a Bi₁₂TiO₂₀ crystal with typical values of the acceptor density N _A and of the product of the electron mobility by the recombination time _R is analyzed. An approximate analytical solution of the nonlinear equation is derived in the case in which the amplitudes of spatial harmonics of the spectrum E _sc form a nonmonotonic sequence. The spatial distribution of the induced electric field is calculated numerically for crystals with different parameters N _A and _R .     

Identification of the Dilute Regime in Particle Sedimentation

We investigate the dynamics of rigid, spherical particles of radius R sinking in a viscous fluid. Both the inertia of the particles and the fluid are neglected. We are interested in a large number N of particles with average distance dR. We investigate in which regime (in terms of N and R/d) the particles do not significantly interact and approximately sink like single particles. We rigorously establish the lower bound for the critical number N_crit of particles. This lower bound agrees with the heuristically expected N_crit in terms of its scaling in R/d. The main difficulty lies in showing that the particles cannot get significantly closer over a relevant time scale. We use the method of reflection for the Stokes operator to bound the strength of the particle interaction.     

f(R) Cosmology in the First Order Formalism

In the present work we consider those theories that are obtained from a Lagrangian density _T (R) = f(R){-g} + _M , that depends on the curvature scalar and a matter Lagrangian that does not depend on the connection, and apply Palatini's method to obtain the field equations. We start with a brief discussion of the field equations of the theory and apply them to a cosmological model described by the FRW metric. Then, we introduce an auxiliary metric to put the resultant equations into the form of GR with cosmological constant and coupling constant that are curvature depending. We show that we reproduce known results for the quadratic case. We find relations among the present values of the cosmological parameters q ₀, H ₀, and . Next we use a simple perturbation scheme to find the departure in angular diameter distance with respect to General Relativity. Finally, we use the observational data to estimate the order of magnitude of what is essentially the departure of f(R) from linearity. The bound that we find for f (0) is so huge that permit almost any f(R). This is in the nature of things: the effect of higher order terms in f(R) are strongly suppressed by power of Planck's time 8G ₀. In order to improve these bounds more research on mathematical aspects of these theories and experimental consequences is necessary.     

Conservation of Energy,Entropy Identity,and Local Stability for the Spatially Homogeneous Boltzmann Equation

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f ₀(v)(1+|v|²+|logf ₀(v)|)L ¹(R ³), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L ([0, ); L ¹ ₂(R ³))C ¹([0, ); L ¹(R ³)) [where L ¹ _s (R ³)={ff(v)(1+|v|²) ^s/2L ¹(R ³)}], and in this class, the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy. Moreover, for hard potentials and the hard-sphere model, a local stability result for conservative solutions (i.e., satisfying the conservation of mass, momentum, and energy) is obtained. As an application of the local stability, a sufficient and necessary condition on the initial data f ₀ such that the conservative solutions f belong to L ¹ _loc([0, ); L ¹ ₂₊ (R ³)) is also given.     

Dimensional crossover and finite-size scaling belowT c

Using the formalism developed in earlier work, dimensional crossover on ad-dimensional layered Ising-type system satisfying periodic boundary conditions and of sizeL is considered belowT _c (L), T _c (L) being the critical temperature of the finite-size system. Effective critical exponents _eff and _eff are shown explicitly to crossover between theird- and (d–1)-dimensional values for _L in the limitsL/ _L andL/ _L 0, respectively, _L , being the correlation length in the layers. Using anL-dependent renormalization group, the effective exponents are shown to satisfy natural generalizations of the standard scaling laws. In addition,L-dependent global scaling fields which span the entire crossover are defined and a scaling form of the equation of state in terms of them derived. All the above assertions are verified explicitly to one loop in perturbation theory, in particular effective exponents and a universal crossover equation of state are obtained and shown in the above asymptotic limits to be in good agreement with known results.     

Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.     

Renormalization of binary trees derived from one-dimensional unimodal maps

For one-dimensional unimodal mapsh (x):I I, whereI=[x ₀,x ₁] when =_max, a binary tree which includes all the periodic windows in the chaotic regime is constructed. By associating each element in the tree with the superstable parameter value of the corresponding periodic interval, we define a different unimodal map. After applying a certain renormalization procedure to this new unimodal map, we find the period-doubling fixed point and the scaling constant. The period-doubling fixed point depends on the details of the maph (x), whereas the scaling constant equals the derivative . The thermodynamics and the scaling function of the resulting dynamical system are also discussed. In addition, the total measure of the periodic windows is calculated with results in basic agreement with those obtained previously by Farmer. Up to 13 levels of the tree have been included, and the convergence of the partial sums of the measure is shown explicitly. A new scaling law has been observed, i.e., the product of the length of a periodic interval characterized by sequenceQ and the scaling constant ofQ is found to be approximately 1.     

Numerical Simulations of Random Walk in Random Environment

This paper is concerned with the numerical simulation of a random walk in a random environment in dimension d = 2. Consider a nearest neighbor random walk on the 2-dimensional integer lattice. The transition probabilities at each site are assumed to be themselves random variables, but fixed for all time. This is the random environment. Consider a parallel strip of radius R centered on an axis through the origin. Let X _R be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then X _R is a random variable, depending on the environment. In dimension d = 1, the variable X _R converges in distribution to the Bernoulli variable, X = 0, 1 with equal probability, as R . Here the 2-dimensional problem is studied using Gauss-Seidel and multigrid algorithms.