Exact Exponents Research Articles

On some properties of Buhmann functions

We study functions introduced by Buhmann. The exact exponent of smoothness of these functions is obtained and the problem of positivity of their Hankel transforms is analyzed.

Chaos and Residual Correlations in Pinned Disordered Systems

We study, using functional renormalization, two copies of an elastic system pinned by mutually correlated random potentials. Short scale decorrelation depends on a nontrivial boundary layer regime with (possibly multiple) chaos exponents. Large scale mutual displacement correlations behave as [x - x'](2zeta-mu), mu proportional to the difference between Flory (or mean field) and exact roughness exponents zeta. For short range disorder mu>0 and small; e.g., for random bond interfaces mu=5zeta-epsilon, epsilon=4-d, and mu=epsilon{[(2pi)(2)/36]-1} for the one component Bragg glass. Random field (i.e., long range) disorder exhibits finite residual correlations (no chaos mu=0) described by new functional renormalization fixed points. Temperature and dynamic chaos (depinning) are discussed.

Sub-electron charge relaxation via 2D hopping conductors

We have extended Monte Carlo simulations of hopping transport incompletely disordered two-dimensional (2D) conductors to the processof external charge relaxation. In this situation, a conductor of areaL × W shunts an externalcapacitor C withinitial charge Qi. At low temperatures, the charge relaxation process stops at some ‘residual’charge value corresponding to the effective threshold of the Coulombblockade of hopping. We have calculated the root mean square (rms) valueQR of the residual charge for a statistical ensemble of capacitor-shunting conductorswith random distribution of localized sites in space and energy and randomQi, as a function of macroscopic parameters of the system. Rather unexpectedly,QR has turned out to depend only on some parameter combination: for negligible Coulomb interaction and for substantial interaction. (Hereν0 is the seed density oflocalized states, while κ is the dielectric constant.) For sufficiently large conductors, both functionsQR/e = F(X) follow the powerlaw F(X) = DX−β, but withdifferent exponents: β = 0.41 ± 0.01 for negligible and β = 0.28 ± 0.01 for significant Coulomb interaction. We have been able to derive this law analytically forthe former (most practical) case, and also explain the scaling (but not the exact value ofthe exponent) for the latter case. In conclusion, we discuss possible applications of thesub-electron charge transfer for ‘grounding’ random background charge in single-electrondevices.

Nonlinear dynamics in one dimension: A criterion for coarsening and its temporal law

We develop a general criterion about coarsening for some classes of nonlinear evolution equations describing one-dimensional pattern-forming systems. This criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is fixed in the course of time. An intermediate scenario may occur, namely "interrupted coarsening." The power of the criterion on which a brief account has been given [Politi and Misbah, Phys. Rev. Lett. 92, 090601 (2004)], and which we extend here to more general equations, lies in the fact that the statement about the occurrence of coarsening, or selection of a length scale, can be made by only inspecting the behavior of the branch of steady state periodic solutions. The criterion states that coarsening occurs if lambda'(A)>0 while a length scale selection prevails if lambda'(A)<0, where lambda is the wavelength of the pattern and A is the amplitude of the profile (prime refers to differentiation). This criterion is established thanks to the analysis of the phase diffusion equation of the pattern. We connect the phase diffusion coefficient D(lambda) (which carries a kinetic information) to lambda'(A), which refers to a pure steady state property. The relationship between kinetics and the behavior of the branch of steady state solutions is established fully analytically. Another important and new result which emerges here is that the exploitation of the phase diffusion coefficient enables us to determine in a rather straightforward manner the dynamical coarsening exponent. Our calculation, based on the idea that |D(lambda)| approximately lambda2/t, is exemplified on several nonlinear equations, showing that the exact exponent is captured. We are not aware of another method that so systematically provides the coarsening exponent. Contrary to many situations where the one-dimensional character has proven essential for the derivation of the coarsening exponent, this idea can be used, in principle, at any dimension. Some speculations about the extension of the present results are outlined.

The size of components in continuum nearest-neighbor graphs

We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ${\mathbb{R}}^d$. The connectivity function is shown to decay superexponentially, and we identify the exact exponent. From this we also obtain the decay rate of the maximal number of points of a path through the origin. We define the generation number of a point in a component and establish its asymptotic distribution as the dimension $d$ tends to infinity.

Thermokinetic modeling of the combustion of carbonaceous particulate matter

The scope of this study was to determine the kinetics (conversion model, preexponential factor and activation energy) of the combustion of two diesel sootlike materials (Printex XE-2B and Flammruss 101) in the presence of an excess of oxygen by dynamic thermogravimetry. A composite kinetic analysis procedure was applied to evaluate the full kinetic triplet from nonisothermal kinetic data. First, the activation energy values were obtained from the Kissinger–Akahira–Sunose isoconversional method. The values were 129 kJ mol −1 (Printex XE-2B) and 144 kJ mol −1 (Flammruss 101). The higher reactivity of Printex XE-2B material was attributed to its markedly greater surface area. The activation energy was found to depend slightly on conversion, suggesting that the combustion was a single-step process for both materials. Second, a comparison of the theoretical masterplots deduced by assuming various conversion models with the experimental masterplots obtained from the kinetic data allowed the selection of the appropriate conversion model of the process. Both materials were found to follow a mechanism based on surface nucleation with subsequent movement of the resulting surface, which was consistent with the penetration of oxygen through the porous structure of the solid samples. Also, the preexponential factors and exact kinetic exponents were evaluated on the basis of predetermined activation energies and conversion models. The adequate consistency of the kinetic triplet was assessed by comparing both experimental and calculated thermoanalytical curves at constant heating rate.

Exact scaling in competitive growth models

A competitive growth model (CGM) describes the aggregation of a single type of particle under two different growth rules with occurrence probabilities p and 1-p . We explain the origin of the scaling behavior of the resulting surface roughness at small p for two CGM's which describe random deposition (RD) competing with ballistic deposition and RD competing with the Edward-Wilkinson (EW) growth rule. Exact scaling exponents are derived. The scaling behavior of the coefficients in the corresponding continuum equations are also deduced. Furthermore, we suggest that, in some CGM's, the p dependence on the coefficients of the continuum equation that represents their universality class can be nontrivial. In some cases, the process cannot be represented by a unique universality class. In order to show this, we introduce a CGM describing RD competing with a constrained EW model. This CGM shows a transition in the scaling exponents from RD to a Kardar-Parisi-Zhang behavior when p is close to 0 and to a Edward-Wilkinson one when p is close to 1 at practical time and length scales. Our simulation results are in excellent agreement with the analytic predictions.

Critical dynamics, duality, and the exact dynamic exponent in extreme type-II superconductors

The critical dynamics of superconductors is studied using renormalization group and duality arguments. We show that in extreme type II superconductors the dynamic critical exponent is given exactly by $z=3/2$. This result does not rely on the widely used models of critical dynamics. Instead, it is shown that $z=3/2$ follows from the duality between the extreme type II superconductor and a model with a critically fluctuating gauge field. Our result is in agreement with Monte Carlo simulations.

Exactly solvable scale-free network model

We study a deterministic scale-free network recently proposed by Barabási, Ravasz, and Vicsek. We find that there are two types of nodes: the hub and rim nodes, which form a bipartite structure of the network. We first derive the exact numbers P (k) of nodes with degree k for the hub and rim nodes in each generation of the network, respectively. Using this, we obtain the exact exponents of the distribution function P (k) of nodes with k degree in the asymptotic limit of k-->infinity . We show that the degree distribution for the hub nodes exhibits the scale-free nature, P (k) proportional to k(-gamma) with gamma=ln 3/ln 2=1.584 962 , while the degree distribution for the rim nodes is given by P(k) proportional to e(-gamma'k) with gamma' =ln (3/2) =0.405 465 . Second, we analytically calculate the second-order average degree of nodes, d(-) . Third, we numerically as well as analytically calculate the spectra of the adjacency matrix A for representing topology of the network. We also analytically obtain the exact number of degeneracies at each eigenvalue in the network. The density of states (i.e., the distribution function of eigenvalues) exhibits the fractal nature with respect to the degeneracy. Fourth, we study the mathematical structure of the determinant of the eigenequation for the adjacency matrix. Fifth, we study hidden symmetry, zero modes, and its index theorem in the deterministic scale-free network. Finally, we study the nature of the maximum eigenvalue in the spectrum of the deterministic scale-free network. We will prove several theorems for it, using some mathematical theorems. Thus, we show that most of all important quantities in the network theory can be analytically obtained in the deterministic scale-free network model of Barabási, Ravasz, and Vicsek. Therefore, we may call this network model the exactly solvable scale-free network.

Dynamical real-space renormalization group calculations with a highly connected clustering scheme on disordered networks

We have defined a type of clustering scheme preserving the connectivity of the nodes in a network, ignored by the conventional Migdal-Kadanoff bond moving process. In high dimensions, our clustering scheme performs better for correlation length and dynamical critical exponents than the conventional Migdal-Kadanoff bond moving scheme. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising model to be z=2.13 and z=2.09 , respectively, at the pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We investigate the phase diagram and the critical behavior of randomly bond diluted lattices in d=2 and 3 in the light of this transformation. We also provide exact correlation exponent and dynamical critical exponent values on hierarchical lattices with power-law and Poissonian degree distributions.

Exponents of Diophantine Approximation and Sturmian Continued Fractions

Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n (ξ) and w n * (ξ) defined by Mahler and Koksma. We calculate their six values when n=2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction ξ by quadratic surds.

Exponents and bounds for uniform spanning trees in d dimensions.

Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such trees can be obtained in terms of the Laplacian matrix on the graph, by using Grassmann integrals. We use this to obtain exact exponents that bound those for the power-law decay of the probability that k distinct branches of the tree pass close to each of two distinct points, as the size of the lattice tends to infinity.

Directed polymers and interfaces in random media: free-energy optimization via confinement in a wandering tube.

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension 1+d with 0<d<2 involves at the same time (i) a confinement in a favorable tube of radius R(S) approximately L (nu(S) ) with nu(S) =1/(4-d)<1/2 (ii) a superdiffusive behavior R approximately Lnu with nu=(3-d)/(4-d)>1/2 for the wandering of the best favorable tube available. The corresponding free energy then scales as F approximately Lomega with omega=2nu-1 and the left tail of the probability distribution involves a stretched exponential of exponent eta=(4-d)/2. These results generalize the well known exact exponents nu=2/3, omega=1/3, and eta=3/2 in d=1, where the subleading transverse length R(S) approximately L(1/3) is known as the typical distance between two replicas in the Bethe ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent alpha and/or to manifolds in dimension D+d= d(t) with 0<D<2. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ( alpha<0 ), whereas it is not for growing correlations ( alpha>0 ). In particular, for an interface of dimension ( d(t) -1) in a space of total dimension 5/3< d(t) <3 with random-bond disorder, our approach yields the confinement exponent nu(S) =( d(t) -1)(3- d(t) )/(5 d(t) -7). Finally, we study the exponents in the presence of an algebraic tail 1/ V1+micro in the disorder distribution, and obtain various regimes in the (micro,d) plane.

Conformal field theory of the Flory model of polymer melting.

We study the scaling limit of a fully packed loop model in two dimensions, where the loops are endowed with a bending rigidity. The scaling limit is described by a three-parameter family of conformal field theories, which we characterize via its Coulomb-gas representation. One choice for two of the three parameters reproduces the critical line of the exactly solvable six-vertex model, while another corresponds to the Flory model of polymer melting. Exact central charge and critical exponents are calculated for polymer melting in two dimensions. Contrary to predictions from mean-field theory we show that polymer melting, as described by the Flory model, is continuous. We test our field theoretical results against numerical transfer matrix calculations.

Performance analysis and error exponents of asymmetric watermarking systems

Digital watermarking is an important technique to protect intellectual property right and to transmit useful secondary data. This paper investigates the performance analysis and error exponents of asymmetric watermarking systems. Asymmetric watermarking provides potentially better levels of security. Its detection is much different from commonly used watermarking detectors, particularly when both gain factors and exact values of the watermark are not known to the detector. Optimum detectors are constructed in this paper. To handle unwieldy computations of large matrices, we develop equivalent yet more practical detectors in the frequency domain using asymptotic analysis techniques. Due to the nonlinearity nature of these detectors, their performance analysis is challenging, especially when the size of the watermark becomes large. Gaussian approximations using the central limit theorem is limited in modeling the performance. To obtain fundamental performance limits, we go a step further to make use of asymptotic statistical analysis techniques to derive the exact error exponents. The analytical method gives insights into the performance of asymmetric watermarking, and paves the way toward seeking efficient watermarks for more general asymmetric watermarking systems.

Kinetic Studies of the Calcination of Ammonium Metavanadate by Thermal Methods

The calcination of ammonium metavanadate was investigated using simultaneous thermogravimetric analysis (TGA) and differential scanning calorimetry techniques. The decomposition of ammonium metavanadate proceeded in three steps, and kinetic analysis was performed under nonisothermal conditions using an integral composite method. The integral “model-free” method of TGA data at various heating rates suggested that each step be subjected to a single kinetic process. The integral master plot method showed that the calcination reaction for the three steps was best described by accommodated nucleation and nuclei growth kinetic models. Also, the preexponential factors and exact kinetic exponents for these three steps were finally evaluated on the basis of predetermined activation energies and kinetic models. The results of nonisothermal kinetic analysis of the calcination of ammonium metavanadate suggested that this integral composite method was very successful in evaluating kinetic parameters and describing the...

Loose, Flat Knots in Collapsed Polymers

We consider single ring polymers which are confined on a plane but maintain a fixed three-dimensional knotted topology. The equilibrium statistics of such systems is studied on the basis of a model on square lattice in which the configurations are represented by N-step polygons with a number of self-intersections restricted to the minimum compatible with the topology. This allows to define the size, s, of the flat knots and to study their localization properties. Due to the presence of both excluded volume and attractive interactions, the model undergoes a theta transition. Accurate Monte Carlo results show that, while in the high temperature swollen regime both prime and composite knot components are localized (〈s〉 N ∼N t , with t=0), in the low temperature, compact phase they are fully delocalized (t=1). Right at the theta transition weak localization prevails (t=0.44±0.02). Part of the results can be interpreted by taking into account a dominance of figure eight shapes for the coarse grained knotted polymer configurations, and by applying the scaling theory of polymer networks of fixed topology. In particular t=3/7 can be conjectured as an exact exponent characterizing the weak knot localization at the theta point.

Polymer θ-point as a knot delocalization transition

We study numerically the tightness of prime flat knots in a model of self-attracting polymers with excluded volume. We find that these knots are localized in the high temperature swollen regime, but become delocalized in the low temperature globular phase. Precisely at the collapse transition, the knots are weakly localized. Some of our results can be interpreted in terms of the theory of polymer networks, which allows one to conjecture exact exponents for the knot length probability distributions.

Fractal geometry of critical Potts clusters

Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. However, the approach to the asymptotic region is slow, and the present range of the data does not allow a unique identification of the exact correction exponents

Adsorption Profiles of Long‐Chain Polymer in Semidilute Solutions Physisorbed to a Wall

AbstractEquilibrium concentration profiles for long macromolecules in solution strongly adsorbed to a flat wall were estimated by a combination of persistent reptation and partial chain regrowth using a configurational bias lattice Monte Carlo method. The profiles were obtained for various concentrations in the semidilute regime and different chain lengths. A master curve exists in a plot of the profile reduced by the bulk concentration as a function of the distance to the wall reduced by the correlation length estimated for the bulk concentration. Near the wall, the master curve follows a power law with an exponent predicted in the scaling theory. Compared to depletion profiles at a repulsive wall that require a small penetration length, no correction of the distance to the wall is needed for the adsorption profiles.Reduced plot of ϕ(x)/ϕbulk vs. x/ξ for chain length N = 1 000 and five bulk concentrations above ϕ* indicated in legend. Slope with the exact exponent from scaling theory is shown for comparison.imageReduced plot of ϕ(x)/ϕbulk vs. x/ξ for chain length N = 1 000 and five bulk concentrations above ϕ* indicated in legend. Slope with the exact exponent from scaling theory is shown for comparison.