Scaling limit for interacting Ornstein-Uhlenbeck processes

The problem of describing the bulk behavior of an interacting system consisting of a large number of particles comes up in different contexts. See for example [1] for a recent exposition. In [4] one of the authors considered the case of interacting diffusions on a circle and proved that the density of particles evolves according to a nonlinear diffusion equation. The interacting particles evolved according to a generator that was symmetric in equilibrium. In this article we consider interacting Ornstein-Uhlenbeck processes. Here the diffusion generator is not symmetric relative to the equilibrium and the earlier methods have to be modified considerably. We use some ideas that were employed in [3] to extend the central limit theorem from the symmetric to nonsymmetric cases.This research is supported in part by the National Science Foundation, grant nos. DMS 89-01682 and DMS-88-06727     

Scaling limit of deeply virtual compton scattering

I outline a perturbative QCD approach to the analysis of the deeply virtual Compton scattering process γ^*p → γp′ in the limit of vanishing momentum transfer t = (p′ − p)². The DVCS amplitude in this limit exhibits a scaling behavior described by two-argument distributions F(x,y) which specify the fractions of the initial momentum p and the momentum transfer r ≡ p′ − p carried by the constituents of the nucleon. The kernel R(x,y;ξ,η) governing the evolution of the non-forward distributions F(x,y) has a remarkable property: it produces the GLAPD evolution kernel P(x/ξ) when integrated over y and reduces to the Brodsky-Lepage evolution kernel V(y,η) after the x-integration. This property is used to construct the solution of the one-loop evolution equation for the flavor non-singlet part of the non-forward quark distribution.     

Scaling models of thermodynamic properties on the coexistence curve: Problems and some solutions

Models consistent with the scaling theory of critical phenomena and capable of describing the thermodynamic properties F of substances on the coexistence curve, such as the density of the liquid ρ _l , density of the gas ρ _g , order parameter f _s , mean coexistence curve diameter f _d , and saturation pressure P _s are discussed. The models are presented in the form of equations F = (τ, D, C), where τ = (T _c ? T)/T _c , and D = (α, β, T _c , ρ _c , P _c , ...) are the critical characteristics, such as T _c , ρ _c , and P _c (temperature, density, and pressure, respectively), α and β are the scaling exponents, and C are adjustable coefficients. The authors developed combined models f(τ, D, C) for describing the indicated properties of a number of compounds (CH₄, NH₃, SF₆, water, methanol, ethanol, diethyl ether, and freons R134a, R143a, and R236ea). The coefficients C were determined based on experimental data over a wide temperature range, including the critical point. The equations derived are used to perform practical calculations, including estimates of the first and second derivatives of the saturation pressure with respect to the temperature in the critical region.     

Scaling of fluctuations and critical exponents

We present a new technique to describe the abnormal behavior of certain fluctuation observables in the critical regime of quantum statistical systems which undergo a phase transition. The idea is to rescale the local fluctuation operators by a relevant external parameter of the system, in addition to the usual scaling with the inverse square root of the volume. The scaling indices used in this scaling procedure are directly related to the critical exponents. Furthermore, it is explained that this new method of scaling preserves the CCR structure of the algebra of macroscopic fluctuations. Finally, scaling indices are computed for the relevant microscopic observables at all temperatures in a mean field approximation for a quantum anharmonic crystal. These indices yield the same critical exponents as predicted by mean field theory.     

Dynamics of limit models

Using noncommutative integration theory, we show why certain singular behavior has been appearing in the dynamics of large quantum mechanical systems, and how to isolate the singularities.Research supported by AFOSR under contract F 44620-71-C-0108.     

Scaling functions for critical surface scattering

Scattering of x-rays at grazing angles is studied for semi-infinite systems which exhibit a second order phase transition. We calculate the scaling functions for the diffuse scattering intensity aboveT _c and for the Bragg-intensity belowT _c, using -expansion techniques. Corrections to the leading critical surface behaviour and the size of the asymptotic scaling region are discussed. In addition, we obtain the crossover function for the excess susceptibility _s.This work was supported by Bundesministerium für Forschung und Technologie     

Scaling limit for a mechanical system of interacting particles

A system of a large number of classical particles moving on a onedimensional segment with virtually reflecting boundaries is studied. The particles interact with one another through repulsive pair-potential forces and are subjected to resistance proportional to their velocities. Because of the latter it is only the number of particles that is conserved under the evolution of the system. It is proved that in the hydrodynamic limit of diffusion type scaling the normalized counting measure of particle locations converges and its limiting density is governed by a non-linear diffusion equation which in typical cases is of porous media equation type.     

Scaling properties of models of nonequilibrium phenomena

The renormalization group method proposed by 't Hooft is developed for the study of scaling properties of some models of nonequilibrium phenomena. For one of two models studied in detail, the Langevin equation for the random variables contains a bilinear streaming velocity and the stationary probability distribution is Gaussian. The time-dependent Ginzburg-Landau model is chosen as a second example because it illustrates the advantage of the 't Hooft method of not having to specify a particular renormalization point. The scaling exponents for a model of the liquid-gas phase transition are calculated in lowest order to illustrate application of the method to a multifield system.     

Scaling properties of percolation models for multifragmentation

We have used scaling properties of nuclear multifragmentation, which have been observed with emulsion data, to investigate the properties of some approaches based on percolation. We have studied different percolation models on a cubic lattice and shown that they can rather well reproduce the data except for binary break up. We have described what the mean field approximation would give in this context and showed that it cannot reproduce the experimental results. Most of the paper is focused on the restructured aggregation model introduced earlier which allows to well reproduce the scaling properties observed experimentally. This model has been studied in details and extended to take account of bonds breaking. It is shown that, in some cases, a nucleus can break up in two pieces. This process cannot be obtained in conventional percolation or aggregation but is observed experimentally in the emulsion data. Other features like the dimensionality of the aggregation model, the restructuration of the clusters and a schematic constraint in momentum space have also been investigated.     

Scaling behavior of one-dimensional weakly disordered models

The density of states and various characteristic lengths of one-dimensional tight-binding models and disordered harmonic chains are calculated in the limit of weak disorder at the band edge of the ordered system. The density of states and a localization length of the one-dimensional Anderson model were already calculated by Derrida and Gardner; we recover their results. For the tight-binding models with off-diagonal disorder our results are in agreement with numerical calculations of Krey.     

Self-organized critical limit of autocatalytic surface reactions

We argue that the autocatalytic surface reaction 2CO+O₂→2CO₂ on Pt(110) may show self-organized critical behavior if the appropriate range of parameter values is investigated. Such a self-organized critical state is characterized by a power-law distribution of reconstructed surface regions.     

Scaling in ordered and critical random boolean networks

Random Boolean networks, originally invented as models of genetic regulatory networks, are simple models for a broad class of complex systems that show rich dynamical structures. From a biological perspective, the most interesting networks lie at or near a critical point in parameter space that divides "ordered" from "chaotic" attractor dynamics. We study the scaling of the average number of dynamically relevant nodes and the median number of distinct attractors in such networks. Our calculations indicate that the correct asymptotic scalings emerge only for very large systems.     

Scaling limit of the energy variable for the two-dimensional Ising ferromagnet

The critical point limit law (scaling limit) of the suitably renormalized energy variable is explicitly calculated for the two-dimensional nearest-neighbour Ising cylinder with free edges. It is shown that the renormalization factor has to behave as (2M 2N lnN)^1/2, where 2M denotes the number of rows and 2N the number of columns. By first taking the limitM and thenN, the limit law is proven to be Gaussian.     

Scaling and memory in the non-Poisson process of limit order cancelation

The order submission and cancelation processes are two crucial aspects in the price formation of stocks traded in order-driven markets. We investigate the dynamics of order cancelation by studying the statistical properties of inter-cancelation durations, defined as the waiting times between consecutive order cancelations of 22 liquid stocks traded on the Shenzhen Stock Exchange of China in year 2003. Three types of cancelations are considered, including cancelation of any limit orders, of buy limit orders and of sell limit orders. We find that the distributions of the inter-cancelation durations of individual stocks can be well modeled by Weibulls for each type of cancelation, and the distributions of rescaled durations of each type of cancelations exhibit a scaling behavior for different stocks. Complex intra-day patterns are also unveiled in the inter-cancelation durations. The detrended fluctuation analysis (DFA) and the multifractal DFA show that the inter-cancelation durations possess long-term memory and multifractal nature, which are not influenced by the intra-day patterns. No clear crossover phenomenon is observed in the detrended fluctuation functions with respect to the time scale. These findings indicate that the cancelation of limit orders is a non-Poisson process, which has potential worth in the construction of order-driven market models.     

Scaling and universality of critical fluctuations in granular gases

The total energy fluctuations of a low-density granular gas in the homogeneous cooling state near the threshold of the clustering instability are studied by means of molecular dynamics simulations. The relative dispersion of the fluctuations is shown to exhibit a power-law divergent behavior. Moreover, the probability distribution of the fluctuations presents data collapse as the system approaches the instability, for different values of the inelasticity. The function describing the collapse turns out to be the symmetric of the one found in several molecular equilibrium and nonequilibrium systems.     

Scaling at the critical temperature of a spin glass

It is argued that the critical exponent δ for an Edwards-Anderson spin glass at its critical temperature can be extracted from the magnetic susceptibility in a weak uniform magnetic field.