Microcanonical scaling in small systems

A microcanonical finite-size scaling ansatz is discussed. It exploits the existence of a well-defined transition point for systems of finite size in the microcanonical ensemble. The best data collapse obtained for small systems yields values for the critical exponents in good agreement with other approaches. The exact location of the infinite system critical point is not needed when extracting critical exponents from the microcanonical finite-size scaling theory.     

Random walk on topologically biased anisotropic percolation clusters and transport properties

Unbiased random walks are performed on topologically biased anisotropic percolation clusters (APC). Topologically biased APCs are generated using suitable anisotropic percolation models. New walk dimensions are found to characterize the anisotropic behaviour of the unbiased random walk on the biased topology. Critical properties of electro and magneto conductivities are characterized estimating respective dynamical critical exponents. A dynamical scaling theory relating dynamical and static critical exponents has been developed. The dynamical critical exponents satisfy the scaling relations within error bar.     

Critical behavior of a spherical model with a free surface

Critical phenomena ind-dimensional ferromagnetic spherical models on hypercubic lattices with free surfaces are studied. The surface specific heat and surface susceptibilities are obtained. The exponents characterizing the divergence of these surface quantities at the bulk critical temperature are found to satisfy recently proposed scaling relations. The variation of the susceptibility with distance from the surface is also discussed. The author's recent scaling theory for surface properties is investigated in detail, and found to give an exact representation for the free energy of a three-dimensional spherical model of finite thickness in finite bulk and surface magnetic fields. A scaling form for the surface free energy is derived.     

非平衡相变的临界标度理论及普适性

综述了作者近年来在非平衡相变临界（ Ｎ Ｐ Ｃ） 标度理论及普适性研究的进展。主要包括一般 Ｎ Ｐ Ｃ 系统规格化模型,局域序参量的概率分布,广义势的临界渐近形式,空时有关函数及其临界奇异行为。论证了 Ｎ Ｐ Ｃ 系统的临界可标度性,导出了一组普适的 Ｎ Ｐ Ｃ 标度关系,由之计算出的４ 种 Ｎ Ｐ Ｃ 普适类的临界指数与目前已知的实验及理论结果吻合得非常好。此外,还讨论了非平衡相变临界标度理论的普适性,将平衡相变临界标度理论作为一种特殊极限情况含于同一理论体系中。     

非平衡相变的临界标度理论及普适性

综述了作者近年来在非平衡相变临界（ Ｎ Ｐ Ｃ） 标度理论及普适性研究的进展。主要包括一般 Ｎ Ｐ Ｃ 系统规格化模型,局域序参量的概率分布,广义势的临界渐近形式,空时有关函数及其临界奇异行为。论证了 Ｎ Ｐ Ｃ 系统的临界可标度性,导出了一组普适的 Ｎ Ｐ Ｃ 标度关系,由之计算出的４ 种 Ｎ Ｐ Ｃ 普适类的临界指数与目前已知的实验及理论结果吻合得非常好。此外,还讨论了非平衡相变临界标度理论的普适性,将平衡相变临界标度理论作为一种特殊极限情况含于同一理论体系中。     

Critical Behaviors in a Stochastic One-Dimensional Sand-Pile Model

A one-dimensional sand-pile model (Manna model), which has a stochastic redistribution process, is studied both in discrete and continuous manners. The system evolves into a critical state after a transient period. A detailed analysis of the probability distribution of the avalanche size and duration is numerically investigated. Interestingly,contrary to the deterministic one-dimensional sand-pile model, where multifractal analysis works well, the analysis based on simple finite-size scaling is suited to fitting the data on the distribution of the avalanche size and duration. The exponents characterizing these probability distributions are measured. Scaling relations of these scaling exponents and their universality class are discussed.     

Critical Behaviors in a Stochastic One-Dimensional Sand-Pile Model

A one-dimensional sand-pile model （Manna model）, which has a stochastic redistribution process, is studied both in discrete and continuous manners. The system evolves into a critical state after a transient period. A detailed analysis of the probability distribution of the avalanche size and duration is numerically investigated. Interestingly,contrary to the deterministic one-dimensional sand-pile model, where multifractal analysis works well, the analysis based on simple finite-size scaling is suited to fitting the data on the distribution of the avalanche size and duration. The exponents characterizing these probability distributions are measured. Scaling relations of these scaling exponents and their universality class are discussed.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

Electrical conductance of rings in composites

A scaling theory for describing the electrical conductance of rings in percolating systems is proposed. The theory is formulated for a given ring geometry and composition and the conductivity is expressed in terms of universal critical exponents for the infinite percolating system. Computer simulation results are found to be in good agreement with predictions of the theory.     

Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multiscaling

We develop a consistent closure procedure for the calculation of the scaling exponents ζ _n of the nth-order correlation functions in fully developed hydro-dynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ _n . This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ _n . Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.     

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.     

Statistical-thermodynamic approach to a chaotic dynamical system: Exactly solvable examples

The static and dynamic properties of a chaotic attractor of a two-dimensional map are studied, which belongs to a particular class of piecewise continuous invertible maps. Coverings of a natural size to cover the attractor are introduced, so that the microscopic information of the attractor is written on each box composing the cover. The statistical thermodynamics of the scaling indices and the size indices of the boxes is formulated. Analytic forms of the free energy functions of the scaling indices and the size indices of the boxes are obtained for examples of a hyperbolic and a nonhyperbolic chaotic attractor. The statistical thermodynamics of local Lyapunov exponents is also studied and a relation between the thermodynamics of scaling indices and of local Lyapunov exponents is invetigated. For the nonhyperbolic example, the free energy and entropy functions of local Lyapunov exponents are obtained in analytic forms. These results display the existence of phase transitions. A phase transition is seen in the thermodynamics of scaling indices also.     

Some aspects of phase transitions described by the self consistent current relaxation theory

A mathematical model for a nonlinear equation of motion for correlation functions is considered which describes the essential features of the self consistent current relaxation theory for a system experiencing some interaction with a static random field in addition to some quadratic self interaction. It is shown that the plane spanned by the two coupling parameters is separated into a region of ergodic motion and another region where the motion is nonergodic. At the separation line of the two phases all correlation functions can be discussed in terms of scaling laws. The critical exponents vary along the separation line continuously and they can be evaluated explicitely. The separation line consists of two pieces. Transitions across the first piece are characterized by a polarization catastrophy, by a vanishing of the transport coefficient and by a diverging low frequency spectrum ruled by one critical frequency scale. Transitions across the second piece do not exhibit a polarization divergence but show also a power law decrease to zero of the transport coefficient. The low frequency spectrum is the sum of two diverging parts. Each part is described by a scaling law, but the scaling frequencies and the scaling functions are quite different.     

Phase transitions in a cluster molecular field approximation

Cluster molecular field approximations represent a substantial progress over the simple Weiss theory where only one spin is considered in the molecular field resulting from all the other spins. In this work we discuss a systematic way of improving the molecular field approximation by inserting spin clusters of variable sizes into a homogeneously magnetised background. The density of states of these spin clusters is then computed exactly. We show that the true non-classical critical exponents can be extracted from spin clusters treated in such a manner. For this purpose a molecular field finite size scaling theory is discussed and effective critical exponents are analysed. Reliable values of critical quantities of various Ising and Potts models are extracted from very small system sizes. Received 30 September 2002 / Received in final form 25 November 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: pleim@theorie1.physik.uni-erlangen.de     

Anomalous roughening of Hele-Shaw flows with quenched disorder

The kinetic roughening of a stable oil-air interface moving in a Hele-Shaw cell that contains a quenched columnar disorder (tracks) has been studied. A capillary effect is responsible for the dynamic evolution of the resulting rough interface, which exhibits anomalous scaling. The three independent exponents needed to characterize the anomalous scaling are determined experimentally. The anomalous scaling is explained in terms of the initial acceleration and subsequent deceleration of the interface tips in the tracks coupled by mass conservation. A phenomenological model that reproduces the measured global and local exponents is introduced.     

On the spectrum of multifractal diffusion process

A quasi-multifractal model of stochastic processes is considered. In contrast to the more widely known multifractal random walk model, it is free of such substantial drawbacks as infinite variance of the modeled processes and time-dependent increments. An analysis of a multifractal diffusion-type process is presented, including the moments of increments and local scaling exponents of the process. A quasi-multifractal spectrum of the process is computed. A focus is on two new concepts in the theory of multifractal processes: effective scaling exponent and quasi-multifractal spectrum of a process.     

Computation of critical exponents for two-dimensional ising model on a cellular automaton

Static critical exponents for the two-dimensional Ising model are computed on a cellular automaton. The analysis of the data within the framework of the finite-size scaling theory reproduces their well-established values.     

Scaling and infrared divergences in the replica field theory of the Ising spin glass

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w². Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the -expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w² and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed. Received 23 December 1998 and Received in final form 23 March 1999