The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.     

Time Decay of Scaling Critical Electromagnetic Schrödinger Flows

We obtain a representation formula for solutions to Schrödinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp ${L^1 \to L^\infty}$ time decay estimate for the 3D-inverse square and the 2D-Aharonov–Bohm potentials.     

Bifurcation of Dicke Model Driven by Laser Field and Scaling Theory of Critical Exponents Far from Equilibrium

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The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator

We consider two models of one-dimensional discrete random Schrödinger operators

$(H_n\psi)_\ell =\psi_{\ell -1}+\psi_{\ell +1}+v_\ell \psi_\ell$

, \({\psi_0=\psi_{n+1}=0}\) in the cases \({ v_k=\sigma \omega_k/\sqrt{n}}\) and \({ v_k=\sigma \omega_k/ \sqrt{k}}\) . Here ω _k are independent random variables with mean 0 and variance 1.

We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.     

Dynamical Scaling Implications of Ferrari,Prähofer,and Spohn’s Remarkable Spatial Scaling Results for Facet-Edge Fluctuations

Spurred by theoretical predictions from Ferrari et al. (Phys Rev E 69:035102(R), 2004), we rederived and extended their result heuristically. With experimental colleagues we used STM line scans to corroborate their prediction that the fluctuations of the step bounding a facet exhibit scaling properties distinct from those of isolated steps or steps on vicinal surfaces. The correlation functions was shown to go as $t^{0.15(3)}$ , decidedly different from the $t^{0.26(2)}$ behavior for fluctuations of isolated steps.     

Pattern Structure and Properties of Invasion Percolation in Porous Media

The percolation clusters with varying occupy probability are constructed.Invasion percolation(IP)in percolation cluster is investigated by means of IP alporithm without trapping.The pattern constructions of IP in percolation clusters are obviously different from that of visocous fingering(VF) in percolation clusters.The fractal dimension Df of IP increases with increasing the percolation probability P.The geometry and the topology of the porous media have strong effects on the pattern structure of IP.For large M(the node numbers occupied by the injected fluid),the rescaled value is Rg/M^1/Df,which asymptotically approaches a constant value,and Rg is the gyration radius of IP cluster,Moreover,the chemical dimension Dl and the shortest path exponent dmin are obtained.     

Critical Scaling of Extended Power Law I - V Isotherms near Vortex Glass Transition

In view of the question about the vortex glass theory of the freezing of disordered vortex matter raised by recent experimental observations, we reinvestigate the critical scaling of high Tc superconductors. It is found that the dc current-voltage characteristic of mixed state superconductors has a general form of extended power law which is based on the Ginzburg-Landau （GL） functional in the similar way as the vortex glass theory. Isotherms simulated from this nonlinear equation fit the experimental I- V data of Strachan et al. [Phys. Rev. Left. 87（2001） 067007]. The puzzling question of the derivative plot for the I - V curves and the controversy surrounding the values of critical exponents are discussed.     

Percolation and the electron–electron interaction in an array of antidots

A square lattice of microcontacts with a period of 1 μm in a dense low-mobility two-dimensional electron gas is studied experimentally and numerically. At the variation of the gate voltage V _g , the conductivity of the array varies by five orders of magnitude in the temperature range T from 1.4 to 77 K in good agreement with the formula σ(V _g ) = (V _g ?V _g ^* (T))^β with β = 4. The saturation of σ(T) at low temperatures is absent because of the electron–electron interaction. A random-lattice model with a phenomenological potential in microcontacts reproduces the dependence σ(T, V _g ) and makes it possible to determine the fraction of microcontacts x(V _g , T) with conductances higher than σ. It is found that the dependence x(V _g ) is nonlinear and the critical exponent in the formula σ ∝ ? (x - 1/2) ^t in the range 1.3 < t(T, V _g ) < β.     

Limits of Gaudin Algebras,Quantization of Bending Flows,Jucys–Murphy Elements and Gelfand–Tsetlin Bases

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra \({U(\mathfrak {g})}\) of a semisimple Lie algebra \({\mathfrak {g}}\). This family is parameterized by collections of pairwise distinct complex numbers z ₁, . . . , z _n . We obtain some new commutative subalgebras in \({U(\mathfrak {g})^{\otimes n}}\) as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the Hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.     

Scaling of ac susceptibility and the nonlinear response function of high-temperature superconductors

The amplitude-dependent ac susceptibility of high-temperature superconductors is shown to obey some empirical scaling relations. We try to analyze this behavior by extending a dc nonlinear response function of mixed state to the ac cases. The derived equations for critical current and ac susceptibility X(T) agree with the scaling relations of experimental data.     

Scaling Behaviour and Memory in Heart Rate of Healthy Human

We investigate a set of complex heart rate time series from healthy human in different behaviour states with the detrended fluctuation analysis and diffusion entropy （DE） method. It is proposed that the scaling properties are influenced by behaviour states. The memory detected by DE exhibits an approximately same pattern after a detrending procedure. Both of them demonstrate the long-range strong correlations in heart rate. These findings may be helpful to understand the underlying dynamical evolution process in the heart rate control system, as well as to model the cardiac dynamic process.     

Ion Acceleration in Laser-Plasma Interaction： Shock, Sheath, and Scaling

Some notes and comments on ion acceleration in laser-plasma interaction is given, in particular for the implication of shock, sheath and sealing. A simple model is proposed for ion acceleration by the combination of shock and sheath. The obtained scaling relations between the maximum ion energy and laser parameters （power, pulse duration） as well plasma parameter （plasma density）for example α PL 7/12 Eion,max α TL1/3 and Eion,max α ne2/3,are compared to the previous works. Some deficiencies and implications of model and results are discussed.     

Scaling Properties of Runaway Electrons

Runaway electrons in tokamaks have been widely studied theoretically and experimentally. The runaway confinement time τ1 in ohmic and additionally heated tokamak plasmas presents an anomalous behavior when compared with theoretical predictions based on neoclassical models. Runaway electrons have received lately a great attention due to several reasons： （a） the possibility to study electromagnetic turbulence by measuring the runaway flux fluctuations and its energy spectra, and （ b ） the runaway electrons are powerful diagnostics capable of yielding valuable information on the actual distribution function of fusion experiments.     

Incompressible and Compressible Limits of Coupled Systems of Nonlinear Schrödinger Equations

Recently, coupled systems of nonlinear Schrödinger equations have been used extensively to describe a double condensate, i.e. a binary mixture of Bose-Einstein condensates. In a double condensate, an interface and shock waves may occur due to large intraspecies and interspecies scattering lengths. To know the dynamics of an interface and assure the existence of shock waves in a double condensate, we study the incompressible and the compressible limits respectively of two coupled systems of nonlinear Schrödinger equations. The main idea of our arguments is to define a “H-functional” like a Lyapunov functional which can control the propagation of densities and linear momenta. Such an idea is different from the one using the standard Wigner transform to investigate the incompressible and the compressible limits of a single nonlinear Schrödinger equation. $\mathfrak{V}Recently, coupled systems of nonlinear Schr?dinger equations have been used extensively to describe a double condensate, i.e. a binary mixture of Bose-Einstein condensates. In a double condensate, an interface and shock waves may occur due to large intraspecies and interspecies scattering lengths. To know the dynamics of an interface and assure the existence of shock waves in a double condensate, we study the incompressible and the compressible limits respectively of two coupled systems of nonlinear Schr?dinger equations. The main idea of our arguments is to define a “H-functional” like a Lyapunov functional which can control the propagation of densities and linear momenta. Such an idea is different from the one using the standard Wigner transform to investigate the incompressible and the compressible limits of a single nonlinear Schr?dinger equation.      

Magnetic, structural, and thermal properties of CoV?O?

In this paper, we investigate the electric, magnetic, structural, and thermal properties of spinel CoV(2)O(4). The temperature dependence of magnetization shows that, in addition to the paramagnetic-to-ferrimagnetic transition at T(C) = 142 K, two magnetic anomalies exist at 100 K, T(1) = 59 K. Consistent with the anomalies, the thermal conductivity presents two valleys at 100 K and T(1). At the temperature T(1), the heat capacity shows one peak, which cannot be attributed to the structural transition as revealed by the x-ray diffraction patterns for CoV(2)O(4). Below the transition temperature T(1), the ac susceptibility displays the characteristics of a glass. The series of phenomena at T(1) and the orbital state on V(3+) sites are discussed.     

Modeling and Simulation of Titania Synthesis in Two-dimensional Methane–air Flames

The formation and growth of titanium dioxide nanoparticles in two-dimensional, non-premixed methane–air flames is investigated via direct numerical simulation. The simulations are performed by capturing the spatio-temporal evolution of the fluid, chemical, and particle fields. The fluid is described by the conservation of mass, momentum, and energy equations; species transport is augmented by the effects of methane–air combustion and the oxidation of titanium tetrachloride; and a nodal approximation to the general dynamic equation is used to represent the effects of nucleation, condensation and coagulation. Simulations are performed for two initial reactant concentration levels, 20% and 30% titanium tetrachloride by mass. The evolution of the temperature, chemical and particle fields as a function of space and size are presented. Results indicate that particle formation and growth is mixing limited in this study and the mean particle diameter and geometric standard deviation increase as the concentration level of the initial reactants increases. In general, high geometric standard deviations correspond to a large particle sizes.     

Poincaré–Lelong Approach to Universality and Scaling of Correlations Between Zeros

This note is concerned with the scaling limit as N→∞ of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L ^N of any positive holomorphic line bundle L over a compact K?hler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the “correlation forms” are universal, i.e. independent of the bundle L, manifold M or point on M. Received: 17 March 1999 / Accepted: 5 August 1999