Critical Behavior of Two Interacting Linear Polymer Chains in a Good Solvent

A model of two interacting (chemically different) linear polymer chains is solved exactly using the real-space renormalization group transformation on a family of Sierpinski gasket type fractals and on a truncated 4-simplex lattice. The members of the family of the Sierpinski gasket-type fractals are characterized by an integer scale factorb which runs from 2 to ∞. The Hausdorff dimensiond _F of these fractals tends to 2 from below asb → ∞. We calculate the contact exponenty for the transition from the State of segregation to a State in which the two chains are entangled forb = 2-5. Using arguments based on the finite-size scaling theory, we show that forb→∞, y = 2 - v(b) d _F, wherev is the end-toend distance exponent of a chain. For a truncated 4-simplex lattice it is shown that the system of two chains either remains in a State in which these chains are intermingled in such a way that they cannot be told apart, in the sense that the chemical difference between the polymer chains completely drop out of the thermodynamics of the system, or in a State in which they are either zipped or entangled. We show the region of existence of these different phases separated by tricritical lines. The value of the contact exponenty is calculated at the tricritical points.     

Block persistence

We define a block persistence probability p _l (t) as the probability that the order parameter integrated on a block of linear size l has never changed sign since the initial time in a phase-ordering process at finite temperature T<T _c . We argue that in the scaling limit of large blocks, where z is the growth exponent (), is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent for large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large-n model, and generically it can be used to determine easily from simulations of coarsening models. We also argue that and the scaling function do not depend on temperature, leading to a definition of at finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. These ideas are applied to the study of persistence for conserved models. We illustrate our discussions by extensive numerical results. We also comment on the relation between this method and an alternative definition of at finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)]. Received: 25 February 1998 / Revised: 24 July 1998 / Accepted: 27 July 1998     

Simplex equations and their solutions

We investigate ann-simplex generalization of the classical and quantum Yang-Baxter equation. For the case ofsl(2) we find the most general solution of the classicaln-simplex equation for alln. These classical solutions can be quantized (in the sense of quantum group theory) forn=2,3 and we exhibit a quantum solution to the tetrahedron equations (n=3). The classical nondegenerate solutions cannot be quantized forn=4.     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

A Monte-Carlo study of meanders

We study the statistics of meanders, i.e. configurations of a road crossing a river through n bridges, and possibly winding around the source, as a toy model for compact folding of polymers. We introduce a Monte-Carlo method which allows us to simulate large meanders up to n=400. By performing large n extrapolations, we give asymptotic estimates of the connectivity per bridge R=3.5018(3), the configuration exponent , the winding exponent and other quantities describing the shape of meanders. Received 21 June 1999     

The scaling behavior of the molecular parameters of the hyperbranched polymers made by self-condensing vinyl polymerization

The hyperbranched polymers can be made by self-condensing vinyl polymerization without gelation transition. The average molecular weights, as well as the average sizes, can reach infinite values as the reaction is quantitatively completed, and the scaling forms of the molecular parameters should exist. In the paper, based on a recursion formula, the scaling form of the number fraction distribution and the number of the n-mers are given analytically as the conversion of double bonds is near 1. The mean square radius of gyration for very large hyperbranched polymers is calculated explicitly to give a scaling exponent. Finally, a scaling relation associated with the fractal dimension and the polydispersity exponent is given clearly.     

Trap-size scaling of finite Bose systems within an exact canonical ensemble

Based on an exact canonical partition function, we investigate the trap-size scaling for ideal Bose gases with a finite number of particles N confined in a cubic box or in a harmonic trap. We study the trap-size scaling behaviors of condensate fraction 〈n₀〉/N and specific heat C_N around the transition temperature T_c (i.e., t = T/T_c − 1 → 0) for the three different traps, where a trap exponent θ in dependence of the trapping potential and the universality class of transition are introduced. In the box trap with periodic and Dirichlet boundary conditions, where θ → 1, we find that the scaling functions governing the various critical behaviors are universal but respective of the boundary conditions. The calculated critical exponents are in nice agreement with analytical scaling predictions. The borders of universality validity are obtained numerically. In the case of the harmonic trap, the critical behavior of the system is also found to be universal, and the trap exponent is obtained as θ ? 0.068.     

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     

f-α Spectrum of circle map

We present an analytic perturbative method for calculatingf(α) and the generalized dimensionD _q of the critical invariant circle of the polynomial circle map. The scaling behaviour is found to depend onz, the exponent defining the map. The asymptotic bounds of the scaling constantsα(z) andδ(z) are verified analytically.     

A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms

A mean-field approach (MFA) is proposed for the analysis of orientational order in a two-dimensional system of stochastic self-propelled particles interacting by local velocity alignment mechanism. The treatment is applied to the cases of ferromagnetic (F) and liquid-crystal (LC) alignment. In both cases, MFA yields a second order phase transition for a critical noise strength and a scaling exponent of 1/2 for the respective order parameters. We find that the critical noise amplitude η_c at which orientational order emerges in the LC case is smaller than in the F-alignment case, i.e. η^LC_C<η^F_C. A comparison with simulations of individual-based models with F- resp. LC-alignment shows that the predictions about the critical behavior and the qualitative relation between the respective critical noise amplitudes are correct.     

Using the phenomenon of liquid superheat to measure critical properties of substances

A method to measure the critical temperature and critical pressure of substances, in particular, thermally unstable ones, is briefly described. The method is used to measure the temperature of attainable superheat of liquids with the help of a wire probe heated by electric current pulses. As the pressure increases, the temperature of attainable superheat tends to a critical temperature. The duration of the heating pulses is from 0.03 to 1 ms. A list of about 130 substances for which measurements of the critical properties were made is presented. The results of these measurements confirmed the scaling form of the relation between the critical constants of substances consisting of long chain molecules and the number of molecular units. Two methods to extrapolate the experimental data for the critical properties of the initial members of homologous series to heavier polymer homologs are proposed. One of the methods is based on the equation of state for the fluid of chain molecules. In this method, the extrapolating equations are power series in n, where n is the number of main units in a chain molecule. In the other method, the hypothesis of functional self-similarity and the presentation of scaling behavior of the critical constants of long chain molecules are used. Homologous series with the general formula R₁ (CH₂)_nR₂, where R₁ and R₂ are different end groups, are considered. We obtain equations to calculate, with good accuracy, the critical temperature and critical pressure of any member of any homologous series with the molecule structure R₁(CH₂)_nR₂ if they are known for one compound belonging to this series.     

Dilute gas viscosity of n-alkanes represented by rigid Lennard-Jones chains

ABSTRACT

The shear viscosity in the dilute gas limit has been calculated by means of the classical trajectory method for a gas consisting of chain-like molecules. The molecules were modelled as rigid chains made up of spherical segments that interact through a combination of site–site Lennard-Jones 12-6 potentials. Results are reported for chains consisting of 2, 3, 4, 6, 8, 12 and 16 segments in the reduced temperature range of 0.3–50 for site–site separations of 0.25σ, 0.333σ, 0.40σ, 0.60σ and 0.80σ, where σ is the Lennard-Jones length scaling parameter. The results were used to determine the shear viscosity of n-alkanes in the zero-density limit by representing an n-alkane molecule as a rigid linear chain consisting of n_c ? 1?spherical segments, where n_c?is the number of carbon atoms. We show that for a given n-alkane molecule, the scaling parameters ? and σ are not unique and not transferable from one molecule to another. The commonly used site–site Lennard-Jones 12-6 potential in combination with a rigid-chain molecular representation can only accurately mimic the viscosity if the scaling parameters are fitted. If the scaling parameters are estimated from the scaling parameters of other n-alkanes, the predicted viscosity values have an unacceptably high uncertainty.     

About a possible constancy of the total cross-section of the e+e− annihilation into hadrons

We show how a series of vector mesons can give a constant total cross-section in agreement with the experimental results up to 5 GeV. We discuss also the ratios n_π^°/n_π±,n_K/n_π and the breakdown of scaling in the inclusive spectra.     

二维完全阻挫$lt;i$gt;XY$lt;/i$gt;模型的动力学指数

采用大规模动力学蒙特卡罗模拟方法,对二维完全阻挫XY模型的Kosterlitz-Thouless（KT）型相变展开数值研究.系统从有序初始态出发演化到高于KT相变的温度,以普适的动力学标度形式为基础,通过测量磁化和Binder累积量,得出动力学关联时间和平衡态空间关联长度,确定出更精确的动力学指数z.特别是建议并证实了一种在KT相变温度以上（T>T_KT）,独立判断动力学指数z的方法.模拟结果表明,动力学指数z≈2,这与在相变温度以下（T<T_KT）测量的结果一致. 关键词： 蒙特卡罗法 动力学指数 Kosterlitz-Thouless相变 XY模型')" href="#">二维完全阻挫XY模型     

Equation of state for the Universe from similarity symmetries

In this paper we proposed to use the group of analysis of symmetries of the dynamical system to describe the evolution of the Universe. This method is used in searching for the unknown equation of state. It is shown that group of symmetries enforce the form of the equation of state for noninteracting scaling multifluids. We showed that symmetries give rise to the equation of state in the form p =-Λ + w ₁ρ(a) + w ₂ a ^β + 0 and energy density ρ = Λ+ρ₀₁ a ^-3(1+w) +ρ₀₂ a ^α +ρ₀₃ a ^-3, which is commonly used in cosmology. The FRW model filled with scaling fluid (called homological) is confronted with the observations of distant type Ia supernovae. We found the class of model parameters admissible by the statistical analysis of SNIa data.We showed that the model with scaling fluid fits well to supernovae data. We found that Ω_m,0 ≃ 0.4 and n ≃ -1 (β = -3n), which can correspond to (hyper) phantom fluid, and to a high density universe. However if we assume prior that Ω_m,0 = 0.3 then the favoured model is close to concordance ΛCDM model. Our results predict that in the considered model with scaling fluids distant type Ia supernovae should be brighter than in the ΛCDM model, while intermediate distant SNIa should be fainter than in the ΛCDM model. We also investigate whether the model with scaling fluid is actually preferred by data over ΛCDM model. As a result we find from the Akaike model selection criterion: it prefers the model with noninteracting scaling fluid.     

Scaling properties of dynamic response in orientational glasses

The freezing transition in dipolar and quadrupolar glasses is characterized by the presence of local random electric and strain fields generated by substitutional disorder. The dynamic response in the ergodic phase above the freezing temperature T_F is studied in terms of Langevin dynamics applied to the recently formulated symmetry-adapted random-bond-random-field (SARBRF) model of orientational glasses. Following the theory of spin glasses it is assumed that for T≥T_F the response can be written in a dynamic scaling form by introducing a scaling exponent v and a frequency scaling variable. The value of v(T) is explicitly evaluated for the quadrupolar (100) SARBRF model, and its relation to the experimentally observed effective exponent u_eFF(T) in dipolar and quadrupolar glasses is discussed.     

Scaling behavior of earthquakes’ inter-events time series

In this paper, we investigate the statistical and scaling properties of the California earthquakes’ inter-events over a period of the recent 40 years. To detect long-term correlations behavior, we apply detrended fluctuation analysis (DFA), which can systematically detect and overcome nonstationarities in the data set at all time scales. We calculate for various earthquakes with magnitudes larger than a given M. The results indicate that the Hurst exponent decreases with increasing M; characterized by a Hurst exponent, which is given by, H = 0:34 + 1:53/M, indicating that for events with very large magnitudes M, the Hurst exponent decreases to 0:50, which is for independent events.      

Semiclassical laser theory in the stochastic and thermodynamic frameworks

The thermodynamic properties of the laser distribution in the steadily oscillating state are investigated to determine the minimum characteristic of the entropy production. First, the laser Langevin equation for five random variables is treated in the light of the stochastic calculus to deduce the photon-number rate equationn = – C+(n – n_c) + [A/(1 + sn)](n–n_A), where n_n and n₄ are the two constants of the fluctuation attributed to the noise forces subject to the usual fluctuation-dissipation theorem, withn ₄ < 0 for the inverted atomic population. We then combine the dynamics of the lasing mode with a model open system of the Lebowitz type with two reservoirs for which the entropy production(p) is expressed and made subject to a variational principle: The modified variation scheme, the same as Prigogine's local potential method, is shown to give the exact lasing distributionp as the optimum between two distributions of thermal type with temperatures far from each other.     

Flux noise near the Berezinski\overset{\lower0.5em\hbox{-Kosterlitz-Thouless transition

The flux noise in Josephson junction arrays is studied in the critical regime above the Berezinskii-Kosterlitz-Thouless transition. In proximity-coupled arrays a local Ohmic damping for the phases is relevant, giving rise to anomalous vortex diffusion and a dynamic scaling of the flux noise in the critical region. The flux noise exhibits a crossover from white to 1/f noise at a frequency ω_ξ∝ξ^?z with a dynamic exponent z=2.