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.     

Scattering approach to the von Klitzing effect

Earlier results of the scattering theoretical approach to the quantum Hall effect are simplified and generalized. Finite size corrections to the plateau values are found to be not of order (l/L _x ) but (l ² /L _x L _y )²,l,L _x ,L _y being the magnetic length, and the sample dimensions inx andy-direction respectively. An expression for the current parallel to the electric field in terms of a scattering matrix is derived. In the weak scattering regime this expression leads to a vanishing diagonal conductivity _xx .     

Comments on the computation of Liapunov exponents for the Mixmaster universe

To explain the discrepancy between recently computed vanishing Liapunov exponents for the evolution of Mixmaster universes and the positive Liapunov exponent for the associated 1-dimensional map first discussed by Belinskii, Khalatnikov, and Lifshitz, the Liapunov exponents computed from a numerical universe evolution are compared using several time variables. Previous numerical results of vanishing Liapunov exponents were obtained with time variables which increased roughly exponentially in each epoch. Here it is found that minisuperspace proper time, which increases by a fixed amount during each epoch, yields nonvanishing Liapunov exponents within the limited number of epochs numerically accessible. The map parameteru as measured along the trajectory attains the values predicted by the map to very high accuracy (except near the maximum of expansion) even though the metric coefficients deviate in some cases from idealized Mixmaster behavior. The number of consecutive single epoch eras is shown to be related to the presence of u in an interval bounded by ratios of Fibonacci numbers.     

Casimir Effect on a Finite Lattice

Lattice quantum field theory is a well established branch of modern quantum field theory (QFT). However, it has only peripherally been used for the investigation of Casimir systems — i.e. for systems in which quantum fields are distorted by their interaction with classical background objects. This article presents a Hamiltonian lattice formulation of static Casimir systems at a level of generality appropriate for an introductory investigation. Background structure — represented by a lattice potential V(x) — is introduced along one spatial direction with translation invariance in all other spatial directions. It is simple to extend this formulation to include arbitrary background structure in more than one spatial direction. Following some general analysis two specific finite 1D lattice QFT systems are analyzed in detail. The first has three Dirichlet boundaries at the lattice sites x = 0, l and L (L > l > 0) with vanishing lattice potential V(x) everywhere in between. The vacuum energy and vacuum stress tensor T^μν for this system are calculated in 0 < x < L. Very careful attention must be and is given to renormalization in the (continuum) limit of vanishing lattice constant. Globally and locally this lattice system is seen to closely mimic the corresponding 1D continuum system — as one would hope. Then we introduce a lattice potential V(x) = c/(x — x₀)² centered at x = x₀ < 0 to the left of the boundary at x = 0 and extending through this boundary and the middle Dirichlet boundary at x = l out to the right‐hand boundary x = L > l and beyond. The vacuum energy and T^μν are calculated for this far more complicated system in the region 0 〈 x < L, again with very good results. The internal consistency of the lattice version of this system is carefully examined. Our conclusion is that finite‐lattice formulation provides a powerful and effective tool, capable of solving completely many Casimir systems which could not possibly be handled using continuum methods. This is precisely our reason for introducing it. Future investigations (in one and more dimensions and in dynamical as well as static contexts) will display more fully the power of this method.     

Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips

We present exact calculations of the zero-temperature partition function (chromatic polynomial) P for the q-state Potts antiferromagnet on triangular lattice strips of arbitrarily great length L_x vertices and of width L_y vertices and, in the L_x→∞ limit, the exponent of the ground state entropy, W=e^S₀/k_B. The strips considered, with their boundary conditions (BC), are (a) (FBC_y, PBC_x) = cyclic for L_y=3, 4, (b) (FBC_y, TPBC_x) = Möbius, L_y=3, (c) (PBC_y, PBC_x) = toroidal, L_y=3, (d) (PBC_y, TPBC_x) = Klein bottle, L_y=3, (e) (PBC_y, FBC_x) = cylindrical, L_y=5, 6, and (f) (FBC_y, FBC_x) = free, L_y=5, where F, P, and TP denote free, periodic, and twisted periodic. Several interesting features are found, including the presence of terms in P proportional to cos(2πL_x/3) for case (c). The continuous locus of points where W is nonanalytic in the q plane is discussed for each case and a comparative discussion is given of the respective loci for families with different boundary conditions. Numerical values of W are given for infinite-length strips of various widths and are shown to approach values for the 2D lattice rapidly. A remark is also made concerning a zero-free region for chromatic zeros. Some results are given for strips of other lattices.     

Statistical mechanics of polymer networks of any topology

The statistical mechanics is considered of any polymer network with a prescribed topology, in dimensiond, which was introduced previously. The basic direct renormalization theory of the associated continuum model is established. It has a very simple multiplicative structure in terms of the partition functions of the star polymers constituting the vertices of the network. A calculation is made toO(²), whered=4–, of the basic critical dimensions _L associated with anyL-leg vertex (L1). From this infinite series of critical exponents, any topology-dependent critical exponent can be derived. This is applied to the configuration exponent _G of any networkG toO(²), includingL-leg star polymers. The infinite sets of contact critical exponents between multiple points of polymers or between the cores of several star polymers are also deduced. As a particular case, the three exponents ₀, ₁, ₂ calculated by des Cloizeaux by field-theoretic methods are recovered. The limiting exact logarithmic laws are derived at the upper critical dimensiond=4. The results are generalized to the series of topological exponents of polymer networks near a surface and of tricritical polymers at the-point. Intersection properties of networks of random walks can be studied similarly. The above factorization theory of the partition function of any polymer network over its constitutingL-vertices also applies to two dimensions, where it can be related to conformal invariance. The basic critical exponents _L and thus any topological polymer exponents are then exactly known. Principal results published elsewhere are recalled.     

Dynamics of the formation of two-dimensional ordered structures

The growth of ordered domains in lattice gas models, which occurs after the system is quenched from infinite temperature to a state below the critical temperatureT _c, is studied by Monte Carlo simulation. For a square lattice with repulsion between nearest and next-nearest neighbors, which in equilibrium exhibits fourfold degenerate (2×1) superstructures, the time-dependent energy E(t), domain size L(t), and structure functionS(q, t) are obtained, both for Glauber dynamics (no conservation law) and the case with conserved density (Kawasaki dynamics). At late times the energy excess and halfwidth of the structure factor decrease proportional tot ^–x, whileL(t) t ^x, where the exponent x=1/2 for Glauber dynamics and x1/3 for Kawasaki dynamics. In addition, the structure factor satisfies a scaling lawS(k,t)=t ^2xS(kt^x). The smaller exponent for the conserved density case is traced back to the excess density contained in the walls between ordered domains which must be redistributed during growth. Quenches toT>T _c, T=T_c (where we estimate dynamic critical exponents) andT=0 are also considered. In the latter case, the system becomes frozen in a glasslike domain pattern far from equilibrium when using Kawasaki dynamics. The generalization of our results to other lattices and structures also is briefly discussed.     

Raman studies and optical properties of some (PbO)x(Bi2O3)0.2(B2O3)0.8−x glasses

Five (PbO)_x(Bi₂O₃)_0.2(B₂O₃)_0.8−x glasses, where x = 0, 0.2, 0.3, 0.4 and 0.6, were prepared. The dilatometric glass transition temperature (T_g) was found in the region 470 (x = 0)≥ T_g ( °C) ≥ 347 (x = 0.6), and the density (ρ) varied within 4.57 (x = 0) ≤ ρ (g/cm³) ≤ 8.31 (x = 0.6). Raman spectra indicated the conversion of BO₃ to BO₄ entities for low x values but for x > 0.3, namely, for x → 0.6, back‐conversion occurred, most probably. From the measurements of the optical transmission on very thin bulk samples, the room temperature optical gap values (E_g) were determined to be in the range 4.03 (x = 0)≥ E_g (eV) ≥ 3.08 (x = 0.6). The temperature (T) dependence of the optical gap (E_g(T)) in the region 300 ≤ T(K) ≤ 600 was examined and approximated by a linear relationship of the form of E_g(T) = E_g(0)− γT, where γ × 10⁻⁴(eV/K) varied from 5.1 to 6.8. The non‐linear refractive index (n₂) was estimated from the optical gap values and it was found to correspond to the n₂ values calculated from the experimental third‐order non‐linear optical susceptibility taken from the literature. Copyright © 2008 John Wiley & Sons, Ltd.     

An analytical form for the Raman shift dependence on size of nanocrystals

An analytical form of the Raman shift dependence on the size of nanocrystals is presented. On the basis of the hard confinement model, this form describes the deviations from Raman shifts in infinite crystals as Δω = π²A[1 − exp(−η)]/12x²η², where η = L/12ax and x = (A/Γ₀)^1/2, L standing for the crystal size, Γ₀ for the intrinsic band linewidth, a for the lattice parameter and A for a suitable phonon curve parameter. It works in those cases where the average phonon curve shows a quadratic dependence on the phonon quasi‐momentum in the range of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

Properties of the skeleton of aggregates grown on a Cayley tree

We discuss and analyze a family of trees grown on a Cayley tree, that allows for a variable exponent in the expression for the mass as a function of chemical distance, M(l)l ^dl . For the suggested model, the corresponding exponent for the mass of the skeleton,d _l ^s , can be expressed in terms ofd _l asd _l ^s = 1,d _l d _l ^c = 2;d _l ^s = d _l –1,d ₁ d _l ^c = 2, which implies that the tree is finitely ramified ford _l 2 and infinitely ramified whend _l 2. Our results are derived using a recursion relation that takes advantage of the one-dimensional nature of the problem. We also present results for the diffusion exponents and probability of return to the origin of a random walk on these trees.     

Chain length probability distribution — equivalence of ASYNNNI and 1d Ising model

An expression for the chain length probability distribution p(l) of a one dimensional Ising chain was derived using the cluster variation method formalism, the p(l) being expressed through the pair cluster probabilities. It was shown numerically that the same expression also applies in the case of one dimensional chains formed along one of the next-nearest neighbor interactions included in the two dimensional ASYNNNI (Asymmetric Next-Nearest Neighbor Ising) model, widely used to describe the statistics of oxygen ordering in the basal CuO _x planes of the YBa₂Cu₃O_6+x type high-T _c superconducting materials. Equivalency between ASYNNNI and 1d Ising model is discussed.      

Universal critical properties of the Eulerian bond-cubic model

We investigate the Eulerian bond-cubic model on the square lattice by means of Monte Carlo simulations,using an efficient cluster algorithm and a finite-size scaling analysis.The critical points and four critical exponents of the model are determined for several values of n.Two of the exponents are fractal dimensions,which are obtained numerically for the first time.Our results are consistent with the Coulomb gas predictions for the critical O(n) branch for n < 2 and the results obtained by previous transfer matrix calculations.For n=2,we find that the thermal exponent,the magnetic exponent and the fractal dimension of the largest critical Eulerian bond component are different from those of the critical O(2) loop model.These results confirm that the cubic anisotropy is marginal at n=2 but irrelevant for n < 2.     

Asymptotic and effective coarsening exponents in surface growth models

We consider a class of unstable surface growth models, ?_t z = -?_x J\partial_t z = -\partial_x {\cal J} , developing a mound structure of size λ and displaying a perpetual coarsening process, i.e. an endless increase in time of λ. The coarsening exponents n, defined by the growth law of the mound size λ with time, λ∼tⁿ, were previously found by numerical integration of the growth equations [A. Torcini, P. Politi, Eur. Phys. J. B 25, 519 (2002)]. Recent analytical work now allows to interpret such findings as finite time effective exponents. The asymptotic exponents are shown to appear at so large time that cannot be reached by direct integration of the growth equations. The reason for the appearance of effective exponents is clearly identified.     

First-Passage and Extreme-Value Statistics of a Particle Subject to a Constant Force Plus a Random Force

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point x ₁ of a particle which starts at x ₀ with velocity v ₀. The probability that the particle has not yet arrived at x ₁ after a time t, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement m=max  _t [x(t)].     

Intrusion of fluids into nanogrooves

We study the shape of gas-liquid interfaces forming inside rectangular nanogrooves (i.e., slit-pores capped on one end). On account of purely repulsive fluid-substrate interactions the confining walls are dry (i.e., wet by vapor) and a liquid-vapor interface intrudes into the nanogrooves to a distance determined by the pressure (i.e., chemical potential). By means of Monte Carlo simulations in the grand-canonical ensemble (GCEMC) we obtain the density ρ(z) along the midline (x = 0 of the nanogroove for various geometries (i.e., depths D and widths L of the nanogroove. We analyze the density profiles with the aid of an analytic expression which we obtain through a transfer-matrix treatment of a one-dimensional effective interface Hamiltonian. Besides geometrical parameters such as D and L , the resulting analytic expression depends on temperature T , densities of coexisting gas and liquid phases in the bulk ρ^g,l _x and the interfacial tension γ . The latter three quantities are determined in independent molecular dynamics simulations of planar gas-liquid interfaces. Our results indicate that the analytic formula provides an excellent representation of ρ(z) as long as L is sufficiently small. At larger L the meniscus of the intruding liquid flattens. Under these conditions the transfer-matrix analysis is no longer adequate and the agreement between GCEMC data and the analytic treatment is less satisfactory.     

2D-to-3D percolation crossover of metal-insulator composites

Percolation properties and d.c. conductivity were determined for an L²×h-random resistor network model of metal-insulator composite films. The effects of the thickness h on the percolation threshold and conductivity were studied numerically in the limit of an infinite size of the L²-plane parallel to the film. For thicknesses ranging from h/L=0.01 to h/L=0.24, a crossover between a finite-size regime and a saturation regime was observed at h/L≈0.1. In the finite-size regime (h/L?0.01), the percolation threshold scales as pc(h)−pc3∝h−1/x, the exponent x being compatible with that of the critical exponent of the 3D correlation length, ν₃. The conductivity exponent t appeared to depend linearly on the ratio h/L with a slope νD compatible with 2+ν₂, where ν₂ is the 2D correlation length exponent. In the saturation regime, a scaling correction for the percolation threshold was found with an exponent 1+1/ν₃. In this regime we also observed a logarithmic dependence of the conductivity exponent on h/L.     

Monte Carlo simulation of very large kinetic Ising models

We study the relaxation of Ising models in three and four dimensions aboveT _c , using multi-spin coding for lattices up to 360³ and 40⁴. The nonlinear relaxation time diverges as (T–T _c )^{–1.05±0.04} in three dimensions, where corrections to scaling are taken into account. In four dimensions the effective exponent is about 0.72; logarithmic correction factors make the analysis difficult here. The linear relaxation time for the asymptotic exponential decay is found to be larger, with effective exponents 1.31 (d=2) and 0.97 (d=4). The difference in the linear and nonlinear relaxation exponents is compatible in three dimensions with the orderparameter exponent , as predicted by Racz.Work supported by SFB 125 Aachen-Jülich-KölnWork started at Department de Physique des Systemes Desordonnes, Universite de Provence, Centre St-Jerome, F-13397 Marseille Cedex 13, France     

Self-avoiding walks on randomly diluted lattices

We discuss how the introduction of quenched impurities changes the exponents of a self-avoiding walk on a lattice. We find that , the exponent for the number of walks, does not change. On the other hand the exponent for the mean square end to end distance does change. This is caused by a singular normalization atp=p _c , which is necessary to compensate for the allowed number of walks on the diluted lattice.     

Statistics of photocounts of blinking fluorescence of quantum dots

The blinking of quantum dots under the action of laser radiation is described based on a model of a binary (two-state) renewal process with on (fluorescent) and off (non fluorescent) states. The T _on and T _off sojourn times in the on and off states are random and power-law distributed with exponents 0 < α < 1 and 0 < β < 1; the averages of the on and off times are infinite. As a consequence of this, the Gaussian statistics is inapplicable and the process is described using a more general statistics. An equation for the density of distribution p(t _on|t) of the total on time during the observation time t is derived that contains derivatives of fractional orders α and β. A solution to this equation is found in terms of fractional stable distributions. The Poisson transform of the density p(t _on|t) leads to the photon counting distribution and determines the fluorescence statistics. It is demonstrated that, if a blinking process with exponents α < β is implemented, then, at fairly long times, the on time will considerably prevail over the off time, i.e., blinking will be suppressed. This behavior is evidenced by the types of distributions of the total fluorescence time, the decay of relative fluctuations, and the Monte Carlo simulated trajectories of the process.