Kinetic-growth self-avoiding walks on small-world networks

Kinetically-grown self-avoiding walks have been studied on Watts-Strogatz small-world networks, rewired from a two-dimensional square lattice. The maximum length L of this kind of walks is limited in regular lattices by an attrition effect, which gives finite values for its mean value 〈L 〉. For random networks, this mean attrition length 〈L 〉 scales as a power of the network size, and diverges in the thermodynamic limit (system size N ↦∞). For small-world networks, we find a behavior that interpolates between those corresponding to regular lattices and randon networks, for rewiring probability p ranging from 0 to 1. For p < 1, the mean self-intersection and attrition length of kinetically-grown walks are finite. For p = 1, 〈L 〉 grows with system size as N^1/2, diverging in the thermodynamic limit. In this limit and close to p = 1, the mean attrition length diverges as (1-p)^-4. Results of approximate probabilistic calculations agree well with those derived from numerical simulations.     

Intrinsic degree-correlations in the static model of scale-free networks

We calculate the mean neighboring degree function and the mean clustering function C(k) of vertices with degree k as a function of k in finite scale-free random networks through the static model. While both are independent of k when the degree exponent γ≥3, they show the crossover behavior for 2 < γ< 3 from k-independent behavior for small k to k-dependent behavior for large k. The k-dependent behavior is analytically derived. Such a behavior arises from the prevention of self-loops and multiple edges between each pair of vertices. The analytic results are confirmed by numerical simulations. We also compare our results with those obtained from a growing network model, finding that they behave differently from each other.     

Lattice random walks for sets of random walkers. First passage times

We have studied the mean first passage time for the first of aset of random walkers to reach a given lattice point on infinite lattices ofD dimensions. In contrast to the well-known result ofinfinite mean first passage times for one random walker in all dimensionsD, we findfinite mean first passage times for certain well-specified sets of random walkers in all dimensions, exceptD = 2. The number of walkers required to achieve a finite mean time for the first walker to reach the given lattice point is a function of the lattice dimensionD. ForD > 4, we find that only one random walker is required to yield a finite first passage time, provided that this random walker reaches the given lattice point with unit probability. We have thus found a simple random walk property which sticks atD > 4.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE78-21460.     

Reflection principles for biased random walks and application to escape time distributions

We present a reflection principle for an arbitrarybiased continuous time random walk (comprising both Markovian and non-Markovian processes) in the presence of areflecting barrier on semi-infinite and finite chains. For biased walks in the presence of a reflecting barrier this principle (which cannot be derived from combinatorics) is completely different from its familiar form in the presence of an absorbing barrier. The result enables us to obtain closed-form solutions for the Laplace transform of the conditional probability for biased walks on finite chains for all three combinations of absorbing and reflecting barriers at the two ends. An important application of these solutions is the calculation of various first-passage-time and escape-time distributions. We obtain exact results for the characteristic functions of various kinds of escape time distributions for biased random walks on finite chains. For processes governed by a long-tailed event-time distribution we show that the mean time of escape from bounded regions diverges even in the presence of a bias—suggesting, in a sense, the absence of true long-range diffusion in such frozen processes.     

Shape of the clusters of a percolative system in the presence of tunneling bonds and a finite electric field

We consider a model correlated percolative system on a 2D square lattice with a finite electric field applied accross its two opposite sides. We study the shape of the clusters formed with the addition of a new kind of bond (we call them tunneling bonds) which respond only above a finite threshold voltage. As expected, the clusters do have an overall elongated shape in the-direction of the applied field. Intuitively, one expects the elongation (with the aspect ratio >1) to increased indefinitely with the field. But, in a previous study, we found the model to belong to the same universality clas in the limits of a zero and an infinite electric field. We explain this behavior by studying the change in these elongated shapes as a function of the applied voltage in finite size samples and find that actually the amount of elongation takes on a maximum value at a size (L)-dependent finite voltageV _m (L) and that asV thje overall deviation from isotropy in the field direction tends to zero (i.e., aspect ratio1) again.     

Rigorous analysis of singularities and absence of analytic continuation at first-order phase-transition points in lattice-spin models

We report about two new rigorous results on the nonanalytic properties of thermodynamic potentials at first-order phase transition. For lattice models (d>or=2) with arbitrary finite state space, finite-range interactions which have two ground states, we prove that the pressure has no analytic continuation at the first-order phase-transition point, under the only further assumptions that the Peierls condition is satisfied for the ground states and that the temperature is sufficiently low. For Ising models with Kac potentials J(gamma)(x)=gamma(d)phi(gammax), where 00) and analyticity in the mean field limit (gamma SE pointing arrow 0).     

First passage time densities for random walk spans

A general expression is derived for the Laplace transform of the probability density of the first passage time for the span of a symmetric continuous-time random walk to reach levelS. We show that when the mean time between steps is finite, the mean first passage time toS is proportional toS ₂. When the pausing time density is asymptotic to a stable density we show that the first passage density is also asymptotically stable. Finally when the jump distribution of the random walk has the asymptotic formp(j)A/|j| ₊₁, 0 < < 2 it is shown that the mean first passage time toS goes likeS .     

Tail states in clean superconductors with magnetic impurities

We analyze the behavior of the density of states in a singlet s-wave superconductor with weak magnetic impurities in the clean limit. By using the method of optimal fluctuation and treating the order parameter self-consistently we show that the density of states is finite everywhere in the superconducting gap, and that it varies as ln(N(E) proportional to -/E-Delta(0)/((7-d)/4) near the mean field gap edge Delta(0) in a d-dimensional superconductor. In contrast to most studied cases the optimal fluctuation is strongly anisotropic.     

Critical phenomena and the quantum critical point of ferromagnetic Zr(1-x)NbxZn2

We present a study of the magnetic properties of Zr(1-x)NbxZn2, using an Arrott plot analysis of the magnetization. The Curie temperature Tc is suppressed to zero temperature for Nb concentration xc = 0.083+/-0.002, while the spontaneous moment vanishes linearly with Tc as predicted by the Stoner theory. The initial susceptibility chi displays critical behavior for x or= xc we find that chi(-1) = chi0(-1) + aT(4/3), where chi0(-1) vanishes as x-->xc. The resulting magnetic phase diagram shows that the quantum critical behavior extends over the widest range of temperatures for x=xc, and demonstrates how a finite transition temperature ferromagnet is transformed into a paramagnet, via a quantum critical point.     

Semiflexible polymer confined to a spherical surface

We develop a formalism for describing the kinematics of a wormlike chain confined to the surface of a sphere that simultaneously satisfies the spherical confinement and the inextensibility of the chain contour. We use this formalism to study the statistical behavior of the wormlike chain on a spherical surface. In particular, we provide an exact, closed-form expression for the mean square end-to-end distance that is valid for any value of chain length L, persistence length l(p), and sphere radius R. We predict two qualitatively different behaviors for a long polymer depending on the ratio R/l(p). For R/l(p)>4, the mean square end-to-end distance increases monotonically with the chain length, whereas for R/l(p)<4, a damped oscillatory behavior is predicted.     

The fully finite spherical model

A lattice sum technique is applied to the constraint equation of the finite size mean spherical model. It is shown that this allows the investigation of the model over a wide range of temperatures, for a wide range of system sizes. Correlation lengths and susceptibilities are shown to obey crossover scaling aroundT=0 below the lower critical dimension, and finite size scaling between the lower and upper critical dimensions. Universal scaling forms are suggested for the lower critical dimension. At and above the upper critical dimension, the behavior is identical to that of finite sized mean field theory. The scaling at and above the upper critical dimension is shown to be modified by the existence of a dangerous irrelevant variable which also governs the failure of hyperscaling. Implications for phenomenological renormalization experiments are discussed. Numerical results of scaling are displayed.     

The Stochastic Limit of the Fröhlich Hamiltonian: Relations with the Quantum Hall Effect

We propose a model of an approximatively two-dimensional electron gas in a uniform electric and magnetic field and interacting with a positive background through the Fröhlich Hamiltonian. We consider the stochastic limit of this model and we find the quantum Langevin equation and the generator of the master equation. This allows us to calculate the explicit form of the conductivity and the resistivity tensors and to deduce a fine tuning condition (FTC) between the electric and the magnetic fields. This condition shows that the x-component of the current is zero unless a certain quotient, involving the physical parameters, takes values in a finite set of physically meaningful rational numbers. We argue that this behavior is quite similar to that observed in the quantum Hall effect. We also show that, under some conditions on the form factors entering in the definition of the model, also the plateaux and the almost linear behavior of the Hall resistivity can be recovered. Our FTC does not distinguish between fractional and integer values.     

Nonlinear reaction-diffusion: a microscopic approach

A microscopic theory for reaction-difusion (R-D) processes is developed from Einstein’s master equation including a reactive term. The mean field formulation leads to a generalized R-D equation for the n-th order annihilation reaction A + A + A + ... + A → 0, and the steady state solutions exhibit long range power law behavior showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely short range concentration distribution with finite support describing situations where diffusion is slow and extinction is fast. We apply the theory to analyze experimental data for morphogen gradient formation in the wing disc of the Drosophila embryo.     

Pattern Selection: Determined by Symmetry and Modifiable by Distant Effects

We consider Saffman–Taylor channel flow without surface tension on a high-pressure driven interface, but modify the usual infinite-fluid in infinite-channel configuration. Here we include the treatment of efflux by considering a finite connected body of fluid in an arbitrarily long channel, with its second free interface the efflux of this configuration. We show that there is a uniquely determined translating solution for the driven interface, which is exactly the 1/2 width S–T solution, following from correct symmetry for a finite channel flow. We establish that there exist no perturbations about this solution corresponding to a finger propagating with any other width: Selection is locally unique and isolated. The stability of this solution is anomalous, in that all freely impressible perturbations are stabilities, while unstable modes request power proportional to their strength from the external agencies that drive the flow, and so, in principle, are experimentally controllable. This is very different from the behavior of the usual infinite fluid. We conjecture that surface tension on the efflux interface modifies channel-width according to 1–2=/v (i.e., (2)² B of the literature) with v the velocity of the high-pressure tip, but the surface tension of the efflux. That is, is decreased below 1/2 by the effect of smoothing the distant efflux. The perturbation theory created here to deal with transport between two free boundaries is novel and dependent upon a symmetry implied by the equations of motion.     

Effects of an oscillating field on a diffusion process in the presence of a trap

Consider a diffusion process on an infinite line terminated by a trap and modulated by a periodic field. When the frequency is equal to zero the mean time to trapping will be finite or infinite, depending on the sign of the field. We ask whether this behavior can be changed by an oscillatory field, and show that it cannot for pure Brownian motion. We suggest that transition can appear when the signal propagation velocity is finite as for the telegrapher's equation. We further suggest that the asymptotic time dependence of the survival probability is proportional tot ^–1/2 just as in the case of ordinary diffusion. The same conclusion is shown to hold for a system whose dynamics is governed by the equation , whereL is a constant.     

Single-file diffusion in a box

We study diffusion of (fluorescently) tagged hard-core interacting particles of finite size in a finite one-dimensional system. We find an exact analytical expression for the tagged particle probability density function using a Bethe ansatz, from which the mean square displacement is calculated. The analysis shows the existence of three regimes of drastically different behavior for short, intermediate, and large times. The results are in excellent agreement with stochastic simulations (Gillespie algorithm).     

Interpolating gauges and the importance of a careful treatment of ε-term

We consider the use of interpolating gauges (with a gauge function F[A;]) in gauge theories to connect the results in a set of different gauges in the path-integral formulation. We point out that the results for physical observables are very sensitive to the epsilon term that we have to add to deal with singularities and thus it cannot be left out of a discussion of gauge-independence generally. We further point out, with reasons, that the fact that we can ignore this term in the discussion of gauge independence while varying of the gauge parameter in Lorentz-type covariant gauges is an exception rather than a rule. We show that generally preserving gauge-independence as is varied requires that the -term has to be varied with . We further show that if we make a naive use of the (fixed) epsilon term (that is appropriate for the Feynman gauge) for general interpolating gauges with arbitrary parameter values [i.e. ], we cannot preserve gauge independence [except when we happen to be in the infinitesimal neighborhood of the Lorentz-type gauges]. We show with an explicit example that for such a naive use of an -term, we develop serious pathological behavior in the path-integral as is/are varied. We point out that correct way to fix the -term in a path-integral in a non-Lorentz gauge is by connecting the path-integral to the Lorentz-gauge path-integral with correct -term as has been done using the finite field-dependent BRS transformations in recent years.     

Reprint of : Dynamics of coupled vibration modes in a quantum non-linear mechanical resonator

We investigate the behaviour of two non-linearly coupled flexural modes of a doubly clamped suspended beam (nanomechanical resonator). One of the modes is externally driven. We demonstrate that classically, the behavior of the non-driven mode is reminiscent of that of a parametrically driven linear oscillator: it exhibits a threshold behavior, with the amplitude of this mode below the threshold being exactly zero. Quantum-mechanically, we were able to access the dynamics of this mode below the classical parametric threshold. We show that whereas the mean displacement of this mode is still zero, the mean squared displacement is finite and at the threshold corresponds to the occupation number of 1/2. This finite displacement of the non-driven mode can serve as an experimentally verifiable quantum signature of quantum motion.     

Dynamics of coupled vibration modes in a quantum non-linear mechanical resonator

We investigate the behaviour of two non-linearly coupled flexural modes of a doubly clamped suspended beam (nanomechanical resonator). One of the modes is externally driven. We demonstrate that classically, the behavior of the non-driven mode is reminiscent of that of a parametrically driven linear oscillator: it exhibits a threshold behavior, with the amplitude of this mode below the threshold being exactly zero. Quantum-mechanically, we were able to access the dynamics of this mode below the classical parametric threshold. We show that whereas the mean displacement of this mode is still zero, the mean squared displacement is finite and at the threshold corresponds to the occupation number of 1/2. This finite displacement of the non-driven mode can serve as an experimentally verifiable quantum signature of quantum motion.     

Dramatic change of the self-diffusions of colloidal ellipsoids by hydrodynamic interactions in narrow channels

The self-diffusion problem of Brownian particles under the constraint of quasi-one-dimensional(q1 D) channel has raised wide concern.The hydrodynamic interaction(HI) plays an important role in many practical problems and two-body interactions remain dominant under q1D constraint.We measure the diffusion coefficient of individual ellipsoid when two ellipsoidal particles are close to each other by video-microscopy measurement.Meanwhile, we obtain the numerical simulation results of diffusion coefficient using finite element software.We find that the self-diffusion coefficient of the ellipsoid decreases exponentially with the decrease of their mutual distance X when X X_0, where X_0 is the maximum distance of the ellipsoids to maintain their mutual influence, X_0 and the variation rate are related to the aspect ratio p = a/b.The mean squared displacement(MSD) of the ellipsoids indicates that the self-diffusion appears as a crossover region, in which the diffusion coefficient increases as the time increases in the intermediate time regime, which is proven to be caused by the spatial variations affected by the hydrodynamic interactions.These findings indicate that hydrodynamic interaction can significantly affect the self-diffusion behavior of adjacent particles and has important implications to the research of microfluidic problems in blood vessels and bones, drug delivery, and lab-on-chip.