Algebraic Tail Research Articles

Physical Networks from Multifunctional Telechelic Star Polymers: A Rheological Study by Experiments and Simulations.

The equilibrium mechanical properties of a cross-linked gel of telechelic star polymers are studied by rheology and Brownian dynamics simulations. The Brownian dynamics model consists of cores to which Rouse arms are attached. Forces between the cores are obtained from a potential of mean force model developed by Likos and co-workers. Both experimentally and in the simulations, networks were created by attaching sticker groups to the ends of the arms of the polymers, which were next allowed to form bonds among them in a one to one fashion. Simulations were sped up by solving the Rouse dynamics exactly. Moreover, the Rouse model was extended to allow for different frictions on different beads. In order to describe the rheology of the non-cross-linked polymers, it had to be assumed that bead frictions increase with increasing bead number along the arms. This friction model could be transferred to describe the rheology of the network without any adjustments other than an overall increase of the frictions due to the formation of bonds. The slowing down at intermediate times of the network rheology compared to that of the non-cross-linked polymers is well described by the model. The percentage of stickers involved in forming inter-star bonds in the system was determined to be 25%, both from simulations and from an application of the Green–Tobolsky relation to the experimental plateau value of the shear relaxation modulus. Simulations with increasing cross-link percentages revealed that on approaching the gel transition the shear relaxation modulus develops an algebraic tail, which gets frozen at a percentage of maximum cross-linking of about 11%.

Expansion of a quantum wave packet in a one-dimensional disordered potential in the presence of a uniform bias force

We study numerically the expansion dynamics of an initially confined quantum wave packet in the presence of a disordered potential and a uniform bias force. For white-noise disorder, we find that the wave packet develops asymmetric algebraic tails for any ratio of the force to the disorder strength. The exponent of the algebraic tails decays smoothly with that ratio and no evidence of a critical behavior on the wave density profile is found. Algebraic localization features a series of critical values of the force-to-disorder strength where the m-th position moment of the wave packet diverges. Below the critical value for the m-th moment, we find fair agreement between the asymptotic long-time value of the m-th moment and the predictions of diagrammatic calculations. Above it, we find that the m-th moment grows algebraically in time. For correlated disorder, we find evidence of systematic delocalization, irrespective to the model of disorder. More precisely, we find a two-step dynamics, where both the center-of-mass position and the width of the wave packet show transient localization, similar to the white-noise case, at short time and delocalization at sufficiently long time. This correlation-induced delocalization is interpreted as due to the decrease of the effective de Broglie wave length, which lowers the effective strength of the disorder in the presence of finite-range correlations.

Auto-correlation function and frequency spectrum due to a super-position of uncorrelated exponential pulses

The auto-correlation function and the frequency power spectral density due to a super-position of uncorrelated exponential pulses are considered. These are shown to be independent of the degree of pulse overlap and thereby the intermittency of the stochastic process. For constant pulse duration and a one-sided exponential pulse shape, the power spectral density has a Lorentzian shape which is flat for low frequencies and a power law at high frequencies. The algebraic tail is demonstrated to result from the discontinuity in the pulse function. For a strongly asymmetric two-sided exponential pulse shape, the frequency spectrum is a broken power law with two scaling regions. In the case of a symmetric pulse shape, the power spectral density is the square of a Lorentzian function. The steep algebraic tail at high frequencies in these cases is demonstrated to follow from the discontinuity in the derivative of the pulse function. A random distribution of pulse durations is shown to result in apparently longer correlation times but has no influence on the asymptotic power law tail of the frequency spectrum. The effect of additional random noise is also discussed, leading to a flat spectrum for high frequencies. The probability density function for the fluctuations is shown to be independent of the distribution of pulse durations. The predictions of this model describe the variety of auto-correlation functions and power spectral densities reported from experimental measurements in the scrape-off layer of magnetically confined plasmas.

Spatio-temporal coordination among functional residues in protein

The microscopic basis of communication among the functional sites in bio-macromolecules is a fundamental challenge in uncovering their functions. We study the communication through temporal cross-correlation among the binding sites. We illustrate via Molecular Dynamics simulations the properties of the temporal cross-correlation between the dihedrals of a small protein, ubiquitin which participates in protein degradation in eukaryotes. We show that the dihedral angles of the residues possess non-trivial temporal cross-correlations with asymmetry with respect to exchange of the dihedrals, having peaks at low frequencies with time scales in nano-seconds and an algebraic tail with a universal exponent for large frequencies. We show the existence of path for temporally correlated degrees of freedom among the functional residues. We explain the qualitative features of the cross-correlations through a general mathematical model. The generality of our analysis suggests that temporal cross-correlation functions may provide convenient theoretical framework to understand bio-molecular functions on microscopic basis.

Slowing Allee effect versus accelerating heavy tails in monostable reaction diffusion equations

We focus on the spreading properties of solutions of monostable reaction–diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between ‘no acceleration’ and ’acceleration’. This implies in particular that, for tails exponentially unbounded but lighter than algebraic, acceleration never occurs in the presence of an Allee effect. This is in sharp contrast with the KPP situation []. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.

Modeling of Heavy Tailed Sea Clutter Based on the Generalized Central Limit Theory

For high-resolution radars operating at low grazing angles, sea clutter always exhibits an impulse behavior with heavy tailed statistical property. Presented is a statistical model for spiky sea clutter based on the generalized central limit theory (CLT). Within the compound Gaussian structure, the CLT is adopted in the modeling of the Bragg scattering speckle component, while the generalized CLT is introduced to describe the contribution of spiky scattering component, which has potential in describing the impulse nature of the underlying mean level because of its algebraic (inverse power) tails. Validation results with S- and X-band measured sea clutter data indicate that the proposed model can improve the fitting accuracy of spiky sea clutter effectively, especially in the tail region.

Fujita blow up phenomena and hair trigger effect: The role of dispersal tails

We consider the nonlocal diffusion equation ∂tu=J⁎u−u+u1+p in the whole of RN. We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel J near the origin, which is linked to the tails of J. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation ∂tu=Δu+u1+p. On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of J. As an application of the result in population dynamics models, we discuss the hair trigger effect for ∂tu=J⁎u−u+u1+p(1−u).

Spin dynamics of hopping electrons in quantum wires: Algebraic decay and noise

We study theoretically spin decoherence and intrinsic spin noise in semiconductor quantum wires caused by an interplay of electron hopping between the localized states and the hyperfine interaction of electron and nuclear spins. At a sufficiently low density of localization sites the hopping rates have an exponentially broad distribution. It allows the description of the spin dynamics in terms of closely-situated "pairs" of sites and single "reaching" states, from which the series of hops result in the electron localized inside a "pair". The developed analytical model and numerical simulations demonstrate disorder-dependent algebraic tails in the spin decay and power-law singularity-like features in the low-frequency part of the spin noise spectrum.

Dissipation and velocity distribution at the shear-driven jamming transition

We investigate energy dissipation and the distribution of particle velocities at the jamming transition for overdamped shear-driven frictionless disks in two dimensions at zero temperature. We find that the dissipation is caused by the fastest particles and that the fraction of particles responsible for the dissipation decreases towards zero as jamming is approached. These particles belong to an algebraic tail of the velocity distribution that approaches ∼v^{-3} as jamming is approached. We further find that different measures of the velocity diverge differently, which means that concepts such as typical velocity may no longer be used, a finding that should have implications for analytical approaches to shear-driven jamming.

Aggregation driven by a localized source

We study aggregation driven by a localized source of monomers. The densities become stationary and have algebraic tails far away from the source. We show that in a model with mass-independent reaction rates and diffusion coefficients, the density of monomers decays as in d dimensions. The decay exponent has irrational values in physically relevant dimensions: and . We also study Brownian coagulation with a localized source and establish the behavior of the total cluster density and the total number of of clusters in the system. The latter quantity exhibits a logarithmic growth with time.

Mass exchange processes with input

We investigate a system of interacting clusters evolving through mass exchange and supplemented by input of small clusters. Three possibilities depending on the rate of exchange generically occur when input is homogeneous: continuous growth, gelation, and instantaneous gelation. We mostly study the growth regime using scaling methods. An exchange process with reaction rates equal to the product of reactant masses admits an exact solution which allows us to justify the validity of scaling approaches in this special case. We also investigate exchange processes with a localized input. We show that if the diffusion coefficients are mass-independent, the cluster mass distribution becomes stationary and develops an algebraic tail far away from the source.

On the gap and time interval between the first two maxima of long random walks

.In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, Gn, and the number of time steps, Ln, between the two highest positions of a Markovian one-dimensional random walker, starting from x0 = 0, after n time steps (taking the x-axis vertical). The jumps ηi = xi −xi − 1 are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), f(η), the Fourier transform of which has the small k behavior , with 0 < μ ⩽ 2. For μ = 2, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For 0 < μ < 2, the RW is a Lévy flight of index μ. We show that the joint PDF of Gn and Ln converges to a well defined stationary bi-variate distribution p(g, l) as the RW duration n goes to infinity. We present a thorough analytical study of the limiting joint distribution p(g, l), as well as of its associated marginals pgap(g) and ptime(l), revealing a rich variety of behaviors depending on the tail of f(η) (from slow decreasing algebraic tail to fast decreasing super-exponential tail). We also address the problem for a random bridge where the RW starts and ends at the origin after n time steps. We show that in the large n limit, the PDF of Gn and Ln converges to the same stationary distribution p(g, l) as in the case of the free-end RW. Finally, we present a numerical check of our analytical predictions. Some of these results were announced in a recent letter (Majumdar et al 2013 Phys. Rev. Lett. 111 070601)

Universal Order and Gap Statistics of Critical Branching Brownian Motion

We study the order statistics of one dimensional branching Brownian motion in which particles either diffuse (with diffusion constant $D$), die (with rate $d$) or split into two particles (with rate $b$). At the critical point $b=d$ which we focus on, we show that, at large time $t$, the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale $\sim \sqrt{Dt}$ which controls the collective fluctuations of the whole bunch and (ii) the length scale of the gap between the bunched particles $\sim \sqrt{D/b}$. We compute the probability distribution function $P(g_k,t|n)$ of the $k$th gap $g_k = x_k - x_{k+1}$ between the $k$th and $(k+1)$th particles given that the system contains exactly $n>k$ particles at time $t$. We show that at large $t$, it converges to a stationary distribution $P(g_k,t\to \infty|n) = p(g_k|n)$ with an algebraic tail $p(g_k|n) \sim 8(D/b) g_k^{-3}$, for $g_k \gg 1$, independent of $k$ and $n$. We verify our predictions with Monte Carlo simulations.

Percolation with Constant Freezing

We introduce and study a model of percolation with constant freezing (PCF) where edges open at constant rate \(1\), and clusters freeze at rate \(\alpha \) independently of their size. Our main result is that the infinite volume process can be constructed on any amenable vertex transitive graph. This is in sharp contrast to models of percolation with freezing previously introduced, where the limit is known not to exist. Our interest is in the study of the percolative properties of the final configuration as a function of \(\alpha \). We also obtain more precise results in the case of trees. Surprisingly the algebraic exponent for the cluster size depends on the degree, suggesting that there is no lower critical dimension for the model. Moreover, even for \(\alpha <\alpha _c\), it is shown that finite clusters have algebraic tail decay, which is a signature of self organised criticality. Partial results are obtained on \(\mathbb {Z}^d\), and many open questions are discussed.

Jittery velocity relaxation of an elastic sphere immersed in a viscous incompressible fluid.

Velocity relaxation of an elastic sphere immersed in a viscous incompressible fluid is studied on the basis of the equations of linear elasticity and the linearized Navier-Stokes equations. It is found that both translational motion after a sudden impulse and rotational motion after a sudden twist show jittery behavior in the long-time regime, with many reversals of velocity if the sphere is sufficiently flexible. In the extreme long-time regime the translational and rotational velocity relaxation functions each decay with a universal algebraic long-time tail. The added mass and the added moment of inertia of the sphere are found to vanish.

Large deviation principles for words drawn from correlated letter sequences

When an sequence of letters is cut into words according to i.i.d. renewal times, an i.i.d. sequence of words is obtained. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In Birkner, Greven and den Hollander, the quenched LDP (= conditional on a typical letter sequence) was derived for the case where the renewal times have an algebraic tail. The rate function turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. The purpose of the present paper is to extend both LDP's to letter sequences that are not It is shown that both LDP's carry over when the letter sequence satisfies a mixing condition called summable variation. The rate functions are again given by specific relative entropies w.r.t. the reference law of words, respectively, letters. But since neither of these reference laws is i.i.d., several approximation arguments are needed to obtain the extension.

The influence of inertia and charge relaxation on electrohydrodynamic drop deformation

We quantify the transient deformation of a droplet immersed in a weakly conductive (leaky dielectric) fluid upon exposure to a uniform DC electric field. Capillary forces are assumed to be sufficiently large that the drop only slightly deviates from its equilibrium spherical shape. In particular, we account for transient (or linear) fluid inertia via the unsteady Stokes equations, and also account for a finite electrical relaxation time over which the drop interface charges. The temporal droplet deformation is governed by two dimensionless groups: (i) the ratio of capillary to momentum diffusion time scales: an Ohnesorge number Oh and (ii) the ratio of charge relaxation to momentum diffusion time scales, which we denote by Sa. If charge and momentum relaxation occur quickly compared to interface deformation, Sa ≪ 1 and Oh ≫ 1 for the droplet and medium, a monotonic deformation is acquired. In contrast, Sa &amp;gt; 1 and Oh &amp;lt; 1 for either phase can lead to a non-monotonic development in the deformation. Numerical values for the deformation are calculated by inverting an analytical expression obtained in the Laplace domain, and are corroborated by asymptotic expansions at early and late times. The droplet and medium behave as perfect dielectrics at early times, which always favors an initial prolate (parallel to the applied field) deformation. As a consequence, for a final oblate (normal to the applied field) deformation, there is a shape transition from prolate to oblate at intermediate times. This transition is caused by the accumulation of sufficient charge at the interface to generate electrical and viscous shear stresses. Notably, after the transition, there may be an “overshoot” in the deformation, i.e., the magnitude exceeds its steady-state value, which is proceeded by an algebraic tail describing the arrival towards the final, steady deformation. Our work demonstrates that transient inertia or a non-zero electrical relaxation time can yield non-monotonic electrohydrodynamic drop deformation.

Distinct degrees and their distribution in complex networks

We investigate a variety of statistical properties associated with the number of distinct degrees that exist in a typical network for various classes of networks. For a single realization of a network with N nodes that is drawn from an ensemble in which the number of nodes of degree k has an algebraic tail, Nk ∼ N/kν for k ≫ 1, the number of distinct degrees grows as N1/ν. Such an algebraic growth is also observed in scientific citation data. We also determine the N dependence of statistical quantities associated with the sparse, large-k range of the degree distribution, such as the location of the first hole (where Nk = 0), the last doublet (two consecutive occupied degrees), triplet, dimer (Nk = 2), trimer, etc.

Thermal Noise and Hydrodynamic Memory Effects on Force Measurements at Microscales

The paper is devoted to the problem of the determination of regular forces acting on microscopic and smaller objects in fluids when the presence of thermal noise affects the results of measurements. One of the methods on how these forces are determined is the measurement of the drift velocity of Brownian particles. Traditionally such experiments are interpreted on the basis of the overdamped Langevin equation. An exact expression has been obtained for this velocity within the linear hydrodynamic theory of the Brownian motion in incompressible fluids, when the external force is constant and when the particle is in a harmonic potential well. It is shown that for sufficiently short experimental times the influence of the hydrodynamic memory effects in the drift velocity determination is significant. The obtained solutions contain algebraic long-time tails due to which the results expected from the standard theory are approached very slowly with the increase of time.

A Hybrid Approach for the Simulation of the Thermal Motion of a Nearly Neutrally Buoyant Nanoparticle in an Incompressible Newtonian Fluid Medium

A hybrid scheme based on Markovian fluctuating hydrodynamics of the fluid and a non-Markovian Langevin dynamics with the Ornstein-Uhlenbeck noise perturbing the translational and rotational equations of motion of a nanoparticle is employed to study the thermal motion of a nearly neutrally buoyant nanoparticle in an incompressible Newtonian fluid medium. A direct numerical simulation adopting an arbitrary Lagrangian-Eulerian based finite element method is employed in simulating the thermal motion of the particle suspended in the fluid contained in a cylindrical vessel. The instantaneous flow around the particle and the particle motion are fully resolved. The numerical results show that (a) the calculated temperature of the nearly neutrally buoyant Brownian particle in a quiescent fluid satisfies the equipartition theorem; (b) the translational and rotational decay of the velocity autocorrelation functions result in algebraic tails, over long time; (c) the translational and rotational mean square displacements of the particle obeys Stokes-Einstein and Stokes-Einstein-Debye relations, respectively; and (d) the parallel and perpendicular diffusivities of the particle closer to the wall are consistent with the analytical results, where available. The study has important implications for designing nanocarriers for targeted drug delivery.