Generalized Langevin equation formulation for anomalous diffusion in the Ising model at the critical temperature

We consider the two- (2D) and three-dimensional (3D) Ising models on a square lattice at the critical temperature T_{c}, under Monte Carlo spin flip dynamics. The bulk magnetization and the magnetization of a tagged line in the 2D Ising model, and the bulk magnetization and the magnetization of a tagged plane in the 3D Ising model, exhibit anomalous diffusion. Specifically, their mean-square displacements increase as power laws in time, collectively denoted as ∼t^{c}, where c is the anomalous exponent. We argue that the anomalous diffusion in all these quantities for the Ising model stems from time-dependent restoring forces, decaying as power laws in time-also with exponent c -in striking similarity to anomalous diffusion in polymeric systems. Prompted by our previous work that has established a memory-kernel based generalized Langevin equation (GLE) formulation for polymeric systems, we show that a closely analogous GLE formulation holds for the Ising model as well. We obtain the memory kernels from spin-spin correlation functions, and the formulation allows us to consistently explain anomalous diffusion as well as anomalous response of the Ising model to an externally applied magnetic field in a consistent manner.

Why Clothes Don't Fall Apart: Tension Transmission in Staple Yarns

The problem of how staple yarns transmit tension is addressed within abstract models in which the Amontons-Coulomb friction laws yield a linear programing (LP) problem for the tensions in the fiber elements. We find there is a percolation transition such that above the percolation threshold the transmitted tension is in principle unbounded. We determine that the mean slack in the LP constraints is a suitable order parameter to characterize this supercritical state. We argue the mechanism is generic, and in practical terms, it corresponds to a switch from a ductile to a brittle failure mode accompanied by a significant increase in mechanical strength.

Wavelet Monte Carlo dynamics: A new algorithm for simulating the hydrodynamics of interacting Brownian particles

We develop a new algorithm for the Brownian dynamics of soft matter systems that evolves time by spatially correlated Monte Carlo moves. The algorithm uses vector wavelets as its basic moves and produces hydrodynamics in the low Reynolds number regime propagated according to the Oseen tensor. When small moves are removed, the correlations closely approximate the Rotne-Prager tensor, itself widely used to correct for deficiencies in Oseen. We also include plane wave moves to provide the longest range correlations, which we detail for both infinite and periodic systems. The computational cost of the algorithm scales competitively with the number of particles simulated, N, scaling as N In N in homogeneous systems and as N in dilute systems. In comparisons to established lattice Boltzmann and Brownian dynamics algorithms, the wavelet method was found to be only a factor of order 1 times more expensive than the cheaper lattice Boltzmann algorithm in marginally semi-dilute simulations, while it is significantly faster than both algorithms at large N in dilute simulations. We also validate the algorithm by checking that it reproduces the correct dynamics and equilibrium properties of simple single polymer systems, as well as verifying the effect of periodicity on the mobility tensor.

Nonlinear least-squares method for the inverse droplet coagulation problem

If the rates, K(x,y), at which particles of size x coalesce with particles of size y is known, then the mean-field evolution of the particle size distribution of an ensemble of irreversibly coalescing particles is described by the Smoluchowski equation. We study the corresponding inverse problem which aims to determine the coalescence rates K(x,y) from measurements of the particle size distribution. We assume that K(x,y) is a homogeneous function of its arguments, a case which occurs commonly in practice. The problem of determining K(x,y), a function to two variables, then reduces to the simpler problem of determining a function of a single variable plus two exponents, μ and ν, which characterize the scaling properties of K(x,y). The price of this simplification is that the resulting least-squares problem is nonlinear in the exponents μ and ν. We demonstrate the effectiveness of the method on a selection of coalescence problems arising in polymer physics, cloud science, and astrophysics. The applications include examples in which the particle size distribution is stationary owing to the presence of sources and sinks of particles and examples in which the particle size distribution is undergoing self-similar relaxation in time.

Scale-invariant growth processes in expanding space

Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting particles, and their large scale behavior depends on the overall growth geometry. We establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries, which preserves the local scale invariance and is independent of other properties such as the dimensionality. This relation generalizes standard conformal transformations as the natural symmetry of self-affine growth processes. We illustrate our main result numerically for various structures of coalescing Lévy flights and fractional Brownian motions, including also branching and finite particle sizes. One of the main benefits of this approach is a full, explicit description of the asymptotic statistics in expanding domains, which are often nontrivial and random due to amplification of initial fluctuations.

Collective oscillations in irreversible coagulation driven by monomer inputs and large-cluster outputs

We describe collective oscillatory behavior in the kinetics of irreversible coagulation with a constant input of monomers and removal of large clusters. For a broad class of collision rates, this system reaches a nonequilibrium stationary state at large times and the cluster size distribution tends to a universal form characterized by a constant flux of mass through the space of cluster sizes. Universality, in this context, means that the stationary state becomes independent of the cutoff as the cutoff grows. This universality is lost, however, if the aggregation rate between large and small clusters increases sufficiently steeply as a function of cluster sizes. We identify a transition to a regime in which the stationary state vanishes as the cutoff grows. This nonuniversal stationary state becomes unstable as the cutoff is increased. It undergoes a Hopf bifurcation after which the stationary state is replaced by persistent and periodic collective oscillations. These oscillations, which bear some similarities to relaxation oscillations in excitable media, carry pulses of mass through the space of cluster sizes such that the average mass flux through any cluster size remains constant. Universality is partially restored in the sense that the scaling of the period and amplitude of oscillation is inherited from the dynamical scaling exponents of the universal regime.

Shape of a ponytail and the statistical physics of hair fiber bundles

A general continuum theory for the distribution of hairs in a bundle is developed, treating individual fibers as elastic filaments with random intrinsic curvatures. Applying this formalism to the iconic problem of the ponytail, the combined effects of bending elasticity, gravity, and orientational disorder are recast as a differential equation for the envelope of the bundle, in which the compressibility enters through an "equation of state." From this, we identify the balance of forces in various regions of the ponytail, extract a remarkably simple equation of state from laboratory measurements of human ponytails, and relate the pressure to the measured random curvatures of individual hairs.

Instantaneous gelation in Smoluchowski's coagulation equation revisited

We study the solutions of the Smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularized gelation time decreases as M increases, consistent with the expectation that the gelation time is zero in the unregularized system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cutoff. We also study the case when a source of monomers is introduced in the regularized system. In this case a stationary state is reached. We present a complete analytic description of this regularized stationary state for the model kernel, K(m(1),m(2)) = max{m(1),m(2)}(ν), which gels instantaneously when M → ∞ if ν>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent ν for large cluster sizes. The total particle density in the stationary state slowly vanishes as [(ν-1)log M](-1/2) when M → ∞. The approach to the stationary state is nontrivial: Oscillations about the stationary state emerge from the interplay between the monomer injection and the cutoff, M, which decay very slowly when M is large. A quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers.

Stochastic transitions and jamming in granular pipe flow

We study a model granular suspension driven down a channel by an embedding fluid via computer simulations. We characterize the different system flow regimes and the stochastic nature of the transitions between them. For packing fractions below a threshold ϕ{m}, granular flow is disordered and exhibits an Ostwald-de Waele-type power-law shear-stress constitutive relation. Above ϕ{m}, two asymptotic states exist; disordered flow can persist indefinitely, yet, in a fraction of samples, the system self-organizes in an ordered form of flow where grains move in parallel ordered layers. In the latter regime, the Ostwald-de Waele relationship breaks down and a nearly solid plug appears in the center, with linear shear regions at the boundaries. Above a higher threshold ϕ{g}, an abrupt jamming transition is observed if ordering is avoided.

Reply to the comment on 'Anomalous dynamics of unbiased polymer translocation through a narrow pore' and other recent papers by D Panja, G Barkema and R Ball

We reply to the comment made by Dubbeldam et al (2009 J. Phys.: Condens. Matter 21 098001) on our paper 'Anomalous dynamics of unbiased polymer translocation through a narrow pore' and our other recent papers.

Identifying critical residues in protein folding: Insights from phi-value and P(fold) analysis

We apply a simulational proxy of the phi-value analysis and perform extensive mutagenesis experiments to identify the nucleating residues in the folding "reactions" of two small lattice Go polymers with different native geometries. Our findings show that for the more complex native fold (i.e., the one that is rich in nonlocal, long-range bonds), mutation of the residues that form the folding nucleus leads to a considerably larger increase in the folding time than the corresponding mutations in the geometry that is predominantly local. These results are compared to data obtained from an accurate analysis based on the reaction coordinate folding probability P(fold) and on structural clustering methods. Our study reveals a complex picture of the transition state ensemble. For both protein models, the transition state ensemble is rather heterogeneous and splits up into structurally different populations. For the more complex geometry the identified subpopulations are actually structurally disjoint. For the less complex native geometry we found a broad transition state with microscopic heterogeneity. These findings suggest that the existence of multiple transition state structures may be linked to the geometric complexity of the native fold. For both geometries, the identification of the folding nucleus via the P(fold) analysis agrees with the identification of the folding nucleus carried out with the phi-value analysis. For the most complex geometry, however, the applied methodologies give more consistent results than for the more local geometry. The study of the transition state structure reveals that the nucleus residues are not necessarily fully native in the transition state. Indeed, it is only for the more complex geometry that two of the five critical residues show a considerably high probability of having all its native bonds formed in the transition state. Therefore, one concludes that, in general, the phi-value correlates with the acceleration/deceleration of folding induced by mutation, rather than with the degree of nativeness of the transition state, and that the "traditional" interpretation of phi-values may provide a more realistic picture of the structure of the transition state only for more complex native geometries.

Fermions without fermion fields

It is shown that an arbitrary fermion hopping Hamiltonian can be mapped into a system with no fermion fields, generalizing an earlier model of Levin and Wen. All operators in the Hamiltonian of the resulting description commute (rather than anticommute) when acting at different sites, despite the system having excitations obeying Fermi statistics. While extra conserved degrees of freedom are introduced, they are all locally identified in the representation obtained. The same methods apply to Majorana (half) fermions, which for Cartesian lattices mitigate the fermion doubling problem. The generality of these results suggests that the observation of Fermion excitations in nature does not demand that anticommuting Fermion fields be fundamental.

Anisotropic diffusion limited aggregation in three dimensions: universality and nonuniversality

We explore the macroscopic consequences of lattice anisotropy for diffusion limited aggregation (DLA) in three dimensions. Simple cubic and bcc lattice growths are shown to approach universal asymptotic states in a coherent fashion, and the approach is accelerated by the use of noise reduction. These states are strikingly anisotropic dendrites with a rich hierarchy of structure. For growth on an fcc lattice, our data suggest at least two stable fixed points of anisotropy, one matching the bcc case. Hexagonal growths, favoring six planar and two polar directions, appear to approach a line of asymptotic states with continuously tunable polar anisotropy. The more planar of these growths visually resembles real snowflake morphologies. Our simulations use a new and dimension-independent implementation of the DLA model. The algorithm maintains a hierarchy of sphere coverings of the growth, supporting efficient random walks onto the growth by spherical moves. Anisotropy was introduced by restricting growth to certain preferred directions.

Off-lattice noise reduced diffusion-limited aggregation in three dimensions

Using off-lattice noise reduction, it is possible to estimate the asymptotic properties of diffusion-limited aggregation clusters grown in three dimensions with greater accuracy than would otherwise be possible. The fractal dimension of these aggregates is found to be 2.50+/-0.01 , in agreement with earlier studies, and the asymptotic value of the relative penetration depth is xi/ R(dep) =0.122+/-0.002 . The multipole powers of the growth measure also exhibit universal asymptotes. The fixed point noise reduction is estimated to be epsilon(f) approximately 0.0035 , meaning that large clusters can be identified with a low noise regime. The slowest correction to scaling exponents are measured for a number of properties of the clusters, and the exponent for the relative penetration depth and quadrupole moment are found to be significantly different from each other. The relative penetration depth exhibits the slowest correction to scaling of all quantities, which is consistent with a theoretical result derived in two dimensions. We also note fast corrections to scaling, whose limited relevance is consistent with the requirement that clusters grow far enough in radius to support sufficient scales of ramification.

Growth by random walker sampling and scaling of the dielectric breakdown model

Random walkers absorbing on a boundary sample the harmonic measure linearly and independently: we discuss how the recurrence times between impacts enable nonlinear moments of the measure to be estimated. From this we derive a technique to simulate dielectric breakdown model growth, which is governed nonlinearly by the harmonic measure. For diffusion-limited aggregation, recurrence times are shown to be accurate and effective in probing the multifractal growth measure in its active region. For the dielectric breakdown model our technique grows large clusters efficiently and we are led to significantly revise earlier exponent estimates. Previous results by two conformal mapping techniques were less converged than expected, and in particular a recent theoretical suggestion of superuniversality is firmly refuted.

Theory of growth by differential sedimentation, with application to snowflake formation

A simple model of irreversible aggregation under differential sedimentation of particles in a fluid is presented. The structure of the aggregates produced by this process is found to feed back on the dynamics in such a way as to stabilize both the exponents controlling the growth rate, and the fractal dimension of the clusters produced at readily predictable values. The aggregation of ice crystals to form snowflakes is considered as a potential application of the model.

Folding and form: insights from lattice simulations

Monte Carlo simulations of a Miyazawa-Jernigan lattice-polymer model indicate that, depending on the native structure's geometry, the model exhibits two broad classes of folding mechanisms for two-state folders. Folding to native structures of low contact order is driven by backbone distance and is characterized by a progressive accumulation of structure towards the native fold. By contrast, folding to high contact order targets is dominated by intermediate stage contacts not present in the native fold, yielding a more cooperative folding process.

Characterization of the probabilistic traveling salesman problem

We show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem. The probabilistic "traveling salesman" must decide on an a priori order in which to visit n cities (randomly distributed over a unit square) before learning that some cities can be omitted. We find the optimized average length of the pruned tour follows E(L(pruned))=sqrt[np](0.872-0.105p)f(np), where p is the probability of a city needing to be visited, and f(np)-->1 as np--> infinity. The average length of the a priori tour (before omitting any cities) is found to follow E(L(a priori))=sqrt[n/p]beta(p), where beta(p)=1/[1.25-0.82 ln(p)] is measured for 0.05< or =p< or =0.6. Scaling arguments and indirect measurements suggest that beta(p) tends towards a constant for p<0.03. Our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths, exploiting the sampling error to provide the analog of thermal fluctuations in simulated (thermal) annealing. The method has general application to the optimization of functions whose cost to evaluate rises with the precision required.

Diffusion-limited aggregation in channel geometry

We performed extensive numerical simulation of diffusion-limited aggregation in two-dimensional channel geometry. Contrary to earlier claims, the measured fractal dimension D=1.712+/-0.002 and its leading correction to scaling are the same as in the radial case. The average cluster, defined as the average conformal map, is similar but not identical to Saffman-Taylor fingers.

Stochastic annealing

We show how to simulate a system in thermal equilibrium when the energy cannot be evaluated exactly: the error distribution needs to be symmetric, but it does not need to be known. We also solve the Ceperley-Dewing version of this problem, where the error distribution is taken to be fully known. These underlying ideas give an effective optimization strategy for problems where the evaluation of each design can be sampled only statistically, including an application to protein folding.

From plasticity to a renormalization group

The marginally rigid state is a candidate paradigm for what makes granular material a state of matter distinct from both liquid and solid. The coordination number is identified as a discriminating characteristic, and for rough-surfaced particles we show that the low values predicted are indeed approached in simple two-dimensional experiments. We show calculations of the stress transmission, suggesting that this is governed by local linear equations of constraint between the stress components. These constraints can in turn be related to the generalized forces conjugate to the motion of grains rolling over each other. The lack of a spatially coherent way of imposing a sign convention on these motions is a problem for scaling up the equations, which leads us to attempt a renormalization-group calculation. Finally, we discuss how perturbations propagate through such systems, suggesting a distinction between the behaviour of rough and smooth grains.

Diffusion-controlled growth: theory and closure approximations

We expand upon a new theoretical framework for diffusion-limited aggregation and associated dielectric breakdown models in two dimensions [R. C. Ball and E. Somfai, Phys. Rev. Lett. 89, 135503 (2002)]. Key steps are understanding how these models interrelate when the ultraviolet cut-off strategy is changed, the analogy with turbulence, and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is discussed. We find angular resonances relatable to the cone angle theory, and we are led to predict a new screening transition in the DBM at large eta.

Theory of diffusion controlled growth

We present a new theoretical framework for diffusion limited aggregation and associated dielectric breakdown models in two dimensions. Key steps are understanding how these models interrelate when the ultraviolet cutoff strategy is changed, the analogy with turbulence and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is both predicted and confirmed.

Protein design depends on the size of the amino acid alphabet

We consider the design of proteins to be simultaneously thermodynamically stable in multiple independent and correlated conformations. We first show that a protein can be trained to fold to multiple independent conformations and calculate its capacity. The number of configurations that it can remember is proportional to the logarithm of the number of amino acid species A, independent of chain length. Next we investigate the recognition of correlated conformations, which we apply to funnel design around a single configuration. The maximum basin of attraction, as parametrized in our model, also depends on the number of amino acid species as ln A. We argue that the extent to which the protein energy landscape can be manipulated is fixed, effecting a trade off between well breadth, well depth, and well number. This emerging picture motivates a clearer understanding of the scope and limits of protein and heteropolymer function.

Off-lattice noise reduction and the ultimate scaling of diffusion-limited aggregation in two dimensions

Off-lattice diffusion-limited aggregation (DLA) clusters grown with different levels of noise reduction are found to be consistent with a simple fractal fixed point. Cluster shapes and their ensemble variation exhibit a dominant slowest correction to scaling, and this also accounts for the apparent "multiscaling" in the DLA mass distribution. We interpret the correction to scaling in terms of renormalized noise. The limiting value of this variable is strikingly small and is dominated by fluctuations in cluster shape. Earlier claims of anomalous scaling in DLA were misled by the slow approach to this small fixed point value.

Stress field in granular systems: loop forces and potential formulation

The transmission of stress through a marginally stable granular pile in two dimensions is exactly formulated in terms of a vector field of loop forces, and thence in terms of a single scalar potential. This leads to a local constitutive equation coupling the stress tensor to fluctuations in the local geometry. For a disordered pile of rough grains this means the stress tensor components are coupled in a frustrated manner. In piles of rough grains with long range staggered order, frustration is avoided and a simple linear theory follows. We show that piles of smooth grains can be mapped onto a pile of unfrustrated rough grains, indicating that the problems of rough and smooth grains may be fundamentally distinct.

How many conformations can a protein remember?

We show that a protein can be trained to recognize multiple conformations, analogous to an associative memory, and provide capacity calculations based on energy fluctuations and information theory. Unlike the linear capacity of a Hopfield network, the number of conformations which can be remembered by a protein sequence depends on the size of the amino acid alphabet as lnA, independent of protein length. This admits the possibility of certain proteins, such as prions, evolving to fold to independent stable conformations, as well as novel possibilities for protein and heteropolymer design.

Angular structure of lacunarity, and the renormalization group

We formulate the angular structure of lacunarity in fractals, in terms of a symmetry reduction of the three point correlation function. This provides a rich probe of universality, and first measurements yield new evidence in support of the equivalence between self-avoiding walks (SAW's) and percolation perimeters in two dimensions. We argue that the lacunarity reveals much of the renormalization group in real space. This is supported by exact calculations for random walks and measured data for percolation clusters and SAW's. Relationships follow between exponents governing inward and outward propagating perturbations, and we also find a very general test for the contribution of long-range interactions.

Universal aging features in the restructuring of fractal colloidal gels

We use multispeckle dynamic light scattering to measure the dynamic structure factor, f(q,tau), of gels formed by aggregation of colloids. Although the gel is an elastic solid, f(q,tau) nearly completely decays on long time scales, with an unusual form, f(q, tau) approximately exp{-(tau/tau(f))(mu)}, with mu approximately 1.5 and with tau(f) proportional variant q(-1). A model for restructuring of the gel with aging correctly accounts for this behavior. Aging leads to a dramatic increase in tau(f); however, all data can be scaled on a single master curve, with tau(f) asymptotically growing linearly with age. This behavior is strikingly similar to that predicted for aging in disordered glassy systems, offering convincing proof of the universality of these concepts.

Effects of nonlocal stress on the determination of shear banding flow

We analyze the steady planar shear flow of the modified Johnson-Segalman model, which has an added nonlocal term. We find that the new term allows for unambiguous selection of the stress at which two "phases" coexist, in contrast to the original model. For general differential constitutive models we show the singular nature of stress selection in terms of a saddle connection between fixed points in the equivalent dynamical system. The result means that stress selection is unique under most conditions for space nonlocal models. Finally, illustrated by simple models, we show that stress selection generally depends on the form of the nonlocal terms (weak universality).

Dynamical weight functions for a planar crack

The stress intensity factors are evaluated for a moving planar crack for loadings which vary arbitrarily in time and three dimensions of space. We exploit the adjoint elasticity equation obeyed by the corresponding weight functions, and a new and more universal Wiener-Hopf factorization of the Rayleigh function, this being the central difficulty in such calculations. For the mode II weight function we give further asymptotic results crucial to a subsequent calculation of crack stability with respect to out-of-plane perturbations.