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Nigel Goldenfeld

Publications and source records attributed to Nigel Goldenfeld.

At least 91 records · Page 5Linked to original sources
arXiv2006

Collective evolution and the genetic code

A dynamical theory for the evolution of the genetic code is presented, which accounts for its universality and optimality. The central concept is that a variety of collective, but non-Darwinian, mechanisms likely to be present in early communal life generically lead to refinement and selection of innovation-sharing protocols, such as the genetic code. Our proposal is illustrated using a simplified computer model, and placed within the context of a sequence of transitions that early life may have made, prior to the emergence of vertical descent.

q-bio.PE↗
arXiv2005

Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling

The Anderson model for independent electrons in a disordered potential is transformed analytically and exactly to a basis of random extended states leading to a variant of augmented space. In addition to the widely-accepted phase diagrams in all physical dimensions, a plethora of additional, weaker Anderson transitions are found, characterized by the long-distance behavior of states. Critical disorders are found for Anderson transitions at which the asymptotically dominant sector of augmented space changes for all states at the same disorder. At fixed disorder, critical energies are also found at which the localization properties of states are singular. Under the approximation of single-parameter scaling, this phase diagram reduces to the widely-accepted one in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson transition at infinitesimal disorder, there is a transition between two localized states, characterized by a change in the nature of wave function decay.

cond-mat.str-el↗
arXiv2005

Roughness-induced critical phenomena in a turbulent flow

I present empirical evidence that turbulent flows are closely analogous to critical phenomena, from a reanalysis of friction factor measurements in rough pipes. The data collapse found here corresponds to Widom scaling near critical points, and implies that a full understanding of turbulence requires explicit accounting for boundary roughness.

cond-mat.stat-mech↗
arXiv2005

Renormalization group approach to multiscale modelling in materials science

Dendritic growth, and the formation of material microstructure in general, necessarily involves a wide range of length scales from the atomic up to sample dimensions. The phase field approach of Langer, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with this range of scales, and provides an effective way to move phase boundaries. However, it fails to preserve memory of the underlying crystallographic anisotropy, and thus is ill-suited for problems involving defects or elasticity. The phase field crystal (PFC) equation-- a conserving analogue of the Hohenberg-Swift equation --is a phase field equation with periodic solutions that represent the atomic density. It can natively model elasticity, the formation of solid phases, and accurately reproduces the nonequilibrium dynamics of phase transitions in real materials. However, the PFC models matter at the atomic scale, rendering it unsuitable for coping with the range of length scales in problems of serious interest. Here, we show that a computationally-efficient multiscale approach to the PFC can be developed systematically by using the renormalization group or equivalent techniques to derive appropriate coarse-grained coupled phase and amplitude equations, which are suitable for solution by adaptive mesh refinement algorithms.

cond-mat.mtrl-sci↗
arXiv2005

Coarse-graining of cellular automata, emergence, and the predictability of complex systems

We study the predictability of emergent phenomena in complex systems. Using nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show how to construct local coarse-grained descriptions of CA in all classes of Wolfram's classification. The resulting coarse-grained CA that we construct are capable of emulating the large-scale behavior of the original systems without accounting for small-scale details. Several CA that can be coarse-grained by this construction are known to be universal Turing machines; they can emulate any CA or other computing devices and are therefore undecidable. We thus show that because in practice one only seeks coarse-grained information, complex physical systems can be predictable and even decidable at some level of description. The renormalization group flows that we construct induce a hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method. Finally we argue that the large scale dynamics of CA can be very simple, at least when measured by the Kolmogorov complexity of the large scale update rule, and moreover exhibits a novel scaling law. We show that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity as one goes to increasingly coarser scales. We interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics.

nlin.CG↗
arXiv2005

Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model

We propose a computationally-efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal (PFC) model. The order parameter describing the density profile at the nanoscale is reconstructed from its slowly-varying amplitude and phase, which satisfy rotationally-covariant equations derivable from the renormalization group. We validate the approach using the example of two-dimensional grain nucleation and growth.

cond-mat.mtrl-sci↗
arXiv2005

Global divergence of microbial genome sequences mediated by propagating fronts

We model the competition between recombination and point mutation in microbial genomes, and present evidence for two distinct phases, one uniform, the other genetically diverse. Depending on the specifics of homologous recombination, we find that global sequence divergence can be mediated by fronts propagating along the genome, whose characteristic signature on genome structure is elucidated, and apparently observed in closely-related {\it Bacillus} strains. Front propagation provides an emergent, generic mechanism for microbial "speciation", and suggests a classification of microorganisms on the basis of their propensity to support propagating fronts.

q-bio.GN↗
arXiv2004

Cognitive styles sex the brain, compete neurally, and quantify deficits in autism

Introduction: Two key dimensions of the mind are understanding and responding to another's mental state (empathizing), and analysing lawful behaviour (systemizing). Methods: Two questionnaires, the Systemizing Quotient (SQ) and the Empathy Quotient (EQ), were administered to a normal control group and a group of individuals with Asperger Syndrome (AS) or High-Functioning Autism (HFA). The multivariate correlations of the joint scores were analysed using principal components analysis. Results: The principal components were well-approximated by the sums and differences of the SQ and EQ scores. The differences in the scores corresponded to sex differences within the control group and also separated out the AS/HFA group, which showed stronger systemizing than the control group, but below-average empathy. The sums of the scores did not show sex differences, but did distinguish the AS/HFA group. Conclusions: These tests reliably sex the brain, and their correlations show that empathizing and systemizing are not independent, but compete neurally. Their combined score (EQ + SQ) quantifies the deficit in autism spectrum conditions.

q-bio.NC↗
arXiv2003

On computational irreducibility and the predictability of complex physical systems

Using elementary cellular automata (CA) as an example, we show how to coarse-grain CA in all classes of Wolfram's classification. We find that computationally irreducible (CIR) physical processes can be predictable and even computationally reducible at a coarse-grained level of description. The resulting coarse-grained CA which we construct emulate the large-scale behavior of the original systems without accounting for small-scale details. At least one of the CA that can be coarse-grained is irreducible and known to be a universal Turing machine.

nlin.CG↗
arXiv2003

Computationally efficient phase-field models with interface kinetics

We present a new phase-field model of solidification which allows efficient computations in the regime when interface kinetic effects dominate over capillary effects. The asymptotic analysis required to relate the parameters in the phase-field with those of the original sharp interface model is straightforward, and the resultant phase-field model can be used for a wide range of material parameters.

cond-mat↗
arXiv2003

Critical dynamics of superconducting BSCCO films

We report on a systematic investigation of the critical fluctuations in the complex conductivity of epitaxially-grown Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ films for ${T \lesssim T_c}$ using a two-coil inductive technique at zero applied field. We observe the static 3D-XY critical exponent in the superfluid density near $T_{c}$. Linear scaling analysis close to the critical temperature yields a dynamic critical exponent of $z = 2.0$ for small drive currents, but non-linear effects are seen to be important. At $T_{c}$, a non-linear scaling analysis also yields a 3D dynamic exponent of $z = 2.0$.

cond-mat.supr-con↗
arXiv2001

Critical Dynamics of a Vortex Loop Model for the Superconducting Transition

We calculate analytically the dynamic critical exponent $z_{MC}$ measured in Monte Carlo simulations for a vortex loop model of the superconducting transition, and account for the simulation results. In the weak screening limit, where magnetic fluctuations are neglected, the dynamic exponent is found to be $z_{MC} = 3/2$. In the perfect screening limit, $z_{MC} = 5/2$. We relate $z_{MC}$ to the actual value of $z$ observable in experiments and find that $z \sim 2$, consistent with some experimental results.

cond-mat.supr-con↗
arXiv2001

Phase Field Model for Three-Dimensional Dendritic Growth with Fluid Flow

We study the effect of fluid flow on three-dimensional (3D) dendrite growth using a phase-field model on an adaptive finite element grid. In order to simulate 3D fluid flow, we use an averaging method for the flow problem coupled to the phase-field method and the Semi-Implicit Approximated Projection Method (SIAPM). We describe a parallel implementation for the algorithm, using Charm++ FEM framework, and demonstrate its efficiency. We introduce an improved method for extracting dendrite tip position and tip radius, facilitating accurate comparison to theory. We benchmark our results for two-dimensional (2D) dendrite growth with solvability theory and previous results, finding them to be in good agreement. The physics of dendritic growth with fluid flow in three dimensions is very different from that in two dimensions, and we discuss the origin of this behavior.

cond-mat.mtrl-sci↗
arXiv2000

Renormalization group and perfect operators for stochastic differential equations

We develop renormalization group methods for solving partial and stochastic differential equations on coarse meshes. Renormalization group transformations are used to calculate the precise effect of small scale dynamics on the dynamics at the mesh size. The fixed point of these transformations yields a perfect operator: an exact representation of physical observables on the mesh scale with minimal lattice artifacts. We apply the formalism to simple nonlinear models of critical dynamics, and show how the method leads to an improvement in the computational performance of Monte Carlo methods.

cond-mat.stat-mech↗
arXiv2000

Fluctuations in finite critical and turbulent systems

We show that hyperscaling and finite-size scaling imply that the probability distribution of the order parameter in finite size critical systems exhibit data collapse. We consider the examples of equilibrium critical systems, and a statistical model of ecology. We explain recent observations that the probability distribution of turbulent power fluctuations in closed flows is the same as that of the harmonic 2DXY model.

cond-mat.stat-mech↗
arXiv2000

Computation of dendritic microstructures using a level set method

We compute time-dependent solutions of the sharp-interface model of dendritic solidification in two dimensions by using a level set method. The steady-state results are in agreement with solvability theory. Solutions obtained from the level set algorithm are compared with dendritic growth simulations performed using a phase-field model and the two methods are found to give equivalent results. Furthermore, we perform simulations with unequal diffusivities in the solid and liquid phases and find reasonable agreement with the available theory.

cond-mat.mtrl-sci↗
arXiv1999

Path integral Monte Carlo simulation of helium at negative pressures

Path integral Monte Carlo (PIMC) simulations of liquid helium at negative pressure have been carried out for a temperature range from the critical temperature to below the superfluid transition. We have calculated the temperature dependence of the spinodal line as well as the pressure dependence of the isothermal sound velocity in the region of the spinodal. We discuss the slope of the superfluid transition line and the shape of the dispersion curve at negative pressures.

cond-mat↗