Coarse-graining Procedure Research Articles

Observable signatures of inflaton decays

We numerically compute features in the power-spectrum that originate from the decay of fields during inflation. Using a simple, phenomenological, multi-field setup, we increase the number of fields from a few to thousands. Whenever a field decays, its associated potential energy is transferred into radiation, causing a jump in the equation of state parameter and mode mixing at the perturbed level. We observe discrete steps in the power-spectrum if the number of fields is low, in agreement with analytic arguments in the literature. These features become increasingly smeared out once many fields decay within a given Hubble time. In this regime we confirm the validity of the analytic approach to staggered inflation, which is based on a coarse-graining procedure. Our numerical approach bridges the aforementioned analytic treatments, and can be used in more complicated scenarios.

Study of a Long Range Perturbation of a One-Dimensional Kac Model

We consider a one dimensional ferromagnetic Ising spin system with interactions that correspond to a 1/r 2 long range perturbation of the usual Kac model. We apply a coarse graining procedure widely used for higher-dimensional finite range Kac potentials to describe the basic properties of the system and the relation with the mean field theory.

Development of a coarse-grained model for simulations of tridecanoin liquid–solid phase transitions

Novel coarse-grained models for molecular dynamics of tridecanoin melts are here proposed as results of a coarse-graining step procedure. The procedure is implemented to develop three coarse-grained models of increasing number of particle types from two to four. Force fields are computed by minimization of the deviations of appropriate distribution functions of the coarse-grained models from those of a reference atomistic one. Density, diffusivity and shear viscosity are computed by numerical simulation and compared with experimental values. The ability of each model to describe liquid-solid transitions is also analyzed. In particular, the model with four types of coarse-grained beads shows a transition from a liquid to a crystal phase.

Stability tests on known and misfolded structures with discrete and all atom molecular dynamics simulations

The brevity of molecular dynamics simulations often limits their utility in developing and evaluating structural models of proteins. The duration of simulations can be increased greatly using discrete molecular dynamics (DMD). However, the trade off is that coarse graining, implicit solvent, and other time-saving procedures reduce the accuracy of DMD simulations. Here we address some of these issues by comparing results of DMD and conventional all atom MD simulations on proteins of known structure and misfolded proteins. DMD simulations were performed at a range of temperatures to identify a ‘physiological’ temperature for DMD that mimicked molecular motions of conventional MD simulations at 310 K. We also compared results obtained with a new implicit solvent model developed here based on Miyazawa–Jernigan interaction pair potential to those obtained with a previously used model based on Kyte–Doolittle hydropathy scale. We compared DMD and all atom molecular dynamics with explicit water by simulating both correctly and incorrectly folded structures, and monomeric and dimeric α β-barrel structures to analyze the ability of these procedures to distinguish between good and bad models. Deviations from the correct structures were substantially greater with DMD, as would be expected from coarse-graining and longer simulation time. Deviations were smallest for β-strands and greatest for coiled loops. Structures of the incorrectly folded models were very poorly preserved during the DMD simulations; but both methods were able to distinguish between the correct and the incorrect structures based on differences in the magnitudes of the root mean squared deviation (RMSD) from the starting conformation.

Particle-layering effect in wall-bounded dissipative particle dynamics

Dissipative particle dynamics (DPD) is a mesoscopic simulation method that describes "clusters" of molecules as a single numerical particle. DPD is a very effective method but it introduces numerical artifacts through the coarse-graining procedure, such as particle ordering in the near-wall region. These artifacts can result in nonphysical phenomena during a simulation of a polymer tethered to the wall undergoing shear flow: polymer sticking and overextension for higher shear rates. In this paper we report that a version of DPD with a so-called solidification boundary formulation and conservative-force interactions based on the equation of state allows to reduce number density fluctuations in near-wall region significantly.

Neoclassical equilibria as starting point for global gyrokinetic microturbulence simulations

The implementation of linearized operators describing inter- and like-species collisions in the global gyrokinetic particle-in-cell code ORB5 [S. Jolliet, Comput. Phys. Commun. 177, 409 (2007)] is presented. A neoclassical axisymmetric equilibrium with self-consistent electric field can be obtained with no assumption made on the radial width of the particle trajectories. The formulation thus makes it possible to study collisional transport in regions where the neoclassical approximation breaks down such as near the magnetic axis. The numerical model is validated against both analytical results as well as other simulation codes. The effects of the poloidally asymmetric Fourier modes of the electric field are discussed, and the contribution of collisional kinetic electrons is studied. In view of subsequent gyrokinetic simulations of turbulence started from a neoclassical equilibrium, the problem of numerical noise inherent to the particle-in-cell approach is addressed. A novel algorithm for collisional gyrokinetic simulation switching between a local and a canonical Maxwellian background for, respectively, carrying out the collisional and collisionless dynamics is proposed, and its beneficial effects together with a coarse graining procedure [Y. Chen and S. E. Parker, Phys. Plasmas 14, 082301 (2007)] on noise and weight spreading reduction are discussed.

Global collisional gyrokinetic simulations of ITG microturbulence starting from a neoclassical equilibrium

Linearized operators describing inter-species and like-species collisions have been discretized and implemented in the gyrokinetic Particle-In-Cell (PIC) code ORB5 [S. Jolliet, Comp. Phys. Comm. 177, 409 (2007)] based on the delta-f approach. Simulation results for neoclassical transport are compared with both analytical predictions as well as results from other codes. This new version of ORB5 including collisional dynamics thus makes it possible to carry out simulations of microturbulence starting from a global neoclassical equilibrium including self-consistent electric fields. First results of ITG microturbulence simulations carried out in this way are presented. The issue of numerical noise, inherent to the PIC approach and further accentuated by the implementation of collisions in the delta-f scheme, is addressed. It is shown how a noise reduction scheme based on a coarse graining procedure [Y. Chen and S. E. Parker, Phys. Plasmas 14, 082301 (2007)] ensures the physical relevance of such simulations. Furthermore, a novel delta-f algorithm is presented, which switches between a canonical and a local Maxwellian background for carrying out the collisonless and collisional dynamics respectively, and its advantages are discussed.

Kinetics of phase separation in fluids: A molecular dynamics study

We present results from extensive three-dimensional molecular dynamics (MD) simulations of phase separation kinetics in fluids. A coarse-graining procedure is used to obtain state-of-the-art MD results. We observe an extended period of temporally linear growth in the viscous hydrodynamic regime. The morphological similarity of coarsening in fluids and solids is also quantified. The velocity field is characterized by the presence of monopolelike defects, which yield a generalized Porod tail in the corresponding structure factor.

Dynamics in binary cluster crystals

As a result of the application of coarse-graining procedures to describe complexfluids, the study of systems consisting of particles interacting through bounded,repulsive pair potentials has become of increasing interest in recent years.A well known example is the so-called generalized exponential model (GEM-m), for which the interaction between particles is described by the potentialv(r) = ϵexp[ − (r/σ)m]. Interactionswith m > 2 lead to the formation of a novel phase of soft matter consisting of clustercrystals. Recent studies on the phase behavior of binary mixtures of GEM-m particles have provided evidence for the formation of novel kinds of alloys, depending onthe cross-interactions between the two species. This work aims to study the dynamicbehavior of such binary mixtures by means of extensive molecular dynamics simulations,and in particular to investigate the effect of the addition of non-clustering particles on thedynamic scenario of one-component cluster crystals. Analogies and differences with theone-component case are revealed and discussed by analyzing self- and collective dynamiccorrelators.

Anyonic entanglement renormalization

We introduce a family of variational ansatz states for chains of anyons which optimally exploits the structure of the anyonic Hilbert space. This ansatz is the natural analog of the multiscale entanglement renormalization ansatz for spin chains. In particular, it has the same interpretation as a coarse-graining procedure and is expected to accurately describe critical systems with algebraically decaying correlations. We numerically investigate the validity of this ansatz using the anyonic golden chain and its relatives as a testbed. This demonstrates the power of entanglement renormalization in a setting with non-Abelian exchange statistics, extending previous work on qudits, bosons, and fermions in two dimensions.

Coarse-graining microscopic strains in a harmonic, two-dimensional solid: Elasticity, nonlocal susceptibilities, and nonaffine noise

In soft matter systems the local displacement field can be accessed directly by video microscopy enabling one to compute local strain fields and hence the elastic moduli in these systems using a coarse-graining procedure. We study this process in detail for a simple triangular, harmonic lattice in two dimensions. Coarse-graining local strains obtained from particle configurations in a Monte Carlo simulation generates nontrivial, nonlocal strain correlations (susceptibilities). These may be understood within a generalized, Landau-type elastic Hamiltonian containing up to quartic terms in strain gradients [K. Franzrahe, Phys. Rev. E 78, 026106 (2008)10.1103/PhysRevE.78.026106]. In order to demonstrate the versatility of the analysis of these correlations and to make our calculations directly relevant for experiments on colloidal solids, we systematically study various parameters such as the choice of statistical ensemble, presence of external pressure and boundary conditions. Crucially, we show that special care needs to be taken for an accurate application of our results to actual experiments, where the analyzed area is embedded within a larger system, to which it is mechanically coupled. Apart from the smooth, affine strain fields, the coarse-graining procedure also gives rise to a noise field (χ) made up of nonaffine displacements. Several properties of χ may be rationalized for the harmonic solid using a simple "cell model" calculation. Furthermore the scaling behavior of the probability distribution of the noise field (χ) is studied. We find that for any inverse temperature β, spring constant f, density ρ and coarse-graining length Λ the probability distribution can be obtained from a master curve of the scaling variable X=χβf/ρΛ(2).

Modelling the viscoelasticity and thermal fluctuations of fluids at the nanoscale

Simulation methodologies for modelling the viscoelasticity and thermal fluctuations of molecular fluids at nanoscale are presented. In particular, we bridge the distinct frameworks of fluctuating hydrodynamics (FHD) and molecular dynamics (MD) simulations using a coarse-graining procedure that properly considers the effective size of each fluid molecule in mapping a phase space vector of the all-atom model to field variables in the FHD representation. We also generalise the FHD equations to model non-Markovian rheological responses by incorporating coloured noise. To capture the increasing elastic responses that emerge at the nanoscale, a composite Newtonian–Maxwell rheological model is developed. Numerical simulations using a staggered discretisation scheme demonstrate that the thermodynamic states and dynamic properties (power spectra) of FHD equations match with those of all-atom MD simulations quantitatively. This agreement also unambiguously determines the wavenumber-dependent transport coefficients of the composite Newtonian–Maxwell model. To avoid the complexities associated with using wavenumber-dependent transport coefficients, we characterise the approximation of using a single set of transport coefficients. We find that for collective fluctuations with a wavelength longer than 25 Å, wavenumber-independent models can accurately describe the power spectra. For fluctuations with shorter wavelengths, relative errors in power spectra increase monotonically with wavenumber.

Entropy production and coarse graining in Markov processes

We study the large time fluctuations of entropy production in Markov processes. Inparticular, we consider the effect of a coarse-graining procedure which decimates fast stateswith respect to a given time threshold. Our results provide strong evidence that entropyproduction is not directly affected by this decimation, provided that it does notentirely remove loops carrying a net probability current. After the study of someexamples of random walks on simple graphs, we apply our analysis to a networkmodel for the kinesin cycle, which is an important biomolecular motor. A tentativegeneral theory of these facts, based on Schnakenberg’s network theory, is proposed.

Covariant coarse graining of inhomogeneous dust flow in general relativity

A new definition of coarse-grained quantities describing the dust flow in general relativity is proposed. It assigns the coarse-grained expansion, shear and vorticity to finite-size comoving domains of fluid in a covariant, coordinate-independent manner. The coarse-grained quantities are all quasi-local functionals, depending only on the geometry of the boundary of the considered domain. They can be thought of as relativistic generalizations of simple volume averages of local quantities in a flat space. The procedure is based on the isometric embedding theorem for S2 surfaces and thus requires the boundary of the domain in question to have spherical topology and positive scalar curvature. We prove that in the limit of infinitesimally small volume the proposed quantities reproduce the local expansion, shear and vorticity. In the case of irrotational flow we derive the time evolution equation for the coarse-grained quantities and show that its structure is very similar to the evolution equation for their local counterparts. Additional terms appearing in it may serve as a measure of the backreacton of small-scale inhomogeneities of the flow on the large-scale motion of the fluid inside the domain and therefore the result may be interesting in the context of the cosmological backreaction problem. We also consider the application of the proposed coarse-graining procedure to a number of known exact solutions of Einstein equations with dust and show that it yields reasonable results.

Bridging molecular and continuous descriptions: the case of dynamics in clays

The theory of transport in porous media such as clays depends on the level of description. On the macroscopic scale,hydrodynamics equations are used. These continuous descriptions are convenient to model the fluid motion in a confined system. Nevertheless, they are valid only if the pores of the material are much larger than the molecular size of the components of the system. Another approach consists in using molecular descriptions. These two methods which correspond to different levels of description are complementary. The link between them can be clarified by using a coarse-graining procedure where the microscopic laws are averaged over fast variables to get the long time macroscopic laws. We present such an approach in the case of clays. Firstly, we detail the various levels of description and the relations among them, by emphasizing the validity domain of the hydrodynamic equations. Secondly, we focus on the case of dehydrated clays where hydrodynamics is not relevant. We show that it is possible to derive a simple model for the motion of the cesium ion based on the difference on time scale between the solvent and the solute particles.

Vortex pairing in two-dimensional Bose gases

Recent experiments on ultracold Bose gases in two dimensions have provided evidence for the existence of the Berezinskii-Kosterlitz-Thouless (BKT) phase via analysis of the interference between two independent systems. In this work we study the two-dimensional quantum degenerate Bose gas at finite temperature using the projected Gross-Pitaevskii equation classical field method. While this describes the highly occupied modes of the gas below a momentum cutoff, we have developed a method to incorporate the higher momentum states in our model. We concentrate on finite-sized homogeneous systems in order to simplify the analysis of the vortex pairing. We determine the dependence of the condensate fraction on temperature and compare this to the calculated superfluid fraction. By measuring the first order correlation function we determine the boundary of the Bose-Einstein condensate and BKT phases, and find it is consistent with the superfluid fraction decreasing to zero. We reveal the characteristic unbinding of vortex pairs above the BKT transition via a coarse-graining procedure. Finally, we model the procedure used in experiments to infer system correlations [Hadzibabic et al., Nature 441, 1118 (2006)], and quantify its level of agreement with directly calculated in situ correlation functions.

Bridging length and time scales in sheared demixing systems: From the Cahn-Hilliard to the Doi-Ohta model

We develop a systematic coarse-graining procedure which establishes the connection between models of mixtures of immiscible fluids at different length and time scales. We start from the Cahn-Hilliard model of spinodal decomposition in a binary fluid mixture under flow from which we derive the coarse-grained description. The crucial step in this procedure is to identify the relevant coarse-grained variables and find the appropriate mapping which expresses them in terms of the more microscopic variables. In order to capture the physics of the Doi-Ohta level, we introduce the interfacial width as an additional variable at that level. In this way, we account for the stretching of the interface under flow and derive analytically the convective behavior of the relevant coarse-grained variables, which in the long wavelength limit recovers the familiar phenomenological Doi-Ohta model. In addition, we obtain the expression for the interfacial tension in terms of the Cahn-Hilliard parameters as a direct result of the developed coarse-graining procedure. Finally, by analyzing the numerical results obtained from the simulations on the Cahn-Hilliard level, we discuss that dissipative processes at the Doi-Ohta level are of the same origin as in the Cahn-Hilliard model. The way to estimate the interface relaxation times of the Doi-Ohta model from the underlying morphology dynamics simulated at the Cahn-Hilliard level is established.

Competing micellar and cylindrical phases in semi-dilute diblock copolymer solutions

We develop a “first principles” coarse-graining procedure based on a soft effective segment representation of an athermal AB diblock copolymer model in selective solvent, to map out the self-assembly phase diagram for several asymmetry ratios f using Monte Carlo free energy calculations. We find that the free energy per unit volume is surprisingly insensitive to the aggregation number of monodisperse cubic and cylindrical phases for a given polymer volume fraction. The cylindrical phase is found to pre-empt the cubic micellar phases for nearly symmetric (f ≃ 0.6) copolymers.

Microscopic derivation of discrete hydrodynamics

By using the standard theory of coarse graining based on Zwanzig's projection operator, we derive the dynamic equations for discrete hydrodynamic variables. These hydrodynamic variables are defined in terms of the Delaunay triangulation. The resulting microscopically derived equations can be understood, a posteriori, as a discretization on an arbitrary irregular grid of the Navier-Stokes equations. The microscopic derivation provides a set of discrete equations that exactly conserves mass, momentum, and energy and the dissipative part of the dynamics produces strict entropy increase. In addition, the microscopic derivation provides a practical implementation of thermal fluctuations in a way that the fluctuation-dissipation theorem is satisfied exactly. This paper points toward a close connection between coarse-graining procedures from microscopic dynamics and discretization schemes for partial differential equations.

APPLICATION OF PERCOLATION THEORY AND RG METHOD TO SOIL MOISTURE DYNAMICS

In this work, we present a review of the results obtained in the exploitation of percolation theory and the renormalization group method in the study of soil moisture spatial patterns. In order to capture critical point in soil moisture spatio-temporal dynamics, we developed an algorithm consisting of three steps: (i) dichotomization, i.e., transforming the soil moisture maps into binary maps; (ii) identification of the largest wet cluster; (iii) scaling transformation, i.e., applying an ad hoc implemented coarse-graining procedure to the binary maps. The methodology was explored by means of several applications on soil moisture data coming from field measurement, remote sensing, and hydrological modelling over a wide range of spatial scales. From the relations between the occupation probability in soil moisture spatial patterns and the normalized size of the largest cluster at different scales, as well as the scaling behaviour, it is possible to argue that also for this physical system the critical point theory applies. The critical probability seems to be a structural feature of the catchment, being insensitive to the scale of the analysis, as well as to the parameterization of the methodology.