Coarse-graining Procedure Research Articles

Lattice Boltzmann method for the convection–diffusion equation in curvilinear coordinate systems

A lattice Boltzmann method for simulating convection and diffusion using a curvilinear grid system is presented. The proposed method does not require an interpolation or coarse-graining procedure, and thus maintains the algorithmic simplicity of the original lattice Boltzmann scheme. The lattice Boltzmann scheme is based on uniformly distributed lattice points in a transformed coordinate system, and the apparent anisotropy of diffusion that arises due to the coordinate transformation is properly handled using the multiple-relaxation-time collision operator. An asymptotic analysis of the lattice Boltzmann equation shows that the proposed method appropriately reproduces the transformed convection–diffusion equation. Several specific problems are numerically analyzed in order to validate the proposed method, including an axially symmetric (two-dimensional) problem in which the diffusion flux at an oblate hemispheroid is simulated using a body-fitted orthogonal curvilinear grid system.

Fluctuation theorem for hidden entropy production

Elimination of seemingly unnecessary variables in Markovian models may cause a difference in the value of irreversible entropy production between the original and reduced dynamics. We show that such difference, which we call the hidden entropy production, obeys an integral fluctuation theorem if all variables are time-reversal invariant, or if the density function is symmetric with respect to the change of sign of the time-reversal antisymmetric variables. The theorem has wide applicability, since the proposed condition is mostly satisfied in the case where the hidden fast variables are equilibrated. The main consequence of this theorem is that the entropy production decreases by the coarse-graining procedure. By contrast, in the case where a stochastic process is obtained by coarse-graining a deterministic and reversible dynamics, the entropy production may increase, implying that the integral fluctuation theorem should not hold for such reductions. We reveal, with an explicit example, that the nonequilibrated time-reversal antisymmetric variables play a crucial role in distinguishing these two cases, thus guaranteeing the consistency of the presented theorem.

Communication: Constructing an implicit quantum mechanical/molecular mechanics solvent model by coarse-graining explicit solvent

To avoid repeated, computationally expensive QM solute calculations while sampling MM solvent in QM/MM simulations, a new approach for constructing an implicit solvent model by coarse-graining the solvent properties over many explicit solvent configurations is proposed. The solvent is modeled using a polarizable force field that is parameterized in terms of distributed multipoles (electrostatics), polarizabilities (induction), and frequency-dependent polarizabilities (dispersion). The coarse-graining procedure exploits the ability to translate these properties to the center of each coarse-graining cell and average them over many solvent configurations before interacting them with the solute. A single coarse-grained QM/MM calculation of the interaction between a formamide solute and aqueous solvent reproduces the much more expensive average over many explicit QM/MM calculations with kJ/mol accuracy.

Modified multiscale entropy for short-term time series analysis

Multiscale entropy (MSE) is a prevalent algorithm used to measure the complexity of a time series. Because the coarse-graining procedure reduces the length of a time series, the conventional MSE algorithm applied to a short-term time series may yield an imprecise estimation of entropy or induce undefined entropy. To overcome this obstacle, the modified multiscale entropy (MMSE) was developed. The coarse-graining procedure was replaced with a moving-average procedure and a time delay was incorporated for constructing template vectors in calculating sample entropy. For conducting short-term time series analysis, this study shows that the MMSE algorithm is more reliable than the conventional MSE.

Phase-Field Model for Microstructure Evolution at the Mesoscopic Scale

This review presents a phase-field model that is generally applicable to homogeneous and heterogeneous systems at the mesoscopic scale. Reviewed first are general aspects about first- and second-order phase transitions that need to be considered to understand the theoretical background of a phase field. The mesoscopic model equations are defined by a coarse-graining procedure from a microscopic model in the continuum limit on the atomic scale. Special emphasis is given to the question of how to separate the interface and bulk contributions to the generalized thermodynamic functional, which forms the basis of all phase-field models. Numerical aspects of the discretization are discussed at the lower scale of applicability. The model is applied to spinodal decomposition and ripening in Ag-Cu with realistic thermodynamic and kinetic data from a database.

Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

Diffuse-interface modeling of solute trapping in rapid solidification: Predictions of the hyperbolic phase-field model and parabolic model with finite interface dissipation

Two recently developed phase-field models, a hyperbolic model and a parabolic model with finite interface dissipation, are employed to study the solute trapping in a Si-0.25 at.% As alloy during rapid solidification. The hyperbolic model is applied at the nanometer scale of the interface width δ. The parabolic model is derived by a coarse-graining procedure and is intended to operate with mesoscopic resolution of the interface η. The coarse-graining numerical parameters, namely interface width η and the interface permeability P, are adjusted in the parabolic model to fit the segregation coefficient calculated by the microscopic model on the nanoscale. Based on the optimal sets of η and P selected at small interface velocity, a linear relation between their logarithm values is obtained. This logarithmic relation provides a theoretical basis for choosing the appropriate values of η and P in the numerical phase-field simulation in three spatial dimensions.

Radiative transfer with partial coherence in optically thick plasmas

A quantum transport model for atomic line radiation in plasmas is developed and analyzed. It is found that the Wigner phase space formulation of QED provides a consistent way to address the wave-particle duality in radiative transfer problems. If the photons' thermal de Broglie length is much smaller than all of the spatial scales of the problem under consideration (large-spectral-band limit), the radiation is not coherent and radiative transfer can be addressed with the usual treatments. In the general case, the Heisenberg uncertainty relation yields ambiguities in the description of the radiation-matter interaction mechanisms. We examine this issue and show that an accurate description of radiative transfer should involve a model with nonlocal interactions and requires an appropriate coarse-graining procedure. Calculations of transmission factors and absorption spectra in ideal cases are performed and indicate that significant misinterpretations can be made in spectroscopic diagnostics if the radiation coherence is not well accounted for. Applications to laser physics are also discussed.

Electrokinetic flows about conducting drops

AbstractWe consider electrokinetic flows about a freely suspended liquid drop, deriving a macroscale description in the thin-double-layer limit where the ratio $\delta $ between Debye width and drop size is asymptotically small. In this description, the electrokinetic transport occurring within the diffuse part of the double layer (the ‘Debye layer’) is represented by effective boundary conditions governing the pertinent fields in the electro-neutral bulk, wherein the generally non-uniform distribution of $\zeta $, the dimensionless zeta potential, is a priori unknown. We focus upon highly conducting drops. Since the tangential electric field vanishes at the drop surface, the viscous stress associated with Debye-scale shear, driven by Coulomb body forces, cannot be balanced locally by Maxwell stresses. The requirement of microscale stress continuity therefore brings about a unique velocity scaling, where the standard electrokinetic scale is amplified by a ${\delta }^{- 1} $ factor. This reflects a transition from slip-driven electro-osmotic flows to shear-induced motion. The macroscale boundary conditions display distinct features reflecting this unique scaling. The effective shear-continuity condition introduces a Lippmann-type stress jump, appearing as a product of the local charge density and electric field. This term, representing the excess Debye-layer shear, follows here from a systematic coarse-graining procedure starting from the exact microscale description, rather than from thermodynamic considerations. The Neumann condition governing the bulk electric field is inhomogeneous, representing asymptotic matching with transverse ionic fluxes emanating from the Debye layer; these fluxes, in turn, are associated with non-uniform tangential ‘surface’ currents within this layer. Their appearance at leading order is a manifestation of dominant advection associated with the large velocity scale. For weak fields, the linearized macroscale equations admit an analytic solution, yielding a closed-form expression for the electrophoretic velocity. When scaled by Smoluchowski’s speed, it reads $${\delta }^{- 1} \frac{\sinh ( \overline{\zeta } / 2)/ \overline{\zeta } }{1+ { \textstyle\frac{3}{2} }\mu + 2\alpha {\mathop{\sinh }\nolimits }^{2} ( \overline{\zeta } / 2)} ,$$ wherein $ \overline{\zeta } $, the ‘drop zeta potential’, is the uniform value of $\zeta $ in the absence of an applied field, $\mu $ the ratio of drop to electrolyte viscosities, and $\alpha $ the ionic drag coefficient. The difference from solid-particle electrophoresis is manifested in two key features: the ${\delta }^{- 1} $ scaling, and the effect of ionic advection, as represented by the appearance of $\alpha $. Remarkably, our result differs from the small-$\delta $ limit of the mobility expression predicted by the weak-field model of Ohshima, Healy & White (J. Chem. Soc. Faraday Trans. 2, vol. 80, 1984, pp. 1643–1667). This discrepancy is related to the dominance of advection on the bulk scale, even for weak fields, which feature cannot be captured by a linear theory. The order of the respective limits of thin double layers and weak applied fields is not interchangeable.

Theoretical reconstruction of realistic dynamics of highly coarse-grainedcis-1,4-polybutadiene melts

The theory to reconstruct the atomistic-level chain diffusion from the accelerated dynamics that is measured in mesoscale simulations of the coarse-grained system, is applied here to the dynamics of cis-1,4-polybutadiene melts where each chain is described as a soft interacting colloidal particle. The rescaling formalism accounts for the corrections in the dynamics due to the change in entropy and the change in friction that are a consequence of the coarse-graining procedure. By including these two corrections the dynamics is rescaled to reproduce the realistic dynamics of the system described at the atomistic level. The rescaled diffusion coefficient obtained from mesoscale simulations of coarse-grained cis-1,4-polybutadiene melts shows good agreement with data from united atom simulations performed by Tsolou et al. [Macromolecules 38, 1478 (2005)]10.1021/ma0491210. The derived monomer friction coefficient is used as an input to the theory for cooperative dynamics that describes the internal dynamics of a polymer moving in a transient regions of slow cooperative motion in a liquid of macromolecules. Theoretically predicted time correlation functions show good agreement with simulations in the whole range of length and time scales in which data are available.

Systematic Coarse-Graining of Microscale Polymer Models as Effective Elastic Chains

One of the key goals of macromolecular modeling is to elucidate how macroscale physical properties arise from the microscale behavior of the polymer constituents. For many biological and industrial applications, a direct simulation approach is impractical due to to the wide range of length scales that must be spanned by the model, necessitating physically sound and practically relevant procedures for coarse-graining polymer systems. We present a highly general systematic coarse-graining procedure that maps any detailed polymer model onto effective elastic-chain models at intermediate and large length scales. Our approach defines a continuous flow of coarse-grained models starting from the detailed microscale model, proceeding through an intermediate-scale stretchable, shearable wormlike chain, and finally resolving to a Gaussian chain at the longest lengths. We demonstrate the utility of this procedure by coarse-graining a wormlike chain with periodic rigid kinks, a model relevant for studying DNA−protein arrays and genome packing. The methodology described in this work provides a physically sound foundation for coarse-grained simulation of polymer systems, as well as analytical expressions for chain statistics at intermediate and long length scales.

A Description of Multiscale Modeling for the Head-Disk Interface Focusing on Bottom-Level Lubricant and Carbon Overcoat Models

The challenges in designing future head disk interface (HDI) demand efficient theoretical modeling tools with flexibility in investigating various combinations of perfluoropolyether (PFPE) and carbon overcoat (COC) materials. For broad range of time and length scales, we developed multiscale/multiphysical modeling approach, which can bring paradigm-shifting improvements in advanced HDI design. In this paper, we introduce our multiscale modeling methodology with an effective strategic framework for the HDI system. Our multiscale methodology in this paper adopts a bottom to top approach beginning with the high-resolution modeling, which describes the intramolecular/intermolecular PFPE-COC degrees of freedom governing the functional oligomeric molecular conformations on the carbon surfaces. By introducing methodology for integrating atomistic/molecular/mesoscale levels via coarse-graining procedures, we investigated static and dynamic properties of PFPE-COC combinations with various molecular architectures. By bridging the atomistic and molecular scales, we are able to systematically incorporate first-principle physics into molecular models, thereby demonstrating a pathway for designing materials based on molecular architecture. We also discussed future materials (e.g., graphene for COC, star-like PFPEs) and systems (e.g., heat-assisted magnetic recording (HAMR)) with higher scale modeling methodology, which enables the incorporation of molecular/mesoscale information into the continuum scale models.

Coarse-graining and Thermodynamics in Far-from-equilibrium Systems

Lying at the core of statistical physics is the need to reduce the number of degrees of freedom in a system. Coarse-graining is a frequently-used procedure to bridge molecular modeling with experiments. In equilibrium systems, this task can be readily performed; however in systems outside equilibrium, a possible lack of equilibration of the eliminated degrees of freedom may lead to incomplete or even misleading descriptions. Here, we present some examples showing how an improper coarse-graining procedure may result in linear approaches to nonlinear processes, miscalculations of activation rates and violations of the fluctuation-dissipation theorem.

Origin of similarity of phase diagrams in amphiphilic and colloidal systems with competing interactions

We show that amphiphilic and colloidal systems with competing interactions can be described by the same Landau-Brazovskii functional. The functional is obtained by a systematic coarse-graining procedure applied tosystems with isotropic interaction potentials. Microscopic expressions for the coefficients in the functional are derived. We propose simple criteria to distinguish the effective interparticle potentials that can lead to macro- or microsegregation. Our considerations concern also charged globular proteins in aqueous solutions and other system with effective short-range attraction long-range repulsion interactions.

Coarse-Grained Multifractality Analysis Based on Structure Function Measurements to Discriminate Healthy from Distressed Foetuses

This paper proposes a combined coarse-grained multifractal method to discriminate between distressed and normal foetuses. The coarse-graining operation was performed by means of a coarse-grained procedure and the multifractal operation was based on a structure function. The proposed method was evaluated by one hundred recordings including eighty normal foetuses and twenty distressed foetuses. We found that it was possible to discriminate between distressed and normal foetuses using the Hurst exponent, singularity, and Holder spectra.

Effective-temperature concept: A physical application for nonextensive statistical mechanics

The H-theorem [(df/dt) ≤ 0] for a free-energy functional, f = u-θs (with u and s representing, respectively, the internal energy and a generalized entropy of a given physical system), has been proven previously by making use of a nonlinear Fokker-Planck equation. Herein we focus on a nonlinear Fokker-Planck equation derived by means of a coarse-graining procedure on the equations of motion of a system of interacting vortices, under overdamped motion, in the absence of thermal noise (T = 0). In this case, we show that the parameter θ is directly related to the density as well as to the interactions among vortices. Generalized quantities such as entropy, internal energy, free energy, and heat capacity are analyzed for varying θ: important relations and physical behavior analogous to those of standard thermodynamics are found, showing that θ plays the role of an effective temperature. Estimates of θ in typical physical situations of different type-II superconductors are presented; in addition to this, possible experimental procedures for varying θ are proposed.

From the discrete to the continuous: towards a cylindrically consistent dynamics

Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations exactly mirror the continuum limit. As a standard tool for the kinematics of loop quantum gravity, we propose a coarse-graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamilton's principal functions. The coarse-graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme for this purpose. A crucial role in this coarse-graining scheme is played by the embedding maps that allow interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as the choice of embedding maps will determine the truncation of the renormalization flow.

Chemically transferable coarse-grained potentials from conditional reversible work calculations

The representability and transferability of effective pair potentials used in multiscale simulations of soft matter systems is ill understood. In this paper, we study liquid state systems composed of n-alkanes, the coarse-grained (CG) potential of which may be assumed pairwise additive and has been obtained using the conditional reversible work (CRW) method. The CRW method is a free-energy-based coarse-graining procedure, which, by means of performing the coarse graining at pair level, rigorously provides a pair potential that describes the interaction free energy between two mapped atom groups (beads) embedded in their respective chemical environments. The pairwise nature of the interactions combined with their dependence on the chemically bonded environment makes CRW potentials ideally suited in studies of chemical transferability. We report CRW potentials for hexane using a mapping scheme that merges two heavy atoms in one CG bead. It is shown that the model is chemically and thermodynamically transferable to alkanes of different chain lengths in the liquid phase at temperatures between the melting and the boiling point under atmospheric (1 atm) pressure conditions. It is further shown that CRW-CG potentials may be readily obtained from a single simulation of the liquid state using the free energy perturbation method, thereby providing a fast and versatile molecular coarse graining method for aliphatic molecules.

Thermodynamic Consistency between Analytic Integral Equation Theory and Coarse-Grained Molecular Dynamics Simulations of Homopolymer Melts

We present the equation of state for a coarse-grained model of polymer melts where each chain is represented as a soft colloidal particle centered on its center-of-mass. The formalism is based on the solution of the Ornstein–Zernike equation and is analytical, allowing for the formal investigation of the elements that ensure thermodynamic consistency in coarse-grained models of polymer melts. By comparing predictions from our expressions with those from computer simulations of the coarse-grained system and with atomistic polymer integral equation theory, we demonstrate that both structural and thermodynamic consistency with the atomistic level description is maintained during our coarse-graining procedures.

Coarse-Graining of Ionic Microgels: Theory and Experiment

Abstract In this work, we discuss the statistical mechanics of many-body systems consisting of electrically charged microgels, and we show that their collective behavior is determined by an interplay between the screened electrostatic and the elastic contributions to their effective interaction potential. The former is derived by means of a statistical-mechanical approach due to Denton [A. R. Denton, Phys. Rev. E 67, 011804 (2003)], and it includes the screened electrostatic potential between penetrable spheres and the counterion entropic contribution. The latter is based on the Hertzian model of the theory of elasticity. Comparisons with experimental results demonstrate the realistic nature of the coarse-graining procedure, which makes it possible to put forward theoretical predictions on the phase diagram of ionic microgels and on the behavior of soft, neutral microgels under confinement in narrow pores.