Systematic Coarse-graining Procedure Research Articles

Generalized Langevin equationwith a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme.

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a generalized Langevin equation(GLE) for a general interacting many-body system. The resulting GLE includes (i) explicitly the potential of mean force (PMF) that describes the equilibrium distribution of the system in the chosen space of reaction coordinates, (ii) a random force term that explicitly depends on the initial state of the system, and (iii) a memory friction contribution that splits into two parts: a part that is linear in the past reaction-coordinate velocity and a part that is in general nonlinear in the past reaction coordinates but does not depend on velocities. Our hybrid scheme thus combines all desirable properties of the Zwanzig and Mori projection schemes. The nonlinear memory friction contribution is shown to be related to correlations between the reaction-coordinate velocity and the random force. We present a numerical method to compute all parameters of our GLE, in particular the nonlinear memory friction function and the random force distribution, from a trajectory in reaction coordinate space. We apply our method on the dihedral-angle dynamics of a butane molecule in water obtained from atomistic molecular dynamics simulations. For this example, we demonstrate that nonlinear memory friction is present and that the random force exhibits significant non-Gaussian corrections. We also present the derivation of the GLE for multidimensional reaction coordinates that are general functions of all positions in the phase-space of the underlying many-body system; this corresponds to a systematic coarse-graining procedure that preserves not only the correct equilibrium behavior but also the correct dynamics of the coarse-grained system.

Comparative analysis of continuum angiogenesis models

Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that depend on the microscopic rules of the discrete framework, however, and such nonlinearities can make a model difficult to mathematically analyse. By contrast, models developed using phenomenological approaches are typically easier to investigate but have a more obscure connection to the underlying microscopic system. To our knowledge, there has been little work done to compare solutions of phenomenological and coarse-grained models. Here we address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological “snail-trail” model for angiogenesis to solutions of a nonlinear system of partial differential equations (PDEs) derived via a systematic coarse-graining procedure (Pillay et al. in Phys Rev E 95(1):012410, 2017. https://doi.org/10.1103/PhysRevE.95.012410). For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to identical PDEs within the domain interior. Numerical and analytical results confirm that pointwise differences between solutions to the two continuum models are small if these conditions hold, and demonstrate how perturbation methods can be used to determine when a phenomenological model provides a good approximation to a more detailed coarse-grained system for the same biological process.

Projection operators in statistical mechanics: a pedagogical approach

The Mori–Zwanzig projection operator formalism is one of the central tools of nonequilibrium statistical mechanics, allowing one to derive macroscopic equations of motion from the microscopic dynamics through a systematic coarse-graining procedure. It is important as a method in the physical sciences and gives many insights into the general structure of nonequilibrium transport equations and the general procedure of microscopic derivations. Therefore, it is a valuable ingredient of basic and advanced courses in statistical mechanics. However, accessible introductions to this method—in particular in its more advanced forms—are extremely rare. In this article, we give a simple and systematic introduction to the Mori–Zwanzig formalism, which allows students to understand the methodology in the form it is used in current research. This includes both basic and modern versions of the theory. Moreover, we relate the formalism to more general aspects of statistical mechanics and quantum mechanics. Thereby, we explain how this method can be incorporated into a lecture course on statistical mechanics as a way to give a general introduction to the study of nonequilibrium systems. Applications, in particular to spin relaxation and dynamical density functional theory, are also discussed.

Iterative Reconstruction of Memory Kernels.

In recent years, it has become increasingly popular to construct coarse-grained models with non-Markovian dynamics to account for an incomplete separation of time scales. One challenge of a systematic coarse-graining procedure is the extraction of the dynamical properties, namely, the memory kernel, from equilibrium all-atom simulations. In this article, we propose an iterative method for memory reconstruction from dynamical correlation functions. Compared to previously proposed noniterative techniques, it ensures by construction that the target correlation functions of the original fine-grained systems are reproduced accurately by the coarse-grained system, regardless of time step and discretization effects. Furthermore, we also propose a new numerical integrator for generalized Langevin equations that is significantly more accurate than the more commonly used generalization of the velocity Verlet integrator. We demonstrate the performance of the above-described methods using the example of backflow-induced memory in the Brownian diffusion of a single colloid. For this system, we are able to reconstruct realistic coarse-grained dynamics with time steps about 200 times larger than those used in the original molecular dynamics simulations.

Coarse graining the phase space ofNqubits

We develop a systematic coarse graining procedure for systems of $N$ qubits. We exploit the underlying geometrical structures of the associated discrete phase space to produce a coarse-grained version with reduced effective size. Our coarse-grained spaces inherit key properties of the original ones. In particular, our procedure naturally yields a subset of the original measurement operators, which can be used to construct a coarse discrete Wigner function. These operators also constitute a systematic choice of incomplete measurements for the tomographer wishing to probe an intractably large system.

Coarse-grained electrostatic interactions of coronene: Towards the crystalline phase.

In this article, we present and compare two different, coarse-grained approaches to model electrostatic interactions of disc-shaped aromatic molecules, specifically coronene. Our study builds on our previous work [T. Heinemann et al., J. Chem. Phys. 141, 214110 (2014)], where we proposed, based on a systematic coarse-graining procedure starting from the atomistic level, an anisotropic effective (Gay-Berne-like) potential capable of describing van der Waals contributions to the interaction energy. To take into account electrostatics, we introduce, first, a linear quadrupole moment along the symmetry axis of the coronene disc. The second approach takes into account the fact that the partial charges within the molecules are distributed in a ring-like fashion. We then reparametrize the effective Gay-Berne-like potential such that it matches, at short distances, the ring-ring potential. To investigate the validity of these two approaches, we perform many-particle molecular dynamics simulations, focusing on the crystalline phase (karpatite) where electrostatic interaction effects are expected to be particularly relevant for the formation of tilted stacked columns. Specifically, we investigate various structural parameters as well as the melting transition. We find that the second approach yields consistent results with those from experiments despite the fact that the underlying potential decays with the wrong distance dependence at large molecule separations. Our strategy can be transferred to a broader class of molecules, such as benzene or hexabenzocoronene.

The effect of antagonistic salt on a confined near-critical mixture.

We consider a near-critical binary mixture with addition of antagonistic salt (hydrophilic cations and hydrophobic anions) confined between weakly charged and selective surfaces. A mesoscopic functional for this system is developed from a microscopic description by a systematic coarse-graining procedure. The functional reduces to the Landau-Brazovskii functional for amphiphilic systems for a sufficiently large ratio between the correlation length in the critical binary mixture and the screening length. Our theoretical result agrees with the experimental observation [Sadakane et al., J. Chem. Phys., 2013, 139, 234905] that the antagonistic salt and the surfactant both lead to a similar mesoscopic structure. For very low salt concentration ρion the Casimir potential is the same as in the presence of inorganic salt. For larger ρion the Casimir potential takes a minimum followed by a maximum for separations of order of tens of nanometers, and exhibits an oscillatory decay very close to the critical point. For separations of tens of nanometers the potential between surfaces with a linear size of hundreds of nanometers can be of order of kBT. We have verified that in the experimentally studied samples [Sadakane et al., J. Chem. Phys., 2013, 139, 234905, Leys et al., Soft Matter, 2013, 9, 9326] the decay length is too small compared to the period of oscillations of the Casimir potential, but the oscillatory force could be observed closer to the critical point.

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.

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.

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.

Enhanced Landau–de Gennes potential for nematic liquid crystals from a systematic coarse-graining procedure

The macroscopic theory of nematics is conveniently described in terms of the phenomenological Landau-de Gennes free energy. Here we show how such an effective free energy can be obtained explicitly from a microscopic model via the help of a systematic coarse-graining procedure. We test our approach for the two- and three-dimensional Lebwohl-Lasher model of nematics. The effective free energy that we obtain is consistent with the phenomenological Landau-de Gennes form for weak orientational ordering and the Maier-Saupe theory of the isotropic-nematic transition. For strong orientational ordering, however, the effective free energy increases rapidly and diverges logarithmically near the fully oriented state. The explicit form for the regularized Landau-de Gennes potential proposed here restricts the order parameter to physical admissible values and reproduces our numerical data accurately.

A systematic coarse-graining strategy for semi-dilute copolymer solutions: from monomersto micelles

A systematic coarse-graining procedure is proposed for the description and simulation ofAB diblock copolymers in selective solvents. Each block is represented by a small number,nA ornB, of effective segments or blobs, containing a large number of microscopic monomers.nA andnB are unequivocally determined by imposing that blobs do not, on average, overlap,even if complete copolymer coils interpenetrate (semi-dilute regime). Ultra-softeffective interactions between blobs are determined by a rigorous inversionprocedure in the low concentration limit. The methodology is applied to anathermal copolymer model where A blocks are ideal (theta solvent), B blocksself-avoiding (good solvent), while A and B blocks are mutually avoiding. Themodel leads to aggregation into polydisperse spherical micelles beyond a criticalmicellar concentration determined by Monte Carlo simulations for several size ratiosf of the two blocks. The simulations also provide accurate estimates of the osmotic pressureand of the free energy of the copolymer solutions over a wide range of concentrations. Themean micellar aggregation numbers are found to be significantly lower than those predictedby an earlier, minimal two-blob representation (Capone et al 2009 J. Phys. Chem. B 113 3629).

Mesoscopic theory for inhomogeneous mixtures

Mesoscopic density functional theory for inhomogeneous mixtures of spherical particles is developed in terms of mesoscopic volume fractions by a systematic coarse-graining procedure starting form microscopic theory. Approximate expressions for the correlation functions and for the grand potential are obtained for weak ordering on mesoscopic length scales. Stability analysis of the disordered phase is performed in mean-field approximation (MF) and beyond. MF shows existence of either a spinodal or a λ-surface on the volume fraction–temperature phase diagram. Separation into homogeneous phases or formation of inhomogeneous distribution of particles occurs on the low-temperature side of the former or the latter surface respectively, depending on both the interaction potentials and the size ratios between particles of different species. Beyond MF the spinodal surface is shifted, and the instability at the λ-surface is suppressed by fluctuations. We interpret the λ-surface as a borderline between homogeneous and inhomogeneous (containing clusters or other aggregates) structure of the disordered phase. For two-component systems explicit expressions for the MF spinodal and λ-surfaces are derived. Examples of interaction potentials of simple form are analysed in some detail, in order to identify conditions leading to inhomogeneous structures.

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.

Universal sequence of ordered structures obtained from mesoscopic description of self-assembly

Mesoscopic theory for soft-matter systems that combines density functional and statistical field theory is derived by a systematic coarse-graining procedure. For particles interacting with spherically symmetric potentials of arbitrary form, the grand-thermodynamic potential consists of two terms. The first term is associated with microscopic length-scale fluctuations, and has the form of the standard density functional. The second term is associated with mesoscopic length-scale fluctuations, and has the form known from the statistical field theory. For the correlation function between density fluctuations in mesoscopic regions, a pair of equations similar to the Ornstein-Zernicke equation with a new closure is obtained. In the special case of weak ordering on the mesoscopic length scale, the theory takes a form similar to either the Landau-Ginzburg-Wilson (LGW) or the Landau-Brazovskii (LB) field theory, depending on the form of the interaction potential. Microscopic expressions for the phenomenological parameters that appear in the Landau-type theories are obtained. Within the framework of this theory, we obtain a universal sequence of phases: disordered, bcc, hexagonal, lamellar, inverted hexagonal, inverted bcc, disordered, for increasing density well below the close-packing density. The sequence of phases agrees with experimental observations and with simulations of many self-assembling systems. In addition to the above phases, more complex phases may appear depending on the interaction potentials. For a particular form of the short-range attraction long-range repulsion potential, we find the bicontinuous gyroid phase ( Ia3d symmetry) that may be related to a network-forming cluster of colloids in a mixture of colloids and nonadsorbing polymers.

Accurate Coarse-Grained Modeling of Red Blood Cells

We develop a systematic coarse-graining procedure for modeling red blood cells (RBCs) using arguments based on mean-field theory. The three-dimensional RBC membrane model takes into account the bending energy, in-plane shear energy, and constraints of fixed surface area and fixed enclosed volume. The coarse-graining procedure is general, it can be used for arbitrary level of coarse-graining and does not employ any fitting parameters. The sensitivity of the coarse-grained model is investigated and its behavior is validated against available experimental data and in dissipative particle dynamics (DPD) simulations of RBCs in capillary and shear flows.

Macroscopic thermodynamics of flowing polymers derived from systematic coarse-graining procedure

Macroscopic thermodynamics of flowing polymeric liquids in terms of a single conformation tensor is derived from atomistic or mechanical bead-spring models via a systematic coarse-graining procedure. The macroscopic equations derived here are consistent with earlier works where they have been proposed on more phenomenological grounds. The present approach provides a microscopic justification of these equations including expressions for the transport coefficients. In addition, the macroscopic equations derived here consider second-order dissipative processes that arise due to cross-correlated fluctuations on the macroscopic variables and insofar offer an extension of the previously chosen level of description.

COARSE GRAINING IN SIMULATED CELL POPULATIONS

The main mechanisms that control the organization of multicellular tissues are still largely open. A commonly used tool to study basic control mechanisms are in vitro experiments in which the growth conditions can be widely varied. However, even in vitro experiments are not free from unknown or uncontrolled influences. One reason why mathematical models become more and more a popular complementary tool to experiments is that they permit the study of hypotheses free from unknown or uncontrolled influences that occur in experiments. Many model types have been considered so far to model multicellular organization ranging from detailed individual-cell based models with explicit representations of the cell shape to cellular automata models with no representation of cell shape, and continuum models, which consider a local density averaged over many individual cells. However, how the different model description may be linked, and, how a description on a coarser level may be constructed based on the knowledge of the finer, microscopic level, is still largely unknown. Here, we consider the example of monolayer growth in vitro to illustrate how, in a multi-step process starting from a single-cell based off-lattice-model that subsumes the information on the sub-cellular scale by characteristic cell-biophysical and cell-kinetic properties, a cellular automaton may be constructed whose rules have been chosen based on the findings in the off-lattice model. Finally, we use the cellular automaton model as a starting point to construct a multivariate master equation from a compartment approach from which a continuum model can be derived by a systematic coarse-graining procedure. We find that the resulting continuum equation largely captures the growth behavior of the CA model. The development of our models is guided by experimental observations on growing monolayers.

Brownian dynamics simulations of bead-rod and bead-spring chains: numerical algorithms and coarse-graining issues

The efficiency and robustness of various numerical schemes have been evaluated by performing Brownian dynamics simulations of bead-rod and three popular nonlinear bead-spring chain models in uniaxial extension and simple shear flow. The bead-spring models include finitely extensible nonlinear elastic (FENE) springs, worm-like chain (WLC) springs, and Pade approximation to the inverse Langevin function (ILC) springs. For the bead-spring chains two new predictor–corrector algorithms are proposed, which are much superior to commonly used explicit and other fully implicit schemes. In the case of bead-rod chain models, the mid-point algorithm of Liu [J. Chem. Phys. 90 (1989) 5826] is found to be computationally more efficient than a fully implicit Newton’s method. Furthermore, the accuracy and computational efficiency of two different stress expressions for the bead-rod chains, namely the Kramers–Kirkwood and the modified Giesekus have been evaluated under both transient and steady conditions. It is demonstrated that the Kramers–Kirkwood with stochastic filtering is the preferred choice for transient flow while the Giesekus expression is better suited for steady state calculations. The issue of coarse graining from a bead-rod chain to a bead-spring chain has also been investigated. Though bead-spring chains are shown to capture only semi-quantitatively the response of the bead-rod chains in transient flows, a systematic coarse-graining procedure that provides the best description of bead-rod chains via bead-spring chains is presented.

On survival dynamics of classical systems. Non-chaotic open billiards

We report on decay problem of classical systems. Mesoscopic level consideration is given on the basis of transient dynamics of non-interacting classical particles bounded in billiards. Three distinct decay channels are distinguished through the long-tailed memory effects revealed by temporal behavior of survival probability t − α : (i) the universal (independent of geometry, initial conditions and space dimension) channel with α=1 of Brownian relaxation of non-trapped regular parabolic trajectories and (ii) the non-Brownian channel α<1 associated with subdiffusion relaxation motion of irregular nearly trapped parabolic trajectories. These channels are common of non-fully chaotic systems, including the non-chaotic case. In the fully chaotic billiards the (iii) decay channel is given by α>1 due to “highly chaotic bouncing ball” trajectories. We develop a statistical approach to the problem, earlier proposed for chaotic classical systems (Physica A 275 (2000) 70). A systematic coarse-graining procedure is introduced for non-chaotic systems (exemplified by circle and square geometry), which are characterized by a certain finite characteristic collision time. We demonstrate how the transient dynamics is related to the intrinsic dynamics driven by the preserved Liouville measure. The detailed behavior of the late-time survival probability, including a role of the initial conditions and a system geometry, is studied in detail, both theoretically and numerically.