Coarse-grained Variables Research Articles

Intermittent direction reversals of moving spatially localized turbulence observed in two-dimensional Kolmogorov flow.

We have found that in two-dimensional Kolmogorov flow a spatially localized turbulent state (SLT) exists stably and travels with a constant speed on average switching the moving direction randomly and intermittently for moderate values of the control parameters: Reynolds number and the flow rate. We define the coarse-grained position and velocity of an SLT and separate the motion of the SLT from its internal turbulent dynamics by introducing a co-moving frame. The switching process of an SLT represented by the coarse-grained velocity seems to be a random telegraph signal. Focusing on the asymmetry of the internal turbulence we introduce two coarse-grained variables characterizing the internal dynamics. These quantities follow the switching process reasonably. This suggests that the twin attracting invariant sets each of which corresponds to a one-way traveling SLT are embedded in the attractor of the moving SLT and the connection of the two sets is too complicated to be represented by a few degrees of freedom but the motion of an SLT is correlated with the internal turbulent dynamics.

Coarse analysis of collective behaviors: Bifurcation analysis of the optimal velocity model for traffic jam formation

We present a general method for analyzing macroscopic collective phenomena observed in many-body systems. For this purpose, we employ diffusion maps, which are one of the dimensionality-reduction techniques, and systematically define a few relevant coarse-grained variables for describing macroscopic phenomena. The time evolution of macroscopic behavior is described as a trajectory in the low-dimensional space constructed by these coarse variables. We apply this method to the analysis of the traffic model, called the optimal velocity model, and reveal a bifurcation structure, which features a transition to the emergence of a moving cluster as a traffic jam.

Parameterization of Coarse-Grained Molecular Interactions through Potential of Mean Force Calculations and Cluster Expansion Techniques

We present a systematic coarse-graining (CG) strategy for many particle molecular systems based on cluster expansion techniques. We construct a hierarchy of coarse-grained Hamiltonians with interaction potentials consisting of two, three and higher body interactions. In this way, the suggested model becomes computationally tractable, since no information from long n-body (bulk) simulations is required in order to develop it, while retaining the fluctuations at the coarse-grained level. The accuracy of the derived cluster expansion based on interatomic potentials is examined over a range of various temperatures and densities and compared to direct computation of the pair potential of mean force. The comparison of the coarse-grained simulations is done on the basis of the structural properties, against detailed all-atom data. On the other hand, by construction, the approximate coarse-grained models retain, in principle, the thermodynamic properties of the atomistic model without the need for any further parameter fitting. We give specific examples for methane and ethane molecules in which the coarse-grained variable is the centre of mass of the molecule. We investigate different temperature (T) and density ( ρ ) regimes, and we examine differences between the methane and ethane systems. Results show that the cluster expansion formalism can be used in order to provide accurate effective pair and three-body CG potentials at high T and low ρ regimes. In the liquid regime, the three-body effective CG potentials give a small improvement over the typical pair CG ones; however, in order to get significantly better results, one needs to consider even higher order terms.

The entropy of a complex molecule.

Entropy is a central concept in the theory of coarse-graining. Through Einstein's formula, it provides the equilibrium probability distribution of the coarse-grained variables used to describe the system of interest. We study with molecular dynamics simulations the equilibrium probability distribution of thermal blobs representing at a coarse-grained level star polymer molecules in melt. Thermal blobs are characterized by the positions and momenta of the centers of mass, and internal energies of the molecules. We show that the entropy of the level of description of thermal blobs can be very well approximated as the sum of the thermodynamic entropy of each single molecule considered as isolated thermodynamic systems. The entropy of a single molecule depends on the intrinsic energy, involving only contributions from the atoms that make the molecule and not from the interactions with atoms of other molecules.

Data-driven parameterization of the generalized Langevin equation

We present a data-driven approach to determine the memory kernel and random noise in generalized Langevin equations. To facilitate practical implementations, we parameterize the kernel function in the Laplace domain by a rational function, with coefficients directly linked to the equilibrium statistics of the coarse-grain variables. We show that such an approximation can be constructed to arbitrarily high order and the resulting generalized Langevin dynamics can be embedded in an extended stochastic model without explicit memory. We demonstrate how to introduce the stochastic noise so that the second fluctuation-dissipation theorem is exactly satisfied. Results from several numerical tests are presented to demonstrate the effectiveness of the proposed method.

Reverse Coarse-Graining for Equation-Free Modeling: Application to Multiscale Molecular Dynamics.

Constructing atom-resolved states from low-resolution data is of practical importance in many areas of science and engineering. This problem is addressed in this article in the context of multiscale factorization methods for molecular dynamics. These methods capture the crosstalk between atomic and coarse-grained scales arising in macromolecular systems. This crosstalk is accounted for by Trotter factorization, which is used to separate the all-atom from the coarse-grained phases of the computation. In this approach, short molecular dynamics runs are used to advance in time the coarse-grained variables, which in turn guide the all-atom state. To achieve this coevolution, an all-atom microstate consistent with the updated coarse-grained variables must be recovered. This recovery is cast here as a nonlinear optimization problem that is solved with a quasi-Newton method. The approach yields a Boltzmann-relevant microstate whose coarse-grained representation and some of its fine-scale features are preserved. Embedding this algorithm in multiscale factorization is shown to be accurate and scalable for simulating proteins and their assemblies.

Equation-free analysis of a dynamically evolving multigraph

In order to illustrate the adaptation of traditional continuum numerical techniques to the study of complex network systems, we use the equation-free framework to analyze a dynamically evolving multigraph. This approach is based on coupling short intervals of direct dynamic network simulation with appropriately-defined lifting and restriction operators, mapping the detailed network description to suitable macroscopic (coarse-grained) variables and back. This enables the acceleration of direct simulations through Coarse Projective Integration (CPI), as well as the identification of coarse stationary states via a Newton-GMRES method. We also demonstrate the use of data-mining, both linear (principal component analysis, PCA) and nonlinear (diffusion maps, DMAPS) to determine good macroscopic variables (observables) through which one can coarse-grain the model. These results suggest methods for decreasing simulation times of dynamic real-world systems such as epidemiological network models. Additionally, the data-mining techniques could be applied to a diverse class of problems to search for a succint, low-dimensional description of the system in a small number of variables.

On the representability problem and the physical meaning of coarse-grained models.

In coarse-grained (CG) models where certain fine-grained (FG, i.e., atomistic resolution) observables are not directly represented, one can nonetheless identify indirect the CG observables that capture the FG observable's dependence on CG coordinates. Often, in these cases it appears that a CG observable can be defined by analogy to an all-atom or FG observable, but the similarity is misleading and significantly undermines the interpretation of both bottom-up and top-down CG models. Such problems emerge especially clearly in the framework of the systematic bottom-up CG modeling, where a direct and transparent correspondence between FG and CG variables establishes precise conditions for consistency between CG observables and underlying FG models. Here we present and investigate these representability challenges and illustrate them via the bottom-up conceptual framework for several simple analytically tractable polymer models. The examples provide special focus on the observables of configurational internal energy, entropy, and pressure, which have been at the root of controversy in the CG literature, as well as discuss observables that would seem to be entirely missing in the CG representation but can nonetheless be correlated with CG behavior. Though we investigate these problems in the framework of systematic coarse-graining, the lessons apply to top-down CG modeling also, with crucial implications for simulation at constant pressure and surface tension and for the interpretations of structural and thermodynamic correlations for comparison to experiment.

Holography as a highly efficient renormalization group flow. II. An explicit construction

We complete the reformulation of the holographic correspondence as a \emph{highly efficient RG flow} that can also determine the UV data in the field theory in the strong coupling and large $N$ limit. We introduce a special way to define operators at any given scale in terms of appropriate coarse-grained collective variables, without requiring the use of the elementary fields. The Wilsonian construction is generalised by promoting the cut-off to a functional of these collective variables. We impose three criteria to determine the coarse-graining. The first criterion is that the effective Ward identities for local conservation of energy, momentum, etc. should preserve their standard forms, but in new scale-dependent background metric and sources which are functionals of the effective single trace operators. The second criterion is that the scale-evolution equations of the operators in the actual background metric should be state-independent, implying that the collective variables should not explicitly appear in them. The final criterion is that the endpoint of the scale-evolution of the RG flow can be transformed to a fixed point corresponding to familiar non-relativistic equations with a finite number of parameters, such as incompressible non-relativistic Navier-Stokes, under a certain universal rescaling of the scale and of the time coordinate. Using previous work, we explicitly show that in the hydrodynamic limit each such highly efficient RG flow reproduces a unique classical gravity theory with precise UV data that satisfy our IR criterion. We obtain the explicit coarse-graining which reproduces Einstein's equations. In a simple example, we are also able to compute the beta function. Finally, we show how our construction can be interpolated with the traditional Wilsonian RG flow at a suitable scale, and can be used to develop new non-perturbative frameworks for QCD-like theories.

ProtoMD: A prototyping toolkit for multiscale molecular dynamics

ProtoMD is a toolkit that facilitates the development of algorithms for multiscale molecular dynamics (MD) simulations. It is designed for multiscale methods which capture the dynamic transfer of information across multiple spatial scales, such as the atomic to the mesoscopic scale, via coevolving microscopic and coarse-grained (CG) variables. ProtoMD can be also be used to calibrate parameters needed in traditional CG-MD methods. The toolkit integrates ‘GROMACS wrapper’ to initiate MD simulations, and ‘MDAnalysis’ to analyze and manipulate trajectory files. It facilitates experimentation with a spectrum of coarse-grained variables, prototyping rare events (such as chemical reactions), or simulating nanocharacterization experiments such as terahertz spectroscopy, AFM, nanopore, and time-of-flight mass spectroscopy. ProtoMD is written in python and is freely available under the GNU General Public License from github.com/CTCNano/proto_md. Program summaryProgram title: ProtoMDCatalogue identifier: AEZN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 281345No. of bytes in distributed program, including test data, etc.: 3706191Distribution format: tar.gzProgramming language: Python 2.7.3.Computer: x86/x86 64.Operating system: Linux.RAM: Depends on the size of the system being simulated and duration of the simulation (few MBs to TBs)Classification: 3, 4.12, 16.9, 23.External routines: GROMACS (tested with versions 4.6x and 5.0x), MDAnalysis, GromacsWrapper, numpy, scipy, MPINature of problem: Prototyping multiscale coarse-grained algorithms for molecular dynamics.Solution method: Combining the open-source GROMACS molecular dynamics package and the python-based MDAnalysis library for running, debugging, and analyzing multiscale simulationsRestrictions: The system under study must be characterized by a clear separation of timescales; otherwise, the multiscale algorithm fails to capture the slowly-varying modes.Running time: Depending on the problem size, simulations can take few hours to months.

Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism.

The Mori-Zwanzig formalism for coarse-graining a complex dynamical system typically introduces memory effects. The Markovian assumption of delta-correlated fluctuating forces is often employed to simplify the formulation of coarse-grained (CG) models and numerical implementations. However, when the time scales of a system are not clearly separated, the memory effects become strong and the Markovian assumption becomes inaccurate. To this end, we incorporate memory effects into CG modeling by preserving non-Markovian interactions between CG variables, and the memory kernel is evaluated directly from microscopic dynamics. For a specific example, molecular dynamics (MD) simulations of star polymer melts are performed while the corresponding CG system is defined by grouping many bonded atoms into single clusters. Then, the effective interactions between CG clusters as well as the memory kernel are obtained from the MD simulations. The constructed CG force field with a memory kernel leads to a non-Markovian dissipative particle dynamics (NM-DPD). Quantitative comparisons between the CG models with Markovian and non-Markovian approximations indicate that including the memory effects using NM-DPD yields similar results as the Markovian-based DPD if the system has clear time scale separation. However, for systems with small separation of time scales, NM-DPD can reproduce correct short-time properties that are related to how the system responds to high-frequency disturbances, which cannot be captured by the Markovian-based DPD model.

Parametric reduced models for the nonlinear Schrödinger equation.

Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored-noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parametrization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.

Locating landmarks on high-dimensional free energy surfaces

Coarse graining of complex systems possessing many degrees of freedom can often be a useful approach for analyzing and understanding key features of these systems in terms of just a few variables. The relevant energy landscape in a coarse-grained description is the free energy surface as a function of the coarse-grained variables, which, despite the dimensional reduction, can still be an object of high dimension. Consequently, navigating and exploring this high-dimensional free energy surface is a nontrivial task. In this paper, we use techniques from multiscale modeling, stochastic optimization, and machine learning to devise a strategy for locating minima and saddle points (termed "landmarks") on a high-dimensional free energy surface "on the fly" and without requiring prior knowledge of or an explicit form for the surface. In addition, we propose a compact graph representation of the landmarks and connections between them, and we show that the graph nodes can be subsequently analyzed and clustered based on key attributes that elucidate important properties of the system. Finally, we show that knowledge of landmark locations allows for the efficient determination of their relative free energies via enhanced sampling techniques.

Invariance of steady-state thermodynamics between different scales of description.

By considering general Markov stochastic dynamics and its coarse graining, we study the framework of stochastic thermodynamics for the original and reduced descriptions corresponding to different scales. We are especially concerned with the case where the irreversible entropy production has a finite difference between the scales. We find that the sum of the increment of the nonequilibrium entropy and the excess part of the entropy production, which are key quantities in the construction of steady-state thermodynamics, is essentially kept invariant with respect to the change in the scales of description. This general result justifies experimental approaches toward steady-state thermodynamics based on coarse-grained variables. We demonstrate our result in a mesoscopic heat engine system.

Diffusion Wavelet Decomposition for Coarse-Graining of Polymer Chains

ABSTRACTHere we present an alternative approach to coarse graining, based on the multiresolution diffusion-wavelet approach to operator compression, which does not require explicit atomistic-to-coarse-grained mappings. Our diffusion-wavelet method takes as input the topology and sparsity of the molecular bonding structure of a system, and returns as output a hierarchical set of degrees of freedom (DoFs) of system-specific coarse-grained variables. Importantly, the hierarchical compression provides a clear framework for modeling at many model scales (levels), beyond the common two-level CG representation. Our results show that the resulting hierarchy separates localized modes, such as a single C-C vibrational mode, from larger-scale motions, e.g., long-range concerted backbone vibrational modes. Our approach correctly captures small-scale chemical features, such as cellulose ring structures, and alkane side chains or CH2 units, as well as large-scale features of the backbone. In particular, the new method’s finest-scale modes describe DoFs similar to united atom models and other chemically-defined CG models. Modes at coarser levels describe increasingly large connected portions of the target polymers. For polyethylene and polystyrene, spatial coordinates and their associated forces were compressed by up to two orders of magnitude. The compression in forces is of particular interest as this allows larger timesteps as well as reducing the number of DoFs.

Unraveling the Mystery of ATP Hydrolysis in Actin Filaments

Actin performs its myriad cellular functions by the growth and disassembly of its filamentous form. The hydrolysis of ATP in the actin filament has been shown to modulate properties of the filament, thus making it a pivotal regulator of the actin life cycle. Actin has evolved to selectively hydrolyze ATP in the filamentous form, F-actin, with an experimentally observed rate increase over the monomeric form, G-actin, of 4.3 × 104. The cause of this dramatic increase in rate is investigated in this paper using extensive QM/MM simulations of both G- and F-actin. To compute the free energy of hydrolysis in both systems, metadynamics is employed along two collective variables chosen to describe the reaction coordinates of hydrolysis. F-actin is modeled as a monomer with restraints applied to coarse-grained variables enforced to keep it in a filament-like conformation. The simulations reveal a barrier height reduction for ATP hydrolysis in F-actin as compared to G-actin of 8 ± 1 kcal/mol, in good agreement with...

Momentum conserving Brownian dynamics propagator for complex soft matter fluids.

We present a Galilean invariant, momentum conserving first order Brownian dynamics scheme for coarse-grained simulations of highly frictional soft matter systems. Friction forces are taken to be with respect to moving background material. The motion of the background material is described by locally averaged velocities in the neighborhood of the dissolved coarse coordinates. The velocity variables are updated by a momentum conserving scheme. The properties of the stochastic updates are derived through the Chapman-Kolmogorov and Fokker-Planck equations for the evolution of the probability distribution of coarse-grained position and velocity variables, by requiring the equilibrium distribution to be a stationary solution. We test our new scheme on concentrated star polymer solutions and find that the transverse current and velocity time auto-correlation functions behave as expected from hydrodynamics. In particular, the velocity auto-correlation functions display a long time tail in complete agreement with hydrodynamics.

A tracer-based inversion method for diagnosing eddy-induced diffusivity and advection

A diagnosis method is presented which inverts a set of tracer flux statistics into an eddy-induced transport intended to apply for all tracers. The underlying assumption is that a linear flux-gradient relationship describes eddy-induced tracer transport, but a full tensor coefficient is assumed rather than a scalar coefficient which allows for down-gradient and skew transports. Thus, Lagrangian advection and anisotropic diffusion not necessarily aligned with the tracer gradient can be diagnosed. In this method, multiple passive tracers are initialized in an eddy-resolving flow simulation. Their spatially-averaged gradients form a matrix, where the gradient of each tracer is assumed to satisfy an identical flux-gradient relationship. The resulting linear system, which is overdetermined when using more than three tracers, is then solved to obtain an eddy transport tensor R which describes the eddy advection (antisymmetric part of R) and potentially anisotropic diffusion (symmetric part of R) in terms of coarse-grained variables. The mathematical basis for this inversion method is presented here, along with practical guidelines for its implementation. We present recommendations for initialization of the passive tracers, maintaining the required misalignment of the tracer gradients, correcting for nonconservative effects, and quantifying the error in the diagnosed transport tensor. A method is proposed to find unique, tracer-independent, distinct rotational and divergent Lagrangian transport operators, but the results indicate that these operators are not meaningfully relatable to tracer-independent eddy advection or diffusion. With the optimal method of diagnosis, the diagnosed transport tensor is capable of predicting the fluxes of other tracers that are withheld from the diagnosis, including even active tracers such as buoyancy, such that relative errors of 14% or less are found.

Importance of crustal structure and anelastic attenuation for estimating ground motion parameters by finite difference simulation

We carry out simulations of seismic wave propagation in anelastic media in order to study the relative importance of source parameters as well as viscoelastic structures in affecting the decay of the ground motions. First, we verify the efficiency of the implementation in a finite difference code of two coarse-grain memory variables methods to model the anelastic behavior of the soil. We find that both methods are sufficiently consistent for a quality factor (Q) larger than 20. Then, we study the relative importance of the focal mechanism, the magnitude, the source depth, the crustal structure and the quality factor in affecting the decay of the ground motions. We verify that the magnitude and the focal mechanism of the source do not have a significant effect on the decay, whereas the focal depth is more important in explaining the variations in decay. The variations of the decay depending on the crustal structure are more difficult to assess. For the shorter distances (up to about 20 km), the velocity structure does not have a significant effect on the decay of the ground motions. The effect of the quality factor is perceptible but remains less important than the effect of the focal depth. However, for epicentral distances larger than about 20 km, both the velocity structure and the quality factor begin to affect significantly the decay. The effect of the quality factor on the decay becomes then even more important than the effect of the focal depth. Therefore, a reduction in the standard deviation of GMPEs could possibly be achieved through taking into account appropriate anelastic attenuation for each region considered. The effect of a 3D velocity structure is studied introducing a typical basin structure. The presence of a sedimentary basin can affect the decay even outside the basin. The spatial difference in ground motion is more pronounced in the elastic case than in the anelastic case.

Unraveling the mystery of ATP hydrolysis in actin filaments.

Actin performs its myriad cellular functions by the growth and disassembly of its filamentous form. The hydrolysis of ATP in the actin filament has been shown to modulate properties of the filament, thus making it a pivotal regulator of the actin life cycle. Actin has evolved to selectively hydrolyze ATP in the filamentous form, F-actin, with an experimentally observed rate increase over the monomeric form, G-actin, of 4.3 × 104. The cause of this dramatic increase in rate is investigated in this paper using extensive QM/MM simulations of both G- and F-actin. To compute the free energy of hydrolysis in both systems, metadynamics is employed along two collective variables chosen to describe the reaction coordinates of hydrolysis. F-actin is modeled as a monomer with restraints applied to coarse-grained variables enforced to keep it in a filament-like conformation. The simulations reveal a barrier height reduction for ATP hydrolysis in F-actin as compared to G-actin of 8 ± 1 kcal/mol, in good agreement with the experimentally measured barrier height reduction of 7 ± 1 kcal/mol. The barrier height reduction is influenced by an enhanced rotational diffusion of water in F-actin as compared to G-actin and shorter water wires between Asp154 and the nucleophilic water in F-actin, leading to more rapid proton transport.