Coarse-grained Equations Of Motion Research Articles

Microscopic derivation of particle-based coarse-grained dynamics

In this paper we revisit the derivation of equations of motion for coarse-grained (CG) particles from the microscopic Hamiltonian dynamics of the underlying atomistic system in equilibrium. The derivation is based on the projection operator method and time-convolution equation. We demonstrate that due to the energy exchange between CG and intraparticle phase space coordinates in the microscopic system, the choice of projection operator is not unique, leading to different CG equations of motion that have the form of the nonlinear generalized Langevin equation (GLE). We derive the idempotence properties for the projection operators along the system trajectories and show that these properties result in streaming terms of the respective GLEs that are conservative forces and allow the expression of the non-conservative forces explicitly through thermodynamic averages, which can be measured from the microscopic simulations. The difference between GLEs that are presented herein lies in how the non-conservative forces are partitioned into dissipative and projected contributions. We compute the projected force and analyze conditions under which the projected (stochastic) force is orthogonal to (uncorrelated with) the momenta of CG particles, therefore justifying a transition to a framework of stochastic differential equations. We show that a position- and momentum-independent memory function appears only if the projected force is fully decoupled from the past CG positions and momenta, respectively. In the case of non-vanishing correlations between the projected force and the CG coordinates in past times, we derive explicitly the position- and momentum-dependent memory function in a form of projection onto a space spanned by N-order Hermite polynomials. The expressions presented herein can be used to construct a hierarchy of thermodynamically consistent CG models with momentum-dependent memory functions. They can also be used to design computational schemes for obtaining the parameters for GLEs and their variants such as dissipative particle dynamics equations from the microscopic data. We illustrate these applications by presenting the GLE with a memory function that is quadratic in the particle momenta.

Analyses of coarse graining procedure for mesoscale simulation

In order to meet great progress in multi-scale modelling, one of the important issues is to know about detailed process of coarse graining. The coarse graining is a standard technique of statistical mechanics to extract specific degree(s) of freedom from a many component system. Many kinds of coarse-graining simulation methods have already been proposed. Because most of them were constructed based on a priori assumptions, making detailed analyses has been difficult from the microscopic points of view. Recently, a generalised coarse-grained (CG) equation of motion has been reported for many particle centre of mass systems straightforwardly derived from the explicit information about all the degrees of freedom at the microscopic level. This equation is mostly exact for the dynamics of CG particles defined in classical many particle systems. Therefore, making discussions about characteristics of this CG equation of motion is useful for obtaining detailed knowledge on the coarse graining from the microscopic points of view. In the present article, a review on recent discussions about characteristics of this generalised CG equation of motion is given. Previous conventional coarse-graining particle dynamics simulations can be categorised as some special cases. Generalisation, confirmation and progress of the fundamental knowledge of coarse graining are partly realised from the present discussions.

Coarse-grained equation of motion for many particle system containing internal degrees of freedom

In order to numerically investigate dynamics in large and complicated systems, it is important to derive the explicit equation of motion of an extracted tagged degree of freedom under the statistical influence of the other parts. This corresponds to adopting the procedure usually called “coarse-graining”. Up-to-now, many kinds of coarse grained (CG) simulation method have been proposed based on phenomenological equations of motion and investigated about their numerical efficiencies for practical applications. We can now investigate a dynamics of appropriate degrees of freedom, i.e., the center of mass, the orientation, the internal vibration modes, etc. of CG particles, for suitably chosen characteristics in a system. Although each of the phenomenological CG methods is well established for practical usages, it is still not sufficient to construct a multiscale simulation procedure for large and complicated systems. Some explicit relation between the coarser and finer descriptions should be grasped for deducing the CG expression based on atomistic pictures. Although Langevin equation, a typical CG equation of motion, has been derived from an atomistic equation of motion in previous literatures, most of these descriptions are limited within the equation of center-of-mass motions. In the present study, a microscopic description for arbitrarily extracted degree(s) of freedom was derived by restricting the system of CG particles constructed by tightly bounded finer particles, like polyatomic molecules. The derived equation of motion was shown to be applicable to dynamics simulation of reorientation and/or internal vibration motions of CG-particle systems by analyses of the equation for realistic conditions.

Tension dynamics in semiflexible polymers. I. Coarse-grained equations of motion

Based on the wormlike chain model, a coarse-grained description of the nonlinear dynamics of a weakly bending semiflexible polymer is developed. By means of a multiple-scale perturbation analysis, a length-scale separation inherent to the weakly bending limit is exploited to reveal the deterministic nature of the spatio temporal relaxation of the backbone tension and to deduce the corresponding coarse-grained equation of motion. From this partial integro-differential equation, some detailed analytical predictions for the nonlinear response of a weakly bending polymer are derived in an accompanying paper [O. Hallatschek, following paper, Phys. Rev. E 75, 031906 (2007)].

Theory of disordered flux-line liquids

We study the equilibrium statics and nonequilibrium driven dynamics of flux-line liquids in the presence of a random pinning potential. Under the assumption of replica symmetry, we find in the static case using a replica Gaussian variational method that the only effect of disorder is to increase the tilt modulus and the confining ``mass'' of the internal modes of the flux lines, thus decreasing their thermal wandering. In the nonequilibrium driven case, we derive the long-scale, coarse-grained equation of motion of the vortices in the presence of disorder, which, apart from new Kardar-Parisi-Zhang nonlinearities, has the same form as the equation of motion for unpinned vortices, with renormalized coefficients. We also compute the structure factor of a disordered vortex liquid, and show that the disorder contributes an additive, Lorentzian squared term, to the structure factor of the center of mass mode, otherwise leaving the functional form of terms describing the internal modes unchanged. The expression of the static structure factor derived within our approach is consistent both with experimental data and with the standard theory of elasticity of vortices in high temperature superconductors.

Imbibition in disordered media

The physics of liquids in porous media gives rise to many interesting phenomena, including imbibition where a viscous fluid displaces a less viscous one. Here we discuss the theoretical and experimental progress made in recent years in this field. The emphasis is on an interfacial description, akin to the focus of a statistical physics approach. Coarse-grained equations of motion have been recently presented in the literature. These contain terms that take into account the pertinent features of imbibition: non-locality and the quenched noise that arises from the random environment, fluctuations of the fluid flow and capillary forces. The theoretical progress has highlighted the presence of intrinsic length-scales that invalidate scale invariance often assumed to be present in kinetic roughening processes such as that of a two-phase boundary in liquid penetration. Another important fact is that the macroscopic fluid flow, the kinetic roughening properties, and the effective noise in the problem are all coupled. Many possible deviations from simple scaling behaviour exist, and we outline the experimental evidence. Finally, prospects for further work, both theoretical and experimental, are discussed.

Driven periodic structures in random environment

We consider a lattice driven in a random environment and analyse its large-scale dynamics in the context of the driven vortex configuration. Using a perturbative coarse-graing procedure, we derive explicitely renormalized equation of motion, including Kardar-Parizi-Zhang nonlinearities and dynamic strain terms. From the generalized Lindemann criterion for the fluctuations of the neighboring sites the temperature shift of the dynamic melting is found to scale proportional to the strength of the disorder and inversly proportional to large driving forces. The presence of dislocations leads to a characteristic anisotropic distortion of the vortex density that is controlled by a Kardar-Parisi-Zhang nonlinearity in the coarse-grained equation of motion. This nonlinearity also implies a screening of the interaction between dislocations and thereby an instability of the vortex lattice to the proliferation of free dislocations.