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.

[1]  Eric Darve,et al.  Computing generalized Langevin equations and generalized Fokker–Planck equations , 2009, Proceedings of the National Academy of Sciences.

[2]  R. Tolman.  A general theory of energy partition with applications to quantum theory , 1918 .

[3]  Avisek Das,et al.  The multiscale coarse-graining method. V. Isothermal-isobaric ensemble. , 2010, The Journal of chemical physics.

[4]  P. Español.  Hybrid description of complex molecules , 2009 .

[5]  Alexandre J. Chorin,et al.  Optimal prediction with memory , 2002 .

[6]  A J Chorin,et al.  Optimal prediction and the Mori-Zwanzig representation of irreversible processes. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Avisek Das,et al.  The multiscale coarse-graining method. VIII. Multiresolution hierarchical basis functions and basis function selection in the construction of coarse-grained force fields. , 2012, The Journal of chemical physics.

[8]  T. Komatsuzaki,et al.  Derivation of the generalized Langevin equation in nonstationary environments. , 2011, The Journal of chemical physics.

[9]  Jhih-Wei Chu,et al.  Emerging methods for multiscale simulation of biomolecular systems , 2007 .

[10]  James D. Gunton,et al.  Theory of Nonlinear Transport Processes: Nonlinear Shear Viscosity and Normal Stress Effects , 1973 .

[11]  Oliver F. Lange,et al.  Collective Langevin dynamics of conformational motions in proteins. , 2006, The Journal of chemical physics.

[12]  Microscopic derivation of causal diffusion equation using the projection operator method. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Sergei Izvekov,et al.  The multiscale coarse-graining method: assessing its accuracy and introducing density dependent coarse-grain potentials. , 2010, The Journal of chemical physics.

[14]  Transport coefficients of non-Newtonian fluid and causal dissipative hydrodynamics. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Sture Nordholm,et al.  A systematic derivation of exact generalized Brownian motion theory , 1975 .

[16]  Avisek Das,et al.  The multiscale coarse-graining method. IX. A general method for construction of three body coarse-grained force fields. , 2012, The Journal of chemical physics.

[17]  H. Mori,et al.  Statistical-Mechanical Theory of Random Frequency Modulations and Generalized Brownian Motions , 1976 .

[18]  Master equations for coupled systems , 1979 .

[19]  Kyozi Kawasaki,et al.  Simple derivations of generalized linear and nonlinear Langevin equations , 1973 .

[20]  M Scott Shell,et al.  Relative entropy as a universal metric for multiscale errors. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  C. R. Willis,et al.  Time-dependent projection-operator approach to master equations for coupled systems , 1974 .

[22]  J. Andrew McCammon,et al.  Generalized Langevin dynamics simulations with arbitrary time‐dependent memory kernels , 1983 .

[23]  P. Español,et al.  Derivation of dynamical density functional theory using the projection operator technique. , 2009, The Journal of chemical physics.

[24]  M Scott Shell,et al.  Systematic coarse-graining of potential energy landscapes and dynamics in liquids. , 2012, The Journal of chemical physics.

[25]  George Em Karniadakis,et al.  Direct construction of mesoscopic models from microscopic simulations. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Dissipative particle dynamics for a harmonic chain: A first-principles derivation. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  Reinier L C Akkermans,et al.  Coarse-grained dynamics of one chain in a polymer melt , 2000 .

[28]  Gregory A Voth,et al.  A multiscale coarse-graining method for biomolecular systems. , 2005, The journal of physical chemistry. B.

[29]  Gregory A Voth,et al.  The multiscale coarse-graining method. II. Numerical implementation for coarse-grained molecular models. , 2008, The Journal of chemical physics.

[30]  B. Berne,et al.  Non‐Markovian activated rate processes: Comparison of current theories with numerical simulation data , 1986 .

[31]  Gregory A. Voth,et al.  The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models. , 2008, The Journal of chemical physics.

[32]  Avisek Das,et al.  The multiscale coarse-graining method. III. A test of pairwise additivity of the coarse-grained potential and of new basis functions for the variational calculation. , 2009, The Journal of chemical physics.

[33]  P. B. Warren,et al.  DISSIPATIVE PARTICLE DYNAMICS : BRIDGING THE GAP BETWEEN ATOMISTIC AND MESOSCOPIC SIMULATION , 1997 .

[34]  S. Nakajima.  On Quantum Theory of Transport Phenomena Steady Diffusion , 1958 .

[35]  C. Colosqui.  High-order hydrodynamics via lattice Boltzmann methods. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Gregory A Voth,et al.  Multiscale coarse graining of liquid-state systems. , 2005, The Journal of chemical physics.

[37]  P. Español,et al.  Microscopic derivation of discrete hydrodynamics. , 2009, The Journal of chemical physics.

[38]  Sauro Succi,et al.  Bottom-up coarse-graining of a simple graphene model: the blob picture. , 2011, The Journal of chemical physics.

[39]  Gregory A. Voth,et al.  The Multiscale Coarse- Graining Method: A Systematic Approach to Coarse-Graining , 2008 .

[40]  R. Zwanzig.  Ensemble Method in the Theory of Irreversibility , 1960 .

[41]  Margaret E. Johnson,et al.  Representability problems for coarse-grained water potentials. , 2007, The Journal of chemical physics.

[42]  Baldwin Robertson,et al.  Equations of Motion in Nonequilibrium Statistical Mechanics , 1966 .

[43]  Shi-aki Hyodo,et al.  Equation of motion for coarse-grained simulation based on microscopic description. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Gregory A Voth,et al.  The multiscale coarse-graining method. X. Improved algorithms for constructing coarse-grained potentials for molecular systems. , 2012, The Journal of chemical physics.

[45]  Florian Müller-Plathe,et al.  Coarse-graining in polymer simulation: from the atomistic to the mesoscopic scale and back. , 2002, Chemphyschem : a European journal of chemical physics and physical chemistry.

[46]  Gregory A Voth,et al.  The multiscale coarse-graining method. VI. Implementation of three-body coarse-grained potentials. , 2010, The Journal of chemical physics.

[47]  Weihai Fang,et al.  Semi-bottom-up coarse graining of water based on microscopic simulations. , 2011, The Journal of chemical physics.

[48]  Kurt Kremer,et al.  Multiscale simulation of soft matter systems. , 2010, Faraday discussions.

[49]  E. Vanden-Eijnden,et al.  Mori-Zwanzig formalism as a practical computational tool. , 2010, Faraday discussions.

[50]  Gregory A Voth,et al.  The multiscale coarse-graining method. IV. Transferring coarse-grained potentials between temperatures. , 2009, The Journal of chemical physics.

[51]  Microscopic formula for transport coefficients of causal hydrodynamics. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  M Scott Shell,et al.  Coarse-graining errors and numerical optimization using a relative entropy framework. , 2011, The Journal of chemical physics.

[53]  Robert Zwanzig,et al.  Memory Effects in Irreversible Thermodynamics , 1961 .

[54]  Robert Zwanzig,et al.  On the identity of three generalized master equations , 1964 .

[55]  Jianhua Xing,et al.  Application of the projection operator formalism to non-hamiltonian dynamics. , 2009, The Journal of chemical physics.

[56]  H. Mori.  Transport, Collective Motion, and Brownian Motion , 1965 .