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\begin{document} |
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\title{An algorithm for performing Langevin dynamics on rigid bodies of arbitrary shape } |
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\author{Teng Lin, Xiuquan Sun and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle \doublespacing |
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\begin{abstract} |
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\end{abstract} |
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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%applications of langevin dynamics |
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As alternative to Newtonian dynamics, Langevin dynamics, which |
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mimics a simple heat bath with stochastic and dissipative forces, |
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has been applied in a variety of studies. The stochastic treatment |
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of the solvent enables us to carry out substantially longer time |
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simulations. Implicit solvent Langevin dynamics simulations of |
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met-enkephalin not only outperform explicit solvent simulations for |
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computational efficiency, but also agrees very well with explicit |
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solvent simulations for dynamical properties.\cite{Shen2002} |
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Recently, applying Langevin dynamics with the UNRES model, Liow and |
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his coworkers suggest that protein folding pathways can be possibly |
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explored within a reasonable amount of time.\cite{Liwo2005} The |
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stochastic nature of the Langevin dynamics also enhances the |
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sampling of the system and increases the probability of crossing |
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energy barriers.\cite{Banerjee2004, Cui2003} Combining Langevin |
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dynamics with Kramers's theory, Klimov and Thirumalai identified |
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free-energy barriers by studying the viscosity dependence of the |
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protein folding rates.\cite{Klimov1997} In order to account for |
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solvent induced interactions missing from implicit solvent model, |
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Kaya incorporated desolvation free energy barrier into implicit |
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coarse-grained solvent model in protein folding/unfolding studies |
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and discovered a higher free energy barrier between the native and |
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denatured states. Because of its stability against noise, Langevin |
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dynamics is very suitable for studying remagnetization processes in |
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various systems.\cite{Palacios1998,Berkov2002,Denisov2003} For |
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instance, the oscillation power spectrum of nanoparticles from |
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Langevin dynamics simulation has the same peak frequencies for |
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different wave vectors, which recovers the property of magnetic |
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excitations in small finite structures.\cite{Berkov2005a} |
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|
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%review rigid body dynamics |
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Rigid bodies are frequently involved in the modeling of different |
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areas, from engineering, physics, to chemistry. For example, |
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missiles and vehicle are usually modeled by rigid bodies. The |
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movement of the objects in 3D gaming engine or other physics |
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simulator is governed by the rigid body dynamics. In molecular |
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simulation, rigid body is used to simplify the model in |
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protein-protein docking study{\cite{Gray2003}}. |
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It is very important to develop stable and efficient methods to |
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integrate the equations of motion for orientational degrees of |
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freedom. Euler angles are the natural choice to describe the |
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rotational degrees of freedom. However, due to $\frac {1}{sin |
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\theta}$ singularities, the numerical integration of corresponding |
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equations of these motion is very inefficient and inaccurate. |
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Although an alternative integrator using multiple sets of Euler |
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angles can overcome this difficulty\cite{Barojas1973}, the |
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computational penalty and the loss of angular momentum conservation |
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still remain. A singularity-free representation utilizing |
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quaternions was developed by Evans in 1977.\cite{Evans1977} |
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Unfortunately, this approach used a nonseparable Hamiltonian |
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resulting from the quaternion representation, which prevented the |
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symplectic algorithm from being utilized. Another different approach |
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is to apply holonomic constraints to the atoms belonging to the |
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rigid body. Each atom moves independently under the normal forces |
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deriving from potential energy and constraint forces which are used |
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to guarantee the rigidness. However, due to their iterative nature, |
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the SHAKE and Rattle algorithms also converge very slowly when the |
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number of constraints increases.\cite{Ryckaert1977, Andersen1983} |
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A break-through in geometric literature suggests that, in order to |
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develop a long-term integration scheme, one should preserve the |
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symplectic structure of the propagator. By introducing a conjugate |
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momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
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equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
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proposed to evolve the Hamiltonian system in a constraint manifold |
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by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
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An alternative method using the quaternion representation was |
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developed by Omelyan.\cite{Omelyan1998} However, both of these |
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methods are iterative and inefficient. In this section, we descibe a |
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symplectic Lie-Poisson integrator for rigid bodies developed by |
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Dullweber and his coworkers\cite{Dullweber1997} in depth. |
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%review langevin/browninan dynamics for arbitrarily shaped rigid body |
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Combining Langevin or Brownian dynamics with rigid body dynamics, |
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one can study slow processes in biomolecular systems. Modeling DNA |
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as a chain of rigid beads, which are subject to harmonic potentials |
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as well as excluded volume potentials, Mielke and his coworkers |
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discovered rapid superhelical stress generations from the stochastic |
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simulation of twin supercoiling DNA with response to induced |
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torques.\cite{Mielke2004} Membrane fusion is another key biological |
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process which controls a variety of physiological functions, such as |
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release of neurotransmitters \textit{etc}. A typical fusion event |
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happens on the time scale of a millisecond, which is impractical to |
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study using atomistic models with newtonian mechanics. With the help |
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of coarse-grained rigid body model and stochastic dynamics, the |
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fusion pathways were explored by many |
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researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the |
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difficulty of numerical integration of anisotropic rotation, most of |
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the rigid body models are simply modeled using spheres, cylinders, |
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ellipsoids or other regular shapes in stochastic simulations. In an |
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effort to account for the diffusion anisotropy of arbitrary |
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particles, Fernandes and de la Torre improved the original Brownian |
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dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
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incorporating a generalized $6\times6$ diffusion tensor and |
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introducing a simple rotation evolution scheme consisting of three |
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consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected |
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errors and biases are introduced into the system due to the |
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arbitrary order of applying the noncommuting rotation |
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operators.\cite{Beard2003} Based on the observation the momentum |
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relaxation time is much less than the time step, one may ignore the |
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inertia in Brownian dynamics. However, the assumption of zero |
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average acceleration is not always true for cooperative motion which |
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is common in protein motion. An inertial Brownian dynamics (IBD) was |
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proposed to address this issue by adding an inertial correction |
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term.\cite{Beard2000} As a complement to IBD which has a lower bound |
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in time step because of the inertial relaxation time, long-time-step |
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inertial dynamics (LTID) can be used to investigate the inertial |
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behavior of the polymer segments in low friction |
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regime.\cite{Beard2000} LTID can also deal with the rotational |
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dynamics for nonskew bodies without translation-rotation coupling by |
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separating the translation and rotation motion and taking advantage |
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of the analytical solution of hydrodynamics properties. However, |
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typical nonskew bodies like cylinders and ellipsoids are inadequate |
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to represent most complex macromolecule assemblies. These intricate |
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molecules have been represented by a set of beads and their |
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hydrodynamic properties can be calculated using variants on the |
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standard hydrodynamic interaction tensors. |
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The goal of the present work is to develop a Langevin dynamics |
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algorithm for arbitrary-shaped rigid particles by integrating the |
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accurate estimation of friction tensor from hydrodynamics theory |
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into the sophisticated rigid body dynamics algorithms. |
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\section{Computational Methods{\label{methodSec}}} |
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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
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Theoretically, the friction kernel can be determined using the |
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velocity autocorrelation function. However, this approach becomes |
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impractical when the system becomes more and more complicated. |
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Instead, various approaches based on hydrodynamics have been |
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developed to calculate the friction coefficients. In general, the |
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friction tensor $\Xi$ is a $6\times 6$ matrix given by |
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\[ |
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\Xi = \left( {\begin{array}{*{20}c} |
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{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
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{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
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\end{array}} \right). |
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\] |
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Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
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translational friction tensor and rotational resistance (friction) |
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tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
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coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
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tensor. When a particle moves in a fluid, it may experience friction |
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force or torque along the opposite direction of the velocity or |
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angular velocity, |
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\[ |
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\left( \begin{array}{l} |
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F_R \\ |
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\tau _R \\ |
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\end{array} \right) = - \left( {\begin{array}{*{20}c} |
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{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
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{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
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\end{array}} \right)\left( \begin{array}{l} |
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v \\ |
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w \\ |
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\end{array} \right) |
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\] |
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where $F_r$ is the friction force and $\tau _R$ is the friction |
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torque. |
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
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For a spherical particle with slip boundary conditions, the |
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translational and rotational friction constant can be calculated |
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from Stoke's law, |
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\[ |
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\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
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{6\pi \eta R} & 0 & 0 \\ |
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0 & {6\pi \eta R} & 0 \\ |
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0 & 0 & {6\pi \eta R} \\ |
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\end{array}} \right) |
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\] |
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and |
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\[ |
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\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
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{8\pi \eta R^3 } & 0 & 0 \\ |
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0 & {8\pi \eta R^3 } & 0 \\ |
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0 & 0 & {8\pi \eta R^3 } \\ |
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\end{array}} \right) |
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\] |
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where $\eta$ is the viscosity of the solvent and $R$ is the |
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hydrodynamic radius. |
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Other non-spherical shapes, such as cylinders and ellipsoids, are |
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widely used as references for developing new hydrodynamics theory, |
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because their properties can be calculated exactly. In 1936, Perrin |
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extended Stokes's law to general ellipsoids, also called a triaxial |
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ellipsoid, which is given in Cartesian coordinates |
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by\cite{Perrin1934, Perrin1936} |
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\[ |
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\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
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}} = 1 |
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\] |
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where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
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due to the complexity of the elliptic integral, only the ellipsoid |
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with the restriction of two axes being equal, \textit{i.e.} |
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prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
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exactly. Introducing an elliptic integral parameter $S$ for prolate |
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ellipsoids : |
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\[ |
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S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
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} }}{b}, |
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\] |
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and oblate ellipsoids: |
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\[ |
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S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
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}}{a}, |
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\] |
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one can write down the translational and rotational resistance |
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tensors |
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\begin{eqnarray*} |
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\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
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\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
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2a}}, |
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\end{eqnarray*} |
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and |
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\begin{eqnarray*} |
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\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
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\Xi _b^{rr} & = & \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
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\end{eqnarray*} |
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
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Unlike spherical and other simply shaped molecules, there is no |
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analytical solution for the friction tensor for arbitrarily shaped |
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rigid molecules. The ellipsoid of revolution model and general |
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triaxial ellipsoid model have been used to approximate the |
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hydrodynamic properties of rigid bodies. However, since the mapping |
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from all possible ellipsoidal spaces, $r$-space, to all possible |
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combination of rotational diffusion coefficients, $D$-space, is not |
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unique\cite{Wegener1979} as well as the intrinsic coupling between |
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translational and rotational motion of rigid bodies, general |
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ellipsoids are not always suitable for modeling arbitrarily shaped |
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rigid molecules. A number of studies have been devoted to |
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determining the friction tensor for irregularly shaped rigid bodies |
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using more advanced methods where the molecule of interest was |
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modeled by a combinations of spheres\cite{Carrasco1999} and the |
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hydrodynamics properties of the molecule can be calculated using the |
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hydrodynamic interaction tensor. Let us consider a rigid assembly of |
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$N$ beads immersed in a continuous medium. Due to hydrodynamic |
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interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
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than its unperturbed velocity $v_i$, |
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\[ |
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v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
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\] |
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where $F_i$ is the frictional force, and $T_{ij}$ is the |
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hydrodynamic interaction tensor. The friction force of $i$th bead is |
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proportional to its ``net'' velocity |
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\begin{equation} |
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F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
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\label{introEquation:tensorExpression} |
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\end{equation} |
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This equation is the basis for deriving the hydrodynamic tensor. In |
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1930, Oseen and Burgers gave a simple solution to |
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Eq.~\ref{introEquation:tensorExpression} |
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\begin{equation} |
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T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
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R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} |
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\end{equation} |
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Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
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A second order expression for element of different size was |
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introduced by Rotne and Prager\cite{Rotne1969} and improved by |
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Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977} |
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\begin{equation} |
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T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
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\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
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_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
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\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
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\label{introEquation:RPTensorNonOverlapped} |
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\end{equation} |
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Both of the Eq.~\ref{introEquation:oseenTensor} and |
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Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption |
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$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for |
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overlapping beads with the same radius, $\sigma$, is given by |
325 |
tim |
2746 |
\begin{equation} |
326 |
tim |
2999 |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
327 |
|
|
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
328 |
|
|
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
329 |
|
|
\label{introEquation:RPTensorOverlapped} |
330 |
tim |
2746 |
\end{equation} |
331 |
tim |
2999 |
To calculate the resistance tensor at an arbitrary origin $O$, we |
332 |
|
|
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
333 |
|
|
$B_{ij}$ blocks |
334 |
|
|
\begin{equation} |
335 |
|
|
B = \left( {\begin{array}{*{20}c} |
336 |
|
|
{B_{11} } & \ldots & {B_{1N} } \\ |
337 |
|
|
\vdots & \ddots & \vdots \\ |
338 |
|
|
{B_{N1} } & \cdots & {B_{NN} } \\ |
339 |
|
|
\end{array}} \right), |
340 |
|
|
\end{equation} |
341 |
|
|
where $B_{ij}$ is given by |
342 |
tim |
2746 |
\[ |
343 |
tim |
2999 |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
344 |
|
|
)T_{ij} |
345 |
tim |
2746 |
\] |
346 |
tim |
2999 |
where $\delta _{ij}$ is the Kronecker delta function. Inverting the |
347 |
|
|
$B$ matrix, we obtain |
348 |
tim |
2746 |
\[ |
349 |
tim |
2999 |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
350 |
|
|
{C_{11} } & \ldots & {C_{1N} } \\ |
351 |
|
|
\vdots & \ddots & \vdots \\ |
352 |
|
|
{C_{N1} } & \cdots & {C_{NN} } \\ |
353 |
|
|
\end{array}} \right), |
354 |
tim |
2746 |
\] |
355 |
tim |
2999 |
which can be partitioned into $N \times N$ $3 \times 3$ block |
356 |
|
|
$C_{ij}$. With the help of $C_{ij}$ and the skew matrix $U_i$ |
357 |
tim |
2746 |
\[ |
358 |
tim |
2999 |
U_i = \left( {\begin{array}{*{20}c} |
359 |
|
|
0 & { - z_i } & {y_i } \\ |
360 |
|
|
{z_i } & 0 & { - x_i } \\ |
361 |
|
|
{ - y_i } & {x_i } & 0 \\ |
362 |
|
|
\end{array}} \right) |
363 |
tim |
2746 |
\] |
364 |
tim |
2999 |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
365 |
|
|
bead $i$ and origin $O$, the elements of resistance tensor at |
366 |
|
|
arbitrary origin $O$ can be written as |
367 |
|
|
\begin{eqnarray} |
368 |
|
|
\Xi _{}^{tt} & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
369 |
|
|
\Xi _{}^{tr} & = & \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
370 |
|
|
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
371 |
|
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
372 |
|
|
\end{eqnarray} |
373 |
|
|
The resistance tensor depends on the origin to which they refer. The |
374 |
|
|
proper location for applying the friction force is the center of |
375 |
|
|
resistance (or center of reaction), at which the trace of rotational |
376 |
|
|
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
377 |
|
|
Mathematically, the center of resistance is defined as an unique |
378 |
|
|
point of the rigid body at which the translation-rotation coupling |
379 |
|
|
tensors are symmetric, |
380 |
|
|
\begin{equation} |
381 |
|
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
382 |
|
|
\label{introEquation:definitionCR} |
383 |
|
|
\end{equation} |
384 |
|
|
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
385 |
|
|
we can easily derive that the translational resistance tensor is |
386 |
|
|
origin independent, while the rotational resistance tensor and |
387 |
|
|
translation-rotation coupling resistance tensor depend on the |
388 |
|
|
origin. Given the resistance tensor at an arbitrary origin $O$, and |
389 |
|
|
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
390 |
|
|
obtain the resistance tensor at $P$ by |
391 |
|
|
\begin{equation} |
392 |
|
|
\begin{array}{l} |
393 |
|
|
\Xi _P^{tt} = \Xi _O^{tt} \\ |
394 |
|
|
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
395 |
|
|
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
396 |
|
|
\end{array} |
397 |
|
|
\label{introEquation:resistanceTensorTransformation} |
398 |
|
|
\end{equation} |
399 |
|
|
where |
400 |
tim |
2746 |
\[ |
401 |
tim |
2999 |
U_{OP} = \left( {\begin{array}{*{20}c} |
402 |
|
|
0 & { - z_{OP} } & {y_{OP} } \\ |
403 |
|
|
{z_i } & 0 & { - x_{OP} } \\ |
404 |
|
|
{ - y_{OP} } & {x_{OP} } & 0 \\ |
405 |
|
|
\end{array}} \right) |
406 |
tim |
2746 |
\] |
407 |
tim |
2999 |
Using Eq.~\ref{introEquation:definitionCR} and |
408 |
|
|
Eq.~\ref{introEquation:resistanceTensorTransformation}, one can |
409 |
|
|
locate the position of center of resistance, |
410 |
|
|
\begin{eqnarray*} |
411 |
|
|
\left( \begin{array}{l} |
412 |
|
|
x_{OR} \\ |
413 |
|
|
y_{OR} \\ |
414 |
|
|
z_{OR} \\ |
415 |
|
|
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
416 |
|
|
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
417 |
|
|
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
418 |
|
|
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
419 |
|
|
\end{array}} \right)^{ - 1} \\ |
420 |
|
|
& & \left( \begin{array}{l} |
421 |
|
|
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
422 |
|
|
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
423 |
|
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
424 |
|
|
\end{array} \right) \\ |
425 |
|
|
\end{eqnarray*} |
426 |
|
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
427 |
|
|
joining center of resistance $R$ and origin $O$. |
428 |
tim |
2746 |
|
429 |
tim |
2999 |
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
430 |
tim |
2746 |
|
431 |
tim |
2999 |
Consider the Langevin equations of motion in generalized coordinates |
432 |
tim |
2746 |
\begin{equation} |
433 |
|
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
434 |
|
|
\label{LDGeneralizedForm} |
435 |
|
|
\end{equation} |
436 |
|
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
437 |
|
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
438 |
tim |
2999 |
$V_i = V_i(v_i,\omega _i)$. The right side of |
439 |
|
|
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in |
440 |
tim |
2746 |
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
441 |
|
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
442 |
|
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
443 |
|
|
it is also convenient to handle the rotation of rigid body in |
444 |
|
|
body-fixed frame. Thus the friction and random forces are calculated |
445 |
|
|
in body-fixed frame and converted back to lab-fixed frame by: |
446 |
|
|
\[ |
447 |
|
|
\begin{array}{l} |
448 |
tim |
2999 |
F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\ |
449 |
|
|
F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\ |
450 |
|
|
\end{array} |
451 |
tim |
2746 |
\] |
452 |
|
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
453 |
|
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
454 |
tim |
2999 |
angular velocity $\omega _i$ |
455 |
tim |
2746 |
\begin{equation} |
456 |
|
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
457 |
|
|
f_{r,i}^b (t) \\ |
458 |
|
|
\tau _{r,i}^b (t) \\ |
459 |
|
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
460 |
|
|
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
461 |
|
|
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
462 |
|
|
\end{array}} \right)\left( \begin{array}{l} |
463 |
|
|
v_{R,i}^b (t) \\ |
464 |
|
|
\omega _i (t) \\ |
465 |
|
|
\end{array} \right), |
466 |
|
|
\end{equation} |
467 |
|
|
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
468 |
|
|
with zero mean and variance |
469 |
|
|
\begin{equation} |
470 |
|
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
471 |
|
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
472 |
tim |
2999 |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
473 |
tim |
2746 |
\end{equation} |
474 |
|
|
The equation of motion for $v_i$ can be written as |
475 |
|
|
\begin{equation} |
476 |
|
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
477 |
|
|
f_{r,i}^l (t) |
478 |
|
|
\end{equation} |
479 |
|
|
Since the frictional force is applied at the center of resistance |
480 |
|
|
which generally does not coincide with the center of mass, an extra |
481 |
|
|
torque is exerted at the center of mass. Thus, the net body-fixed |
482 |
|
|
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
483 |
|
|
given by |
484 |
|
|
\begin{equation} |
485 |
|
|
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
486 |
|
|
\end{equation} |
487 |
|
|
where $r_{MR}$ is the vector from the center of mass to the center |
488 |
tim |
2999 |
of the resistance. Instead of integrating the angular velocity in |
489 |
|
|
lab-fixed frame, we consider the equation of angular momentum in |
490 |
|
|
body-fixed frame |
491 |
tim |
2746 |
\begin{equation} |
492 |
tim |
2999 |
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
493 |
|
|
+ \tau _{r,i}^b(t) |
494 |
tim |
2746 |
\end{equation} |
495 |
|
|
Embedding the friction terms into force and torque, one can |
496 |
|
|
integrate the langevin equations of motion for rigid body of |
497 |
|
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
498 |
|
|
$h= \delta t$: |
499 |
|
|
|
500 |
tim |
2999 |
{\tt moveA:} |
501 |
tim |
2746 |
\begin{align*} |
502 |
tim |
2999 |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
503 |
|
|
+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
504 |
|
|
% |
505 |
|
|
{\bf r}(t + h) &\leftarrow {\bf r}(t) |
506 |
|
|
+ h {\bf v}\left(t + h / 2 \right), \\ |
507 |
|
|
% |
508 |
|
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
509 |
|
|
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
510 |
|
|
% |
511 |
|
|
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
512 |
|
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
513 |
tim |
2746 |
\end{align*} |
514 |
|
|
In this context, the $\mathrm{rotate}$ function is the reversible |
515 |
tim |
2999 |
product of the three body-fixed rotations, |
516 |
tim |
2746 |
\begin{equation} |
517 |
|
|
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
518 |
|
|
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
519 |
|
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
520 |
|
|
\end{equation} |
521 |
|
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
522 |
tim |
2999 |
rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed |
523 |
|
|
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
524 |
|
|
axis $\alpha$, |
525 |
tim |
2746 |
\begin{equation} |
526 |
|
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
527 |
|
|
\begin{array}{lcl} |
528 |
tim |
2999 |
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
529 |
tim |
2746 |
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
530 |
|
|
j}(0). |
531 |
|
|
\end{array} |
532 |
|
|
\right. |
533 |
|
|
\end{equation} |
534 |
|
|
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
535 |
|
|
rotation matrix. For example, in the small-angle limit, the |
536 |
|
|
rotation matrix around the body-fixed x-axis can be approximated as |
537 |
|
|
\begin{equation} |
538 |
|
|
\mathsf{R}_x(\theta) \approx \left( |
539 |
|
|
\begin{array}{ccc} |
540 |
|
|
1 & 0 & 0 \\ |
541 |
|
|
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
542 |
|
|
\theta^2 / 4} \\ |
543 |
|
|
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
544 |
|
|
\theta^2 / 4} |
545 |
|
|
\end{array} |
546 |
|
|
\right). |
547 |
|
|
\end{equation} |
548 |
tim |
2999 |
All other rotations follow in a straightforward manner. After the |
549 |
|
|
first part of the propagation, the forces and body-fixed torques are |
550 |
|
|
calculated at the new positions and orientations |
551 |
tim |
2746 |
|
552 |
tim |
2999 |
{\tt doForces:} |
553 |
|
|
\begin{align*} |
554 |
|
|
{\bf f}(t + h) &\leftarrow |
555 |
|
|
- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
556 |
|
|
% |
557 |
|
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
558 |
|
|
\times \frac{\partial V}{\partial {\bf u}}, \\ |
559 |
|
|
% |
560 |
|
|
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
561 |
|
|
\cdot {\bf \tau}^s(t + h). |
562 |
|
|
\end{align*} |
563 |
tim |
2746 |
Once the forces and torques have been obtained at the new time step, |
564 |
|
|
the velocities can be advanced to the same time value. |
565 |
|
|
|
566 |
tim |
2999 |
{\tt moveB:} |
567 |
tim |
2746 |
\begin{align*} |
568 |
tim |
2999 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
569 |
|
|
\right) |
570 |
|
|
+ \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\ |
571 |
|
|
% |
572 |
|
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 |
573 |
|
|
\right) |
574 |
|
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
575 |
tim |
2746 |
\end{align*} |
576 |
|
|
|
577 |
tim |
2999 |
\section{Results and Discussion} |
578 |
tim |
2746 |
|
579 |
tim |
2999 |
The Langevin algorithm described in previous section has been |
580 |
|
|
implemented in {\sc oopse}\cite{Meineke2005} and applied to studies |
581 |
|
|
of the static and dynamic properties in several systems. |
582 |
tim |
2746 |
|
583 |
tim |
2999 |
\subsection{Temperature Control} |
584 |
tim |
2746 |
|
585 |
tim |
2999 |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
586 |
|
|
the solvent's thermal motions is controlled by the external |
587 |
|
|
temperature. The capability to maintain the temperature of the whole |
588 |
|
|
system was usually used to measure the stability and efficiency of |
589 |
|
|
the algorithm. In order to verify the stability of this new |
590 |
|
|
algorithm, a series of simulations are performed on system |
591 |
|
|
consisiting of 256 SSD water molecules with different viscosities. |
592 |
|
|
The initial configuration for the simulations is taken from a 1ns |
593 |
|
|
NVT simulation with a cubic box of 19.7166~\AA. All simulation are |
594 |
|
|
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
595 |
|
|
with reference temperature at 300~K. The average temperature as a |
596 |
|
|
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
597 |
|
|
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
598 |
|
|
1$ poise. The better temperature control at higher viscosity can be |
599 |
|
|
explained by the finite size effect and relative slow relaxation |
600 |
|
|
rate at lower viscosity regime. |
601 |
|
|
\begin{table} |
602 |
|
|
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
603 |
|
|
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
604 |
|
|
\label{langevin:viscosity} |
605 |
|
|
\begin{center} |
606 |
|
|
\begin{tabular}{lll} |
607 |
|
|
\hline |
608 |
|
|
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
609 |
|
|
\hline |
610 |
|
|
1 & 300.47 & 10.99 \\ |
611 |
|
|
0.1 & 301.19 & 11.136 \\ |
612 |
|
|
0.01 & 303.04 & 11.796 \\ |
613 |
|
|
\hline |
614 |
|
|
\end{tabular} |
615 |
|
|
\end{center} |
616 |
|
|
\end{table} |
617 |
tim |
2746 |
|
618 |
tim |
2999 |
Another set of calculations were performed to study the efficiency of |
619 |
|
|
temperature control using different temperature coupling schemes. |
620 |
|
|
The starting configuration is cooled to 173~K and evolved using NVE, |
621 |
|
|
NVT, and Langevin dynamic with time step of 2 fs. |
622 |
|
|
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
623 |
|
|
the systems reach equilibrium. The orange curve in |
624 |
|
|
Fig.~\ref{langevin:temperature} represents the simulation using |
625 |
|
|
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
626 |
|
|
which gives reasonable tight coupling, while the blue one from |
627 |
|
|
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
628 |
|
|
scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal |
629 |
|
|
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
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|
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loses the temperature control ability. |
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|
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|
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|
|
\begin{figure} |
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|
|
\centering |
634 |
gezelter |
3000 |
\includegraphics[width=\linewidth]{temperature.pdf} |
635 |
tim |
2999 |
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
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|
|
temperature fluctuation versus time.} \label{langevin:temperature} |
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|
|
\end{figure} |
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|
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|
|
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
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|
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|
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In order to verify that Langevin dynamics can mimic the dynamics of |
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the systems absent of explicit solvents, we carried out two sets of |
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simulations and compare their dynamic properties. |
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Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
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|
|
made of 256 pentane molecules and two banana shaped molecules at |
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|
|
273~K. It has an equivalent implicit solvent system containing only |
647 |
|
|
two banana shaped molecules with viscosity of 0.289 center poise. To |
648 |
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calculate the hydrodynamic properties of the banana shaped molecule, |
649 |
|
|
we created a rough shell model (see Fig.~\ref{langevin:roughShell}), |
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|
|
in which the banana shaped molecule is represented as a ``shell'' |
651 |
|
|
made of 2266 small identical beads with size of 0.3 \AA on the |
652 |
|
|
surface. Applying the procedure described in |
653 |
|
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
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|
|
identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$, |
655 |
|
|
-0.1988 $\rm{\AA}$), as well as the resistance tensor, |
656 |
|
|
\[ |
657 |
|
|
\left( {\begin{array}{*{20}c} |
658 |
|
|
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
659 |
|
|
0& 0.9270&-0.007063& 0.08585&0&0\\ |
660 |
|
|
0&-0.007063&0.7494&0.2057&0&0\\ |
661 |
|
|
0&0.0858&0.2057& 58.64& 0&0\\ |
662 |
|
|
0.08585&0&0&0&48.30&3.219&\\ |
663 |
|
|
0.2057&0&0&0&3.219&10.7373\\ |
664 |
|
|
\end{array}} \right). |
665 |
|
|
\] |
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|
|
where the units for translational, translation-rotation coupling and rotational tensors are $\frac{kcal \cdot fs}{mol \cdot \rm{\AA}^2}$, $\frac{kcal \cdot fs}{mol \cdot \rm{\AA} \cdot rad}$ and $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respectively. |
667 |
|
|
Curves of the velocity auto-correlation functions in |
668 |
|
|
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
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|
|
However, because of the stochastic nature, simulation using Langevin |
670 |
|
|
dynamics was shown to decay slightly faster than MD. In order to |
671 |
|
|
study the rotational motion of the molecules, we also calculated the |
672 |
|
|
auto-correlation function of the principle axis of the second GB |
673 |
|
|
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
674 |
|
|
probably due to the reason that we used the experimental viscosity directly instead of calculating bulk viscosity from simulation. |
675 |
|
|
|
676 |
|
|
\begin{figure} |
677 |
|
|
\centering |
678 |
gezelter |
3000 |
\includegraphics[width=\linewidth]{roughShell.pdf} |
679 |
tim |
2999 |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
680 |
|
|
model for banana shaped molecule.} \label{langevin:roughShell} |
681 |
|
|
\end{figure} |
682 |
|
|
|
683 |
|
|
\begin{figure} |
684 |
|
|
\centering |
685 |
gezelter |
3000 |
\includegraphics[width=\linewidth]{twoBanana.pdf} |
686 |
tim |
2999 |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
687 |
|
|
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
688 |
|
|
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
689 |
|
|
\end{figure} |
690 |
|
|
|
691 |
|
|
\begin{figure} |
692 |
|
|
\centering |
693 |
gezelter |
3000 |
\includegraphics[width=\linewidth]{vacf.pdf} |
694 |
tim |
2999 |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
695 |
|
|
auto-correlation functions of NVE (explicit solvent) in blue and |
696 |
|
|
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
697 |
|
|
\end{figure} |
698 |
|
|
|
699 |
|
|
\begin{figure} |
700 |
|
|
\centering |
701 |
gezelter |
3000 |
\includegraphics[width=\linewidth]{uacf.pdf} |
702 |
tim |
2999 |
\caption[Auto-correlation functions of the principle axis of the |
703 |
|
|
middle GB particle]{Auto-correlation functions of the principle axis |
704 |
|
|
of the middle GB particle of NVE (blue) and Langevin dynamics |
705 |
|
|
(red).} \label{langevin:uacf} |
706 |
|
|
\end{figure} |
707 |
|
|
|
708 |
tim |
2746 |
\section{Conclusions} |
709 |
|
|
|
710 |
tim |
2999 |
We have presented a new Langevin algorithm by incorporating the |
711 |
|
|
hydrodynamics properties of arbitrary shaped molecules into an |
712 |
|
|
advanced symplectic integration scheme. The temperature control |
713 |
|
|
ability of this algorithm was demonstrated by a set of simulations |
714 |
|
|
with different viscosities. It was also shown to have significant |
715 |
|
|
advantage of producing rapid thermal equilibration over |
716 |
|
|
Nos\'{e}-Hoover method. Further studies in systems involving banana |
717 |
|
|
shaped molecules illustrated that the dynamic properties could be |
718 |
|
|
preserved by using this new algorithm as an implicit solvent model. |
719 |
|
|
|
720 |
|
|
|
721 |
tim |
2746 |
\section{Acknowledgments} |
722 |
|
|
Support for this project was provided by the National Science |
723 |
|
|
Foundation under grant CHE-0134881. T.L. also acknowledges the |
724 |
|
|
financial support from center of applied mathematics at University |
725 |
|
|
of Notre Dame. |
726 |
|
|
\newpage |
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|
|
|
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|
|
\bibliographystyle{jcp2} |
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|
|
\bibliography{langevin} |
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|
|
|
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|
|
\end{document} |