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\begin{document} |
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\title{Langevin Dynamics for Rigid Body of Arbitrary Shape } |
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\author{Teng Lin 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 an excellent alternative to newtonian dynamics, Langevin |
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dynamics, which mimics a simple heat bath with stochastic and |
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dissipative forces, has been applied in a variety of studies. The |
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stochastic treatment of the solvent enables us to carry out |
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substantially longer time simulation. Implicit solvent Langevin |
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dynamics simulation of met-enkephalin not only outperforms explicit |
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solvent simulation on computation efficiency, but also agrees very |
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well with explicit solvent simulation on dynamics |
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properties\cite{Shen2002}. Recently, applying Langevin dynamics with |
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UNRES model, Liow and his coworkers suggest that protein folding |
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pathways can be possibly exploited within a reasonable amount of |
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time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics |
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also enhances the sampling of the system and increases the |
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probability of crossing energy barrier\cite{Banerjee2004, Cui2003}. |
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Combining Langevin dynamics with Kramers's theory, Klimov and |
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Thirumalai identified the free-energy barrier by studying the |
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viscosity dependence of the protein folding rates\cite{Klimov1997}. |
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In order to account for solvent induced interactions missing from |
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implicit solvent model, Kaya incorporated desolvation free energy |
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barrier into implicit coarse-grained solvent model in protein |
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folding/unfolding study and discovered a higher free energy barrier |
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between the native and denatured states. Because of its stability |
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against noise, Langevin dynamics is very suitable for studying |
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remagnetization processes in various |
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systems\cite{Garcia-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}. In an |
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attempt to reduce the computational cost of simulation, multiple |
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time stepping (MTS) methods have been introduced and have been of |
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great interest to macromolecule and protein |
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community\cite{Tuckerman1992}. Relying on the observation that |
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forces between distant atoms generally demonstrate slower |
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fluctuations than forces between close atoms, MTS method are |
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generally implemented by evaluating the slowly fluctuating forces |
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less frequently than the fast ones. Unfortunately, nonlinear |
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instability resulting from increasing timestep in MTS simulation |
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have became a critical obstruction preventing the long time |
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simulation. Due to the coupling to the heat bath, Langevin dynamics |
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has been shown to be able to damp out the resonance artifact more |
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efficiently\cite{Sandu1999}. |
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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 of orientational degrees of |
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freedom. Euler angles are the nature choice to describe the |
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rotational degrees of freedom. However, due to its singularity, the |
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numerical integration of corresponding equations of motion is very |
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inefficient and inaccurate. Although an alternative integrator using |
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different sets of Euler angles can overcome this difficulty\cite{}, |
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the computational penalty and the lost of angular momentum |
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conservation still remain. In 1977, a singularity free |
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representation utilizing quaternions was developed by |
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Evans\cite{Evans1977}. Unfortunately, this approach suffer from the |
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nonseparable Hamiltonian resulted from quaternion representation, |
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which prevents the symplectic algorithm to be utilized. Another |
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different approach is to apply holonomic constraints to the atoms |
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belonging to the rigid body\cite{}. Each atom moves independently |
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under the normal forces deriving from potential energy and |
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constraint forces which are used to guarantee the rigidness. |
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However, due to their iterative nature, SHAKE and Rattle algorithm |
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converge very slowly when the number of constraint increases. |
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The 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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geometric structure of the flow. Matubayasi and Nakahara developed a |
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time-reversible integrator for rigid bodies in quaternion |
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representation. Although it is not symplectic, this integrator still |
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demonstrates a better long-time energy conservation than traditional |
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methods because of the time-reversible nature. Extending |
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Trotter-Suzuki to general system with a flat phase space, Miller and |
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his colleagues devised an novel symplectic, time-reversible and |
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volume-preserving integrator in quaternion representation. However, |
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all of the integrators in quaternion representation suffer from the |
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computational penalty of constructing a rotation matrix from |
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quaternions to evolve coordinates and velocities at every time step. |
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An alternative integration scheme utilizing rotation matrix directly |
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is RSHAKE , in which a conjugate momentum to rotation matrix is |
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introduced to re-formulate the Hamiltonian's equation and the |
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Hamiltonian is evolved in a constraint manifold by iteratively |
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satisfying the orthogonality constraint. However, RSHAKE is |
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inefficient because of the iterative procedure. An extremely |
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efficient integration scheme in rotation matrix representation, |
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which also preserves the same structural properties of the |
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Hamiltonian flow as Miller's integrator, is proposed by Dullweber, |
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Leimkuhler and McLachlan (DLM)\cite{Dullweber1997}. |
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|
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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 the slow processes in biomolecular systems. Modeling |
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the DNA as a chain of rigid spheres beads, which subject to harmonic |
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potentials as well as excluded volume potentials, Mielke and his |
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coworkers discover rapid superhelical stress generations from the |
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stochastic simulation of twin supercoiling DNA with response to |
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induced torques\cite{Mielke2004}. Membrane fusion is another key |
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biological process which controls a variety of physiological |
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functions, such as release of neurotransmitters \textit{etc}. A |
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typical fusion event happens on the time scale of millisecond, which |
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is impracticable to study using all atomistic model with newtonian |
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mechanics. With the help of coarse-grained rigid body model and |
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stochastic dynamics, the fusion pathways were exploited by many |
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researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the |
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difficulty of numerical integration of anisotropy rotation, most of |
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the rigid body models are simply modeled by sphere, cylinder, |
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ellipsoid or other regular shapes in stochastic simulations. In an |
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effort to account for the diffusion anisotropy of the 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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error and bias are introduced into the system due to the arbitrary |
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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, assumption of the 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{Beard2001}. 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{Beard2001}. 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 cylinder and ellipsoid are inadequate to |
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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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hydrodynamics properties can be calculated using variant |
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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 rigid particles by integrating the accurate |
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estimation of friction tensor from hydrodynamics theory into the |
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sophisticated rigid body dynamics. |
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\section{Method{\label{methodSec}}} |
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\subsection{Friction Tensor} |
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For an arbitrary rigid body moves in a fluid, it may experience |
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friction force $f_r$ or friction torque $\tau _r$ along the opposite |
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direction of the velocity $v$ or angular velocity $\omega$ at |
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arbitrary origin $P$, |
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\begin{equation} |
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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 _{P,t} } & {\Xi _{P,c}^T } \\ |
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{\Xi _{P,c} } & {\Xi _{P,r} } \\ |
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\end{array}} \right)\left( \begin{array}{l} |
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\nu \\ |
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\omega \\ |
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\end{array} \right) |
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\end{equation} |
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where $\Xi _{P,t}t$ is the translation friction tensor, $\Xi _{P,r}$ |
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is the rotational friction tensor and $\Xi _{P,c}$ is the |
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translation-rotation coupling tensor. The procedure of calculating |
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friction tensor using hydrodynamic tensor and comparison between |
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bead model and shell model were elaborated by Carrasco \textit{et |
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al}\cite{Carrasco1999}. An important property of the friction tensor |
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is that the translational friction tensor is independent of origin |
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while the rotational and coupling are sensitive to the choice of the |
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origin \cite{Brenner1967}, which can be described by |
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\begin{equation} |
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\begin{array}{c} |
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\Xi _{P,t} = \Xi _{O,t} = \Xi _t \\ |
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\Xi _{P,c} = \Xi _{O,c} - r_{OP} \times \Xi _t \\ |
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\Xi _{P,r} = \Xi _{O,r} - r_{OP} \times \Xi _t \times r_{OP} + \Xi _{O,c} \times r_{OP} - r_{OP} \times \Xi _{O,c}^T \\ |
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\end{array} |
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\end{equation} |
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Where $O$ is another origin and $r_{OP}$ is the vector joining $O$ |
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and $P$. It is also worthy of mention that both of translational and |
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rotational frictional tensors are always symmetric. In contrast, |
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coupling tensor is only symmetric at center of reaction: |
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\begin{equation} |
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\Xi _{R,c} = \Xi _{R,c}^T |
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\end{equation} |
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The proper location for applying friction force is the center of |
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reaction, at which the trace of rotational resistance tensor reaches |
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minimum. |
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\subsection{Rigid body dynamics} |
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The Hamiltonian of rigid body can be separated in terms of potential |
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energy $V(r,A)$ and kinetic energy $T(p,\pi)$, |
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\[ |
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H = V(r,A) + T(v,\pi ) |
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\] |
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A second-order symplectic method is now obtained by the composition |
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of the flow maps, |
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\[ |
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\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
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_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
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\] |
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Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
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sub-flows which corresponding to force and torque respectively, |
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\[ |
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\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
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_{\Delta t/2,\tau }. |
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\] |
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Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
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$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
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order inside $\varphi _{\Delta t/2,V}$ does not matter. |
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Furthermore, kinetic potential can be separated to translational |
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kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
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\begin{equation} |
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T(p,\pi ) =T^t (p) + T^r (\pi ). |
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\end{equation} |
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where $ T^t (p) = \frac{1}{2}v^T m v $ and $T^r (\pi )$ is defined |
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by \ref{introEquation:rotationalKineticRB}. Therefore, the |
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corresponding flow maps are given by |
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\[ |
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\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
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_{\Delta t,T^r }. |
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\] |
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The free rigid body is an example of Lie-Poisson system with |
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Hamiltonian function |
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\begin{equation} |
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T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
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\label{introEquation:rotationalKineticRB} |
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\end{equation} |
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where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
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Lie-Poisson structure matrix, |
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\begin{equation} |
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J(\pi ) = \left( {\begin{array}{*{20}c} |
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0 & {\pi _3 } & { - \pi _2 } \\ |
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{ - \pi _3 } & 0 & {\pi _1 } \\ |
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{\pi _2 } & { - \pi _1 } & 0 \\ |
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\end{array}} \right) |
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\end{equation} |
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Thus, the dynamics of free rigid body is governed by |
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\begin{equation} |
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\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
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\end{equation} |
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One may notice that each $T_i^r$ in Equation |
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\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
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instance, the equations of motion due to $T_1^r$ are given by |
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\begin{equation} |
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\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}A = AR_1 |
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\label{introEqaution:RBMotionSingleTerm} |
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\end{equation} |
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where |
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\[ R_1 = \left( {\begin{array}{*{20}c} |
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0 & 0 & 0 \\ |
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0 & 0 & {\pi _1 } \\ |
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0 & { - \pi _1 } & 0 \\ |
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\end{array}} \right). |
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\] |
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The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
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\[ |
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\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),A(\Delta t) = |
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A(0)e^{\Delta tR_1 } |
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\] |
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with |
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\[ |
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e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
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0 & 0 & 0 \\ |
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0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
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0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
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\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
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\] |
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To reduce the cost of computing expensive functions in $e^{\Delta |
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tR_1 }$, we can use Cayley transformation, |
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\[ |
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e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
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) |
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\] |
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The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
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manner. |
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In order to construct a second-order symplectic method, we split the |
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angular kinetic Hamiltonian function into five terms |
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\[ |
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T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
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) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
341 |
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(\pi _1 ) |
342 |
|
|
\]. |
343 |
|
|
Concatenating flows corresponding to these five terms, we can obtain |
344 |
|
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the flow map for free rigid body, |
345 |
|
|
\[ |
346 |
|
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
347 |
|
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
348 |
|
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
349 |
|
|
_1 }. |
350 |
|
|
\] |
351 |
|
|
|
352 |
|
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The equations of motion corresponding to potential energy and |
353 |
|
|
kinetic energy are listed in the below table, |
354 |
|
|
\begin{center} |
355 |
|
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\begin{tabular}{|l|l|} |
356 |
|
|
\hline |
357 |
|
|
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
358 |
|
|
Potential & Kinetic \\ |
359 |
|
|
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
360 |
|
|
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
361 |
|
|
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
362 |
|
|
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
363 |
|
|
\hline |
364 |
|
|
\end{tabular} |
365 |
|
|
\end{center} |
366 |
|
|
|
367 |
|
|
Finally, we obtain the overall symplectic flow maps for free moving |
368 |
|
|
rigid body |
369 |
|
|
\begin{align*} |
370 |
|
|
\varphi _{\Delta t} = &\varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \circ \\ |
371 |
|
|
&\varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \circ \\ |
372 |
|
|
&\varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
373 |
|
|
\label{introEquation:overallRBFlowMaps} |
374 |
|
|
\end{align*} |
375 |
|
|
|
376 |
|
|
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
377 |
|
|
|
378 |
|
|
Consider a Langevin equation of motions in generalized coordinates |
379 |
|
|
\begin{equation} |
380 |
|
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
381 |
|
|
\label{LDGeneralizedForm} |
382 |
|
|
\end{equation} |
383 |
|
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
384 |
|
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
385 |
|
|
$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
386 |
|
|
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
387 |
|
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
388 |
|
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
389 |
|
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
390 |
|
|
it is also convenient to handle the rotation of rigid body in |
391 |
|
|
body-fixed frame. Thus the friction and random forces are calculated |
392 |
|
|
in body-fixed frame and converted back to lab-fixed frame by: |
393 |
|
|
\[ |
394 |
|
|
\begin{array}{l} |
395 |
|
|
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
396 |
|
|
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
397 |
|
|
\end{array}. |
398 |
|
|
\] |
399 |
|
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
400 |
|
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
401 |
|
|
angular velocity $\omega _i$, |
402 |
|
|
\begin{equation} |
403 |
|
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
404 |
|
|
f_{r,i}^b (t) \\ |
405 |
|
|
\tau _{r,i}^b (t) \\ |
406 |
|
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
407 |
|
|
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
408 |
|
|
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
409 |
|
|
\end{array}} \right)\left( \begin{array}{l} |
410 |
|
|
v_{R,i}^b (t) \\ |
411 |
|
|
\omega _i (t) \\ |
412 |
|
|
\end{array} \right), |
413 |
|
|
\end{equation} |
414 |
|
|
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
415 |
|
|
with zero mean and variance |
416 |
|
|
\begin{equation} |
417 |
|
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
418 |
|
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
419 |
|
|
2k_B T\Xi _R \delta (t - t'). |
420 |
|
|
\end{equation} |
421 |
|
|
The equation of motion for $v_i$ can be written as |
422 |
|
|
\begin{equation} |
423 |
|
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
424 |
|
|
f_{r,i}^l (t) |
425 |
|
|
\end{equation} |
426 |
|
|
Since the frictional force is applied at the center of resistance |
427 |
|
|
which generally does not coincide with the center of mass, an extra |
428 |
|
|
torque is exerted at the center of mass. Thus, the net body-fixed |
429 |
|
|
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
430 |
|
|
given by |
431 |
|
|
\begin{equation} |
432 |
|
|
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
433 |
|
|
\end{equation} |
434 |
|
|
where $r_{MR}$ is the vector from the center of mass to the center |
435 |
|
|
of the resistance. Instead of integrating angular velocity in |
436 |
|
|
lab-fixed frame, we consider the equation of motion of angular |
437 |
|
|
momentum in body-fixed frame |
438 |
|
|
\begin{equation} |
439 |
|
|
\dot \pi _i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b |
440 |
|
|
(t) + \tau _{r,i}^b(t) |
441 |
|
|
\end{equation} |
442 |
|
|
|
443 |
|
|
Embedding the friction terms into force and torque, one can |
444 |
|
|
integrate the langevin equations of motion for rigid body of |
445 |
|
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
446 |
|
|
$h= \delta t$: |
447 |
|
|
|
448 |
|
|
{\tt part one:} |
449 |
|
|
\begin{align*} |
450 |
|
|
v_i (t + h/2) &\leftarrow v_i (t) + \frac{{hf_{t,i}^l (t)}}{{2m_i }} \\ |
451 |
|
|
\pi _i (t + h/2) &\leftarrow \pi _i (t) + \frac{{h\tau _{t,i}^b (t)}}{2} \\ |
452 |
|
|
r_i (t + h) &\leftarrow r_i (t) + hv_i (t + h/2) \\ |
453 |
|
|
A_i (t + h) &\leftarrow rotate\left( {h\pi _i (t + h/2)I^{ - 1} } \right) \\ |
454 |
|
|
\end{align*} |
455 |
|
|
In this context, the $\mathrm{rotate}$ function is the reversible |
456 |
|
|
product of five consecutive body-fixed rotations, |
457 |
|
|
\begin{equation} |
458 |
|
|
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
459 |
|
|
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
460 |
|
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
461 |
|
|
\end{equation} |
462 |
|
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
463 |
|
|
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
464 |
|
|
angular momentum ($\pi$) by an angle $\theta$ around body-fixed axis |
465 |
|
|
$\alpha$, |
466 |
|
|
\begin{equation} |
467 |
|
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
468 |
|
|
\begin{array}{lcl} |
469 |
|
|
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
470 |
|
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
471 |
|
|
j}(0). |
472 |
|
|
\end{array} |
473 |
|
|
\right. |
474 |
|
|
\end{equation} |
475 |
|
|
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
476 |
|
|
rotation matrix. For example, in the small-angle limit, the |
477 |
|
|
rotation matrix around the body-fixed x-axis can be approximated as |
478 |
|
|
\begin{equation} |
479 |
|
|
\mathsf{R}_x(\theta) \approx \left( |
480 |
|
|
\begin{array}{ccc} |
481 |
|
|
1 & 0 & 0 \\ |
482 |
|
|
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
483 |
|
|
\theta^2 / 4} \\ |
484 |
|
|
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
485 |
|
|
\theta^2 / 4} |
486 |
|
|
\end{array} |
487 |
|
|
\right). |
488 |
|
|
\end{equation} |
489 |
|
|
All other rotations follow in a straightforward manner. |
490 |
|
|
|
491 |
|
|
After the first part of the propagation, the friction and random |
492 |
|
|
forces are generated at the center of resistance in body-fixed frame |
493 |
|
|
and converted back into lab-fixed frame |
494 |
|
|
\[ |
495 |
|
|
f_{t,i}^l (t + h) = - \left( {\frac{{\partial V}}{{\partial r_i }}} |
496 |
|
|
\right)_{r_i (t + h)} + A_i^T (t + h)[F_{f,i}^b (t + h) + F_{r,i}^b |
497 |
|
|
(t + h)], |
498 |
|
|
\] |
499 |
|
|
while the system torque in lab-fixed frame is transformed into |
500 |
|
|
body-fixed frame, |
501 |
|
|
\[ |
502 |
|
|
\tau _{t,i}^b (t + h) = A\tau _{s,i}^l (t) + \tau _{n,i}^b (t) + |
503 |
|
|
\tau _{r,i}^b (t). |
504 |
|
|
\] |
505 |
|
|
Once the forces and torques have been obtained at the new time step, |
506 |
|
|
the velocities can be advanced to the same time value. |
507 |
|
|
|
508 |
|
|
{\tt part two:} |
509 |
|
|
\begin{align*} |
510 |
|
|
v_i (t) &\leftarrow v_i (t + h/2) + \frac{{hf_{t,i}^l (t + h)}}{{2m_i }} \\ |
511 |
|
|
\pi _i (t) &\leftarrow \pi _i (t + h/2) + \frac{{h\tau _{t,i}^b (t + h)}}{2} \\ |
512 |
|
|
\end{align*} |
513 |
|
|
|
514 |
|
|
\section{Results and discussion} |
515 |
|
|
|
516 |
|
|
\subsection{} |
517 |
|
|
|
518 |
|
|
\subsection{Lipid bilayer} |
519 |
|
|
|
520 |
|
|
\subsection{Liquid crystal} |
521 |
|
|
|
522 |
|
|
\section{Conclusions} |
523 |
|
|
|
524 |
|
|
\section{Acknowledgments} |
525 |
|
|
Support for this project was provided by the National Science |
526 |
|
|
Foundation under grant CHE-0134881. T.L. also acknowledges the |
527 |
|
|
financial support from center of applied mathematics at University |
528 |
|
|
of Notre Dame. |
529 |
|
|
\newpage |
530 |
|
|
|
531 |
|
|
\bibliographystyle{jcp2} |
532 |
|
|
\bibliography{langevin} |
533 |
|
|
|
534 |
|
|
\end{document} |