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\chapter{\label{chapt:methodology}Langevin Dynamics for Rigid Bodies of Arbitrary Shape} |
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\chapter{\label{chapt:methodology}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\section{Introduction} |
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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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%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). |
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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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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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\section{Computational methods{\label{methodSec}}} |
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\subsection{\label{introSection:frictionTensor} Friction Tensor} |
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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
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Theoretically, the friction kernel can be determined using velocity |
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autocorrelation function. However, this approach become impractical |
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when the system become more and more complicate. Instead, various |
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where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
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joining center of resistance $R$ and origin $O$. |
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\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
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\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
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Consider a Langevin equation of motions in generalized coordinates |
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\begin{equation} |
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\end{equation} |
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where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
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and moment of inertial) matrix and $V_i$ is a generalized velocity, |
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$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
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$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
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(\ref{LDGeneralizedForm}) consists of three generalized forces in |
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lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
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$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
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\section{Results and discussion} |
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The Langevin algorithm described in Sec.~\ref{LDRB} has been |
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implemented in {\sc oopse}\cite{Meineke2005} and applied to several |
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test systems. |
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\subsection{Langevin dynamics of} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{temperature.eps} |
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\caption[]{.} \label{langevin:temperature} |
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\end{figure} |
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\subsection{LD of banana-shaped molecule} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{one_banana.eps} |
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\caption[]{.} \label{langevin:banana} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{roughShell.eps} |
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\caption[Rough Shell]{Rough Shell.} \label{langevin:roughShell} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{twoBanana.eps} |
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\caption[Two Banana Shaped Molecules]{Two Banana Shaped Molecules.} |
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\label{langevin:twoBanana} |
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\end{figure} |
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\section{Conclusions} |