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\chapter{\label{chapt:methodology}Langevin Dynamics for Rigid Bodies of Arbitrary Shape} |
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\chapter{\label{chapt:langevin}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\section{Introduction} |
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|
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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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|
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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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|
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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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systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
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oscillation power spectrum of nanoparticles from Langevin dynamics |
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simulation has the same peak frequencies for different wave |
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vectors,which recovers the property of magnetic excitations in small |
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finite structures\cite{Berkov2005a}. In an attempt to reduce the |
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computational cost of simulation, multiple time stepping (MTS) |
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methods have been introduced and have been of great interest to |
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macromolecule and protein community\cite{Tuckerman1992}. Relying on |
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the observation that forces between distant atoms generally |
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demonstrate slower fluctuations than forces between close atoms, MTS |
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method are generally implemented by evaluating the slowly |
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fluctuating forces less frequently than the fast ones. |
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Unfortunately, nonlinear instability resulting from increasing |
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timestep in MTS simulation have became a critical obstruction |
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preventing the long time simulation. Due to the coupling to the heat |
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bath, Langevin dynamics has been shown to be able to damp out the |
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resonance artifact more efficiently\cite{Sandu1999}. |
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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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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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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{Beard2001}. LTID can also deal with the rotational |
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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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estimation of friction tensor from hydrodynamics theory into the |
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sophisticated rigid body dynamics. |
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|
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\section{Method{\label{methodSec}}} |
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\section{Computational Methods{\label{methodSec}}} |
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|
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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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approaches based on hydrodynamics have been developed to calculate |
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the friction coefficients. The friction effect is isotropic in |
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Equation, $\zeta$ can be taken as a scalar. In general, friction |
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tensor $\Xi$ is a $6\times 6$ matrix given by |
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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 become |
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impractical when the system become more and more complicate. |
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Instead, various approaches based on hydrodynamics have been |
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developed to calculate the friction coefficients. In general, |
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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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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
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|
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For a spherical particle, the translational and rotational friction |
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constant can be calculated from Stoke's law, |
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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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Other non-spherical shape, such as cylinder and ellipsoid |
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\textit{etc}, are widely used as reference for developing new |
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hydrodynamics theory, because their properties can be calculated |
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exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
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also called a triaxial ellipsoid, which is given in Cartesian |
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coordinates by\cite{Perrin1934, Perrin1936} |
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exactly. In 1936, Perrin extended Stokes's law to general |
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ellipsoids, also called a triaxial ellipsoid, which is given in |
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Cartesian coordinates 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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due to the complexity of the elliptic integral, only the ellipsoid |
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with the restriction of two axes having to be 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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exactly. Introducing an elliptic integral parameter $S$ for prolate |
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: |
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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, |
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and oblate : |
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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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tensors |
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\[ |
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\begin{array}{l} |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
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\end{array}, |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
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2a}}, |
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\end{array} |
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\] |
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and |
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\[ |
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\begin{array}{l} |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
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\end{array}. |
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\] |
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|
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
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|
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Unlike spherical and other regular shaped molecules, there is not |
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analytical solution for friction tensor of any arbitrary shaped |
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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 space, $r$-space, to all possible |
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combination of rotational diffusion coefficients, $D$-space is not |
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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 body, general ellipsoid |
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is not always suitable for modeling arbitrarily shaped rigid |
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molecule. A number of studies have been devoted to determine the |
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friction tensor for irregularly shaped rigid bodies using more |
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translational and rotational motion of rigid body, general |
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ellipsoids are not always suitable for modeling arbitrarily shaped |
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rigid molecule. A number of studies have been devoted to determine |
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the friction tensor for irregularly shaped rigid bodies using more |
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advanced method where the molecule of interest was modeled by |
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combinations of spheres(beads)\cite{Carrasco1999} and the |
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hydrodynamics properties of the molecule can be calculated using the |
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B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
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)T_{ij} |
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\] |
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where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
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$B$, we obtain |
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where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
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matrix, we obtain |
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|
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\[ |
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C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
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|
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The resistance tensor depends on the origin to which they refer. The |
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proper location for applying friction force is the center of |
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resistance (reaction), at which the trace of rotational resistance |
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tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
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resistance is defined as an unique point of the rigid body at which |
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the translation-rotation coupling tensor are symmetric, |
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resistance (or center of reaction), at which the trace of rotational |
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resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
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Mathematically, the center of resistance is defined as an unique |
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point of the rigid body at which the translation-rotation coupling |
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tensor are symmetric, |
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\begin{equation} |
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\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
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\label{introEquation:definitionCR} |
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\end{equation} |
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Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
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From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
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we can easily find out that the translational resistance tensor is |
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origin independent, while the rotational resistance tensor and |
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translation-rotation coupling resistance tensor depend on the |
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origin. Given resistance tensor at an arbitrary origin $O$, and a |
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vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
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origin. Given the resistance tensor at an arbitrary origin $O$, and |
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a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
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obtain the resistance tensor at $P$ by |
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\begin{equation} |
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\begin{array}{l} |
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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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|
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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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|
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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} |
372 |
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where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
373 |
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and moment of inertial) matrix and $V_i$ is a generalized velocity, |
374 |
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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.~ |
375 |
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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 |
377 |
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$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
405 |
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\begin{equation} |
406 |
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\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
407 |
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\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
408 |
< |
2k_B T\Xi _R \delta (t - t'). |
408 |
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2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
409 |
|
\end{equation} |
410 |
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|
411 |
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The equation of motion for $v_i$ can be written as |
517 |
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+ \frac{h}{2} {\bf \tau}^b(t + h) . |
518 |
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\end{align*} |
519 |
|
|
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\section{Results and discussion} |
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\section{Results and Discussion} |
521 |
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|
522 |
> |
The Langevin algorithm described in previous section has been |
523 |
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implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
524 |
> |
studies of kinetics and thermodynamic properties in several systems. |
525 |
> |
|
526 |
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\subsection{Temperature Control} |
527 |
> |
|
528 |
> |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
529 |
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the solvent's thermal motions is controlled by the external |
530 |
> |
temperature. The capability to maintain the temperature of the whole |
531 |
> |
system was usually used to measure the stability and efficiency of |
532 |
> |
the algorithm. In order to verify the stability of this new |
533 |
> |
algorithm, a series of simulations are performed on system |
534 |
> |
consisiting of 256 SSD water molecules with different viscosities. |
535 |
> |
Initial configuration for the simulations is taken from a 1ns NVT |
536 |
> |
simulation with a cubic box of 19.7166~\AA. All simulation are |
537 |
> |
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
538 |
> |
with reference temperature at 300~K. Average temperature as a |
539 |
> |
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
540 |
> |
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
541 |
> |
1$ poise. The better temperature control at higher viscosity can be |
542 |
> |
explained by the finite size effect and relative slow relaxation |
543 |
> |
rate at lower viscosity regime. |
544 |
> |
\begin{table} |
545 |
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\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
546 |
> |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
547 |
> |
\label{langevin:viscosity} |
548 |
> |
\begin{center} |
549 |
> |
\begin{tabular}{|l|l|l|} |
550 |
> |
\hline |
551 |
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$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
552 |
> |
1 & 300.47 & 10.99 \\ |
553 |
> |
0.1 & 301.19 & 11.136 \\ |
554 |
> |
0.01 & 303.04 & 11.796 \\ |
555 |
> |
\hline |
556 |
> |
\end{tabular} |
557 |
> |
\end{center} |
558 |
> |
\end{table} |
559 |
> |
|
560 |
> |
Another set of calculation were performed to study the efficiency of |
561 |
> |
temperature control using different temperature coupling schemes. |
562 |
> |
The starting configuration is cooled to 173~K and evolved using NVE, |
563 |
> |
NVT, and Langevin dynamic with time step of 2 fs. |
564 |
> |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
565 |
> |
the systems reach equilibrium. The orange curve in |
566 |
> |
Fig.~\ref{langevin:temperature} represents the simulation using |
567 |
> |
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
568 |
> |
which gives reasonable tight coupling, while the blue one from |
569 |
> |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
570 |
> |
scaling to the desire temperature. In extremely lower friction |
571 |
> |
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
572 |
> |
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
573 |
> |
the temperature control ability. |
574 |
> |
|
575 |
> |
\begin{figure} |
576 |
> |
\centering |
577 |
> |
\includegraphics[width=\linewidth]{temperature.eps} |
578 |
> |
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
579 |
> |
temperature fluctuation versus time.} \label{langevin:temperature} |
580 |
> |
\end{figure} |
581 |
|
|
582 |
+ |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
583 |
+ |
|
584 |
+ |
In order to verify that Langevin dynamics can mimic the dynamics of |
585 |
+ |
the systems absent of explicit solvents, we carried out two sets of |
586 |
+ |
simulations and compare their dynamic properties. |
587 |
+ |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
588 |
+ |
made of 256 pentane molecules and two banana shaped molecules at |
589 |
+ |
273~K. It has an equivalent implicit solvent system containing only |
590 |
+ |
two banana shaped molecules with viscosity of 0.289 center poise. To |
591 |
+ |
calculate the hydrodynamic properties of the banana shaped molecule, |
592 |
+ |
we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
593 |
+ |
in which the banana shaped molecule is represented as a ``shell'' |
594 |
+ |
made of 2266 small identical beads with size of 0.3 \AA on the |
595 |
+ |
surface. Applying the procedure described in |
596 |
+ |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
597 |
+ |
identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
598 |
+ |
well as the resistance tensor, |
599 |
+ |
\[ |
600 |
+ |
\left( {\begin{array}{*{20}c} |
601 |
+ |
0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
602 |
+ |
3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
603 |
+ |
-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
604 |
+ |
5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
605 |
+ |
0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
606 |
+ |
0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
607 |
+ |
\end{array}} \right). |
608 |
+ |
\] |
609 |
+ |
Curves of velocity auto-correlation functions in |
610 |
+ |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
611 |
+ |
However, because of the stochastic nature, simulation using Langevin |
612 |
+ |
dynamics was shown to decay slightly fast. In order to study the |
613 |
+ |
rotational motion of the molecules, we also calculated the auto- |
614 |
+ |
correlation function of the principle axis of the second GB |
615 |
+ |
particle, $u$. |
616 |
+ |
|
617 |
+ |
\begin{figure} |
618 |
+ |
\centering |
619 |
+ |
\includegraphics[width=\linewidth]{roughShell.eps} |
620 |
+ |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
621 |
+ |
model for banana shaped molecule.} \label{langevin:roughShell} |
622 |
+ |
\end{figure} |
623 |
+ |
|
624 |
+ |
\begin{figure} |
625 |
+ |
\centering |
626 |
+ |
\includegraphics[width=\linewidth]{twoBanana.eps} |
627 |
+ |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
628 |
+ |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
629 |
+ |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
630 |
+ |
\end{figure} |
631 |
+ |
|
632 |
+ |
\begin{figure} |
633 |
+ |
\centering |
634 |
+ |
\includegraphics[width=\linewidth]{vacf.eps} |
635 |
+ |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
636 |
+ |
auto-correlation functions in NVE (blue) and Langevin dynamics |
637 |
+ |
(red).} \label{langevin:vacf} |
638 |
+ |
\end{figure} |
639 |
+ |
|
640 |
+ |
\begin{figure} |
641 |
+ |
\centering |
642 |
+ |
\includegraphics[width=\linewidth]{uacf.eps} |
643 |
+ |
\caption[Auto-correlation functions of the principle axis of the |
644 |
+ |
middle GB particle]{Auto-correlation functions of the principle axis |
645 |
+ |
of the middle GB particle in NVE (blue) and Langevin dynamics |
646 |
+ |
(red).} \label{langevin:twoBanana} |
647 |
+ |
\end{figure} |
648 |
+ |
|
649 |
|
\section{Conclusions} |
650 |
+ |
|
651 |
+ |
We have presented a new Langevin algorithm by incorporating the |
652 |
+ |
hydrodynamics properties of arbitrary shaped molecules into an |
653 |
+ |
advanced symplectic integration scheme. The temperature control |
654 |
+ |
ability of this algorithm was demonstrated by a set of simulations |
655 |
+ |
with different viscosities. It was also shown to have significant |
656 |
+ |
advantage of producing rapid thermal equilibration over |
657 |
+ |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
658 |
+ |
shaped molecules illustrated that the dynamic properties could be |
659 |
+ |
preserved by using this new algorithm as an implicit solvent model. |