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2 < \chapter{\label{chapt:methodology}Langevin Dynamics for Rigid Bodies of Arbitrary Shape}
2 > \chapter{\label{chapt:methodology}LANGEVIN DYNAMICS for RIGID BODIES of ARBITRARY SHAPE}
3  
4   \section{Introduction}
5  
# Line 47 | Line 47 | efficiently\cite{Sandu1999}.
47   has been shown to be able to damp out the resonance artifact more
48   efficiently\cite{Sandu1999}.
49  
50 – %review rigid body dynamics
51 – Rigid bodies are frequently involved in the modeling of different
52 – areas, from engineering, physics, to chemistry. For example,
53 – missiles and vehicle are usually modeled by rigid bodies.  The
54 – movement of the objects in 3D gaming engine or other physics
55 – simulator is governed by the rigid body dynamics. In molecular
56 – simulation, rigid body is used to simplify the model in
57 – protein-protein docking study{\cite{Gray2003}}.
58 –
59 – It is very important to develop stable and efficient methods to
60 – integrate the equations of motion of orientational degrees of
61 – freedom. Euler angles are the nature choice to describe the
62 – rotational degrees of freedom. However, due to its singularity, the
63 – numerical integration of corresponding equations of motion is very
64 – inefficient and inaccurate. Although an alternative integrator using
65 – different sets of Euler angles can overcome this difficulty\cite{},
66 – the computational penalty and the lost of angular momentum
67 – conservation still remain. In 1977, a singularity free
68 – representation utilizing quaternions was developed by
69 – Evans\cite{Evans1977}. Unfortunately, this approach suffer from the
70 – nonseparable Hamiltonian resulted from quaternion representation,
71 – which prevents the symplectic algorithm to be utilized. Another
72 – different approach is to apply holonomic constraints to the atoms
73 – belonging to the rigid body\cite{}. Each atom moves independently
74 – under the normal forces deriving from potential energy and
75 – constraint forces which are used to guarantee the rigidness.
76 – However, due to their iterative nature, SHAKE and Rattle algorithm
77 – converge very slowly when the number of constraint increases.
78 –
79 – The break through in geometric literature suggests that, in order to
80 – develop a long-term integration scheme, one should preserve the
81 – geometric structure of the flow. Matubayasi and Nakahara developed a
82 – time-reversible integrator for rigid bodies in quaternion
83 – representation. Although it is not symplectic, this integrator still
84 – demonstrates a better long-time energy conservation than traditional
85 – methods because of the time-reversible nature. Extending
86 – Trotter-Suzuki to general system with a flat phase space, Miller and
87 – his colleagues devised an novel symplectic, time-reversible and
88 – volume-preserving integrator in quaternion representation. However,
89 – all of the integrators in quaternion representation suffer from the
90 – computational penalty of constructing a rotation matrix from
91 – quaternions to evolve coordinates and velocities at every time step.
92 – An alternative integration scheme utilizing rotation matrix directly
93 – is RSHAKE , in which a conjugate momentum to rotation matrix is
94 – introduced to re-formulate the Hamiltonian's equation and the
95 – Hamiltonian is evolved in a constraint manifold by iteratively
96 – satisfying the orthogonality constraint. However, RSHAKE is
97 – inefficient because of the iterative procedure. An extremely
98 – efficient integration scheme in rotation matrix representation,
99 – which also preserves the same structural properties of the
100 – Hamiltonian flow as Miller's integrator, is proposed by Dullweber,
101 – Leimkuhler and McLachlan (DLM).
102 –
50   %review langevin/browninan dynamics for arbitrarily shaped rigid body
51   Combining Langevin or Brownian dynamics with rigid body dynamics,
52   one can study the slow processes in biomolecular systems. Modeling
# Line 153 | Line 100 | sophisticated rigid body dynamics.
100  
101   \section{Method{\label{methodSec}}}
102  
103 < \subsection{\label{introSection:frictionTensor} Friction Tensor}
103 > \subsection{\label{introSection:frictionTensor}Friction Tensor}
104   Theoretically, the friction kernel can be determined using velocity
105   autocorrelation function. However, this approach become impractical
106   when the system become more and more complicate. Instead, various

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