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%\documentclass[aps,prb,preprint]{revtex4} |
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\documentclass[11pt]{article} |
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\usepackage{endfloat} |
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\usepackage{amsmath,bm} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\usepackage{times} |
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\usepackage{mathptmx} |
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of motion. |
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Euler angles are a natural choice to describe the rotational degrees |
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of freedom. However, due to $1 \over \sin \theta$ singularities, the |
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of freedom. However, due to $\frac{1}{\sin \theta}$ singularities, the |
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numerical integration of corresponding equations of these motion can |
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become inaccurate (and inefficient). Although the use of multiple |
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sets of Euler angles can overcome this problem,\cite{Barojas1973} the |
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\section{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
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Consider the Langevin equations of motion in generalized coordinates |
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\begin{equation} |
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\mathbf{M}_i \dot \mathbf{V}_i(t) = \mathbf{F}_{s,i}(t) + \mathbf{F}_{f,i}(t) + \mathbf{R}_{i}(t) |
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\mathbf{M} \dot{\mathbf{V}}(t) = \mathbf{F}_{s}(t) + |
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\mathbf{F}_{f}(t) + \mathbf{R}(t) |
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\label{LDGeneralizedForm} |
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\end{equation} |
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where $\mathbf{M}_i$ is a $6\times6$ diagonal mass matrix (which |
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includes the rigid body mass and moments of inertia) and $\mathbf{V}_i$ is a |
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generalized velocity, $\mathbf{V}_i = |
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\left\{\mathbf{v}_i,\mathbf{\omega}_i \right\}$. The right side of |
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where $\mathbf{M}$ is a $6 \times 6$ diagonal mass matrix (which |
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includes the mass of the rigid body as well as the moments of inertia |
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in the body-fixed frame) and $\mathbf{V}$ is a generalized velocity, |
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$\mathbf{V} = |
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\left\{\mathbf{v},\mathbf{\omega}\right\}$. The right side of |
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Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a |
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system force $\mathbf{F}_{s,i}$, a frictional or dissipative force |
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$\mathbf{F}_{f,i}$ and stochastic force $\mathbf{R}_{i}$. While the |
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system force $\mathbf{F}_{s}$, a frictional or dissipative force |
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$\mathbf{F}_{f}$ and stochastic force $\mathbf{R}$. While the |
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evolution of the system in Newtownian mechanics is typically done in the |
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lab-fixed frame, it is convenient to handle the rotation of rigid |
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bodies in the body-fixed frame. Thus the friction and random forces are |
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calculated in body-fixed frame and converted back to lab-fixed frame |
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using the rigid body's rotation matrix ($Q_i$): |
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using the rigid body's rotation matrix ($Q$): |
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\begin{equation} |
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\begin{array}{l} |
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\mathbf{F}_{f,i}(t) = Q_{i}^{T} \mathbf{F}_{f,i}^b (t), \\ |
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\mathbf{R}_{i}(t) = Q_{i}^{T} \mathbf{R}_{i}^b (t). \\ |
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\mathbf{F}_{f}(t) = Q^{T} \mathbf{F}_{f}^b (t), \\ |
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\mathbf{R}(t) = Q^{T} \mathbf{R}^b (t). \\ |
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\end{array} |
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\end{equation} |
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Here, the body-fixed friction force $\mathbf{F}_{f,i}^b$ is proportional to |
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the body-fixed velocity at the center of resistance $\mathbf{v}_{R,i}^b$ and |
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angular velocity $\mathbf{\omega}_i$ |
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\begin{equation} |
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\mathbf{F}_{f,i}^b (t) = \left( \begin{array}{l} |
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\mathbf{f}_{f,i}^b (t) \\ |
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\mathbf{\tau}_{f,i}^b (t) \\ |
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\mathbf{F}_{f}^b (t) = \left( \begin{array}{l} |
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\mathbf{f}_{f}^b (t) \\ |
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\mathbf{\tau}_{f}^b (t) \\ |
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\end{array} \right) = - \left( \begin{array}{*{20}c} |
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\Xi_{R,t} & \Xi_{R,c}^T \\ |
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\Xi_{R,c} & \Xi_{R,r} \\ |
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\end{array} \right)\left( \begin{array}{l} |
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\mathbf{v}_{R,i}^b (t) \\ |
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\mathbf{\omega}_i (t) \\ |
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\mathbf{v}_{R}^b (t) \\ |
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\mathbf{\omega} (t) \\ |
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\end{array} \right), |
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\end{equation} |
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while the random force $\mathbf{R}_{i}^l$ is a Gaussian stochastic variable |
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while the random force $\mathbf{R}^l$ is a Gaussian stochastic variable |
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with zero mean and variance |
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\begin{equation} |
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\left\langle {\mathbf{R}_{i}^l (t) (\mathbf{R}_{i}^l (t'))^T } \right\rangle = |
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\left\langle {\mathbf{R}_{i}^b (t) (\mathbf{R}_{i}^b (t'))^T } \right\rangle = |
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\left\langle {\mathbf{R}^l (t) (\mathbf{R}^l (t'))^T } \right\rangle = |
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\left\langle {\mathbf{R}^b (t) (\mathbf{R}^b (t'))^T } \right\rangle = |
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2 k_B T \Xi_R \delta(t - t'). \label{randomForce} |
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\end{equation} |
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Once the $6\times6$ resistance tensor at the center of resistance |
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vector is needed, a gaussian random vector is generated and then the |
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square root matrix is multiplied onto this vector. |
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The equation of motion for $\mathbf{v}_i$ can be written as |
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The equation of motion for $\mathbf{v}$ can be written as |
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\begin{equation} |
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m\dot \mathbf{v}_i (t) = \mathbf{f}_{s,i} (t) + \mathbf{f}_{f,i}^l (t) + |
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\mathbf{R}_{i}^l (t) |
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m \dot{\mathbf{v}} (t) = \mathbf{f}_{s} (t) + \mathbf{f}_{f}^l (t) + |
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\mathbf{R}^l (t) |
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\end{equation} |
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Since the frictional force is applied at the center of resistance |
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which generally does not coincide with the center of mass, an extra |
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torque is exerted at the center of mass. Thus, the net body-fixed |
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frictional torque at the center of mass, $\tau_{f,i}^b (t)$, is |
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frictional torque at the center of mass, $\tau_{f}^b (t)$, is |
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given by |
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\begin{equation} |
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\tau_{f,i}^b \leftarrow \tau_{f,i}^b + \mathbf{r}_{MR} \times \mathbf{f}_{r,i}^b |
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\tau_{f}^b \leftarrow \tau_{f}^b + \mathbf{r}_{MR} \times \mathbf{f}_{r}^b |
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\end{equation} |
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where $r_{MR}$ is the vector from the center of mass to the center |
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of the resistance. Instead of integrating the angular velocity in |
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lab-fixed frame, we consider the equation of angular momentum in |
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body-fixed frame |
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\begin{equation} |
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\dot j_i (t) = \tau_{s,i} (t) + \tau_{f,i}^b (t) + \mathbf{R}_{i}^b(t) |
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\dot j(t) = \tau_{s} (t) + \tau_{f}^b (t) + \mathbf{R}^b(t) |
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\end{equation} |
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Embedding the friction terms into force and torque, one can integrate |
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the Langevin equations of motion for rigid body of arbitrary shape in |
