--- trunk/tengDissertation/Langevin.tex 2006/06/28 17:36:32 2904 +++ trunk/tengDissertation/Langevin.tex 2006/06/28 19:09:25 2905 @@ -180,8 +180,8 @@ S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt and oblate : \[ S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } -}}{a} -\], +}}{a}. +\] one can write down the translational and rotational resistance tensors \[ @@ -231,8 +231,8 @@ This equation is the basis for deriving the hydrodynam \label{introEquation:tensorExpression} \end{equation} This equation is the basis for deriving the hydrodynamic tensor. In -1930, Oseen and Burgers gave a simple solution to Equation -\ref{introEquation:tensorExpression} +1930, Oseen and Burgers gave a simple solution to +Eq.~\ref{introEquation:tensorExpression} \begin{equation} T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} @@ -248,9 +248,9 @@ Both of the Equation \ref{introEquation:oseenTensor} a \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. \label{introEquation:RPTensorNonOverlapped} \end{equation} -Both of the Equation \ref{introEquation:oseenTensor} and Equation -\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} -\ge \sigma _i + \sigma _j$. An alternative expression for +Both of the Eq.~\ref{introEquation:oseenTensor} and +Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption +$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for overlapping beads with the same radius, $\sigma$, is given by \begin{equation} T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - @@ -258,7 +258,6 @@ T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( \frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] \label{introEquation:RPTensorOverlapped} \end{equation} - To calculate the resistance tensor at an arbitrary origin $O$, we construct a $3N \times 3N$ matrix consisting of $N \times N$ $B_{ij}$ blocks @@ -276,15 +275,14 @@ matrix, we obtain \] where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ matrix, we obtain - \[ C = B^{ - 1} = \left( {\begin{array}{*{20}c} {C_{11} } & \ldots & {C_{1N} } \\ \vdots & \ddots & \vdots \\ {C_{N1} } & \cdots & {C_{NN} } \\ -\end{array}} \right) +\end{array}} \right), \] -, which can be partitioned into $N \times N$ $3 \times 3$ block +which can be partitioned into $N \times N$ $3 \times 3$ block $C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ \[ U_i = \left( {\begin{array}{*{20}c} @@ -296,14 +294,12 @@ arbitrary origin $O$ can be written as where $x_i$, $y_i$, $z_i$ are the components of the vector joining bead $i$ and origin $O$. Hence, the elements of resistance tensor at arbitrary origin $O$ can be written as -\begin{equation} -\begin{array}{l} - \Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ +\begin{eqnarray} + \Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ \Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ - \Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ - \end{array} + \Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ \label{introEquation:ResistanceTensorArbitraryOrigin} -\end{equation} +\end{eqnarray} The resistance tensor depends on the origin to which they refer. The proper location for applying friction force is the center of @@ -339,9 +335,9 @@ Using Equations \ref{introEquation:definitionCR} and { - y_{OP} } & {x_{OP} } & 0 \\ \end{array}} \right) \] -Using Equations \ref{introEquation:definitionCR} and -\ref{introEquation:resistanceTensorTransformation}, one can locate -the position of center of resistance, +Using Eq.~\ref{introEquation:definitionCR} and +Eq.~\ref{introEquation:resistanceTensorTransformation}, one can +locate the position of center of resistance, \begin{eqnarray*} \left( \begin{array}{l} x_{OR} \\ @@ -358,7 +354,6 @@ the position of center of resistance, (\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ \end{array} \right) \\ \end{eqnarray*} - where $x_OR$, $y_OR$, $z_OR$ are the components of the vector joining center of resistance $R$ and origin $O$. @@ -371,8 +366,8 @@ $V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ \end{equation} where $M_i$ is a $6\times6$ generalized diagonal mass (include mass and moment of inertial) matrix and $V_i$ is a generalized velocity, -$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ -(\ref{LDGeneralizedForm}) consists of three generalized forces in +$V_i = V_i(v_i,\omega _i)$. The right side of +Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in lab-fixed frame, systematic force $F_{s,i}$, dissipative force $F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the system in Newtownian mechanics typically refers to lab-fixed frame, @@ -382,12 +377,12 @@ in body-fixed frame and converted back to lab-fixed fr \[ \begin{array}{l} F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ - F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ - \end{array}. + F_{r,i}^l (t) = A^T F_{r,i}^b (t). \\ + \end{array} \] Here, the body-fixed friction force $F_{r,i}^b$ is proportional to the body-fixed velocity at center of resistance $v_{R,i}^b$ and -angular velocity $\omega _i$, +angular velocity $\omega _i$ \begin{equation} F_{r,i}^b (t) = \left( \begin{array}{l} f_{r,i}^b (t) \\ @@ -407,7 +402,6 @@ with zero mean and variance \left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = 2k_B T\Xi _R \delta (t - t'). \label{randomForce} \end{equation} - The equation of motion for $v_i$ can be written as \begin{equation} m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + @@ -429,7 +423,6 @@ + \tau _{r,i}^b(t) \dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) + \tau _{r,i}^b(t) \end{equation} - Embedding the friction terms into force and torque, one can integrate the langevin equations of motion for rigid body of arbitrary shape in a velocity-Verlet style 2-part algorithm, where @@ -449,7 +442,6 @@ $h= \delta t$: \mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} (t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). \end{align*} - In this context, the $\mathrm{rotate}$ function is the reversible product of the three body-fixed rotations, \begin{equation} @@ -484,11 +476,10 @@ All other rotations follow in a straightforward manner \end{array} \right). \end{equation} -All other rotations follow in a straightforward manner. +All other rotations follow in a straightforward manner. After the +first part of the propagation, the forces and body-fixed torques are +calculated at the new positions and orientations -After the first part of the propagation, the forces and body-fixed -torques are calculated at the new positions and orientations - {\tt doForces:} \begin{align*} {\bf f}(t + h) &\leftarrow @@ -500,12 +491,9 @@ torques are calculated at the new positions and orient {\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) \cdot {\bf \tau}^s(t + h). \end{align*} +Once the forces and torques have been obtained at the new time step, +the velocities can be advanced to the same time value. -{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix -$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and -torques have been obtained at the new time step, the velocities can -be advanced to the same time value. - {\tt moveB:} \begin{align*} {\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2