[ViewVC] Diff of: group/trunk/openmdDocs/openmdDoc.tex

Comparing trunk/openmdDocs/openmdDoc.tex (file contents):
Revision 3707 by gezelter, Wed Jun 16 15:38:45 2010 UTC vs.
Revision 3708 by kstocke1, Mon Nov 22 22:34:45 2010 UTC

+that is easy to learn.
+\tableofcontents
-<
+%\listoffigures
-<
+%\listoftables
->
+\listoffigures
->
+\listoftables
+\mainmatter
+{\tt minimizer} & string & Chooses a minimizer & Possible minimizers
+are SD and CG. Either {\tt ensemble} or {\tt minimizer} must be specified. \\
+{\tt ensemble} & string & Sets the ensemble. & Possible ensembles are
-<
+NVE, NVT, NPTi, NPAT, NPTf, NPTxyz, and LD.  Either {\tt ensemble}
->
+NVE, NVT, NPTi, NPAT, NPTf, NPTxyz, LD and LHull.  Either {\tt ensemble}
+or {\tt minimizer} must be specified. \\
+{\tt dt} & fs & Sets the time step. & Selection of {\tt dt} should be
+small enough to sample the fastest motion of the simulation. ({\tt
+ &  (with separate barostats on each box dimension) & \\
+LD & Langevin Dynamics & {\tt ensemble = LD;} \\
+ &  (approximates the effects of an implicit solvent) & \\
-+
+ LangevinHull & Non-periodic Langevin Dynamics  & {\tt ensemble = LHull;} \\
-+
+ & (Langevin Dynamics for molecules on convex hull;\\
-+
+ & Newtonian for interior molecules) & \\
+\end{tabular}
+\end{center}
+in the body-fixed frame) and ${\bf V}$ is a generalized velocity,
+${\bf V} =
+\left\{{\bf v},{\bf \omega}\right\}$. The right side of
-<
+Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a
->
+Eq. \ref{LDGeneralizedForm} consists of three generalized forces: a
+system force (${\bf F}_{s}$), a frictional or dissipative force (${\bf
+F}_{f}$) and a stochastic force (${\bf F}_{r}$). While the evolution
+of the system in Newtonian mechanics is typically done in the lab
+\endhead
+\hline
+\endfoot
-<
+{\tt viscosity} & centipoise & Sets the value of viscosity of the implicit
->
+{\tt viscosity} & poise & Sets the value of viscosity of the implicit
+solvent  \\
+{\tt targetTemp} & K & Sets the target temperature of the system.
+This parameter must be specified to use Langevin dynamics. \\
+{\tt HydroPropFile} & string & Specifies the name of the resistance
+tensor (usually a {\tt .diff} file) which is precalculated by {\tt
-<
+Hydro}. This keyworkd is not necessary if the simulation contains only
->
+Hydro}. This keyword is not necessary if the simulation contains only
+simple bodies (spheres and ellipsoids). \\
+{\tt beadSize} & $\mbox{\AA}$ & Sets the diameter of the beads to use
+when the {\tt RoughShell} model is used to approximate the resistance
+tensor.
+\label{table:ldParameters}
+\end{longtable}
-+
-+
+\section{Langevin Hull Dynamics (LHull)}
-+
-+
+The Langevin Hull uses an external bath at a fixed constant pressure
-+
+($P$) and temperature ($T$) with an effective solvent viscosity
-+
+($\eta$).  This bath interacts only with the objects on the exterior
-+
+hull of the system.  Defining the hull of the atoms in a simulation is
-+
+done in a manner similar to the approach of Kohanoff, Caro and
-+
+Finnis.\cite{Kohanoff:2005qm} That is, any instantaneous configuration
-+
+of the atoms in the system is considered as a point cloud in three
-+
+dimensional space.  Delaunay triangulation is used to find all facets
-+
+between coplanar
-+
+neighbors.\cite{delaunay,springerlink:10.1007/BF00977785} In highly
-+
+symmetric point clouds, facets can contain many atoms, but in all but
-+
+the most symmetric of cases, the facets are simple triangles in
-+
+-space which contain exactly three atoms.
-+
+The convex hull is the set of facets that have {\it no concave
-+
+  corners} at an atomic site.\cite{Barber96,EDELSBRUNNER:1994oq} This
-+
+eliminates all facets on the interior of the point cloud, leaving only
-+
+those exposed to the bath. Sites on the convex hull are dynamic; as
-+
+molecules re-enter the cluster, all interactions between atoms on that
-+
+molecule and the external bath are removed.  Since the edge is
-+
+determined dynamically as the simulation progresses, no {\it a priori}
-+
+geometry is defined. The pressure and temperature bath interacts only
-+
+with the atoms on the edge and not with atoms interior to the
-+
+simulation.
-+
-+
+Atomic sites in the interior of the simulation move under standard
-+
+Newtonian dynamics,
-+
+\begin{equation}
-+
+m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U,
-+
+\label{eq:Newton}
-+
+\end{equation}
-+
+where $m_i$ is the mass of site $i$, ${\mathbf v}_i(t)$ is the
-+
+instantaneous velocity of site $i$ at time $t$, and $U$ is the total
-+
+potential energy.  For atoms on the exterior of the cluster
-+
+(i.e. those that occupy one of the vertices of the convex hull), the
-+
+equation of motion is modified with an external force, ${\mathbf
-+
+  F}_i^{\mathrm ext}$:
-+
+\begin{equation}
-+
+m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}.
-+
+\end{equation}
-+
-+
+The external bath interacts indirectly with the atomic sites through
-+
+the intermediary of the hull facets.  Since each vertex (or atom)
-+
+provides one corner of a triangular facet, the force on the facets are
-+
+divided equally to each vertex.  However, each vertex can participate
-+
+in multiple facets, so the resultant force is a sum over all facets
-+
+$f$ containing vertex $i$:
-+
+\begin{equation}
-+
+{\mathbf F}_{i}^{\mathrm ext} = \sum_{\begin{array}{c}\mathrm{facets\
-+
+    } f \\ \mathrm{containing\ } i\end{array}} \frac{1}{3}\  {\mathbf
-+
+  F}_f^{\mathrm ext}
-+
+\end{equation}
-+
-+
+The external pressure bath applies a force to the facets of the convex
-+
+hull in direct proportion to the area of the facet, while the thermal
-+
+coupling depends on the solvent temperature, viscosity and the size
-+
+and shape of each facet. The thermal interactions are expressed as a
-+
+standard Langevin description of the forces,
-+
+\begin{equation}
-+
+\begin{array}{rclclcl}
-+
+{\mathbf F}_f^{\text{ext}} & = &  \text{external pressure} & + & \text{drag force} & + & \text{random force} \\
-+
+& = &  -\hat{n}_f P A_f  & - & \Xi_f(t) {\mathbf v}_f(t)  & + & {\mathbf R}_f(t)
-+
+\end{array}
-+
+\end{equation}
-+
+Here, $A_f$ and $\hat{n}_f$ are the area and (outward-facing) normal
-+
+vectors for facet $f$, respectively.  ${\mathbf v}_f(t)$ is the
-+
+velocity of the facet centroid,
-+
+\begin{equation}
-+
+{\mathbf v}_f(t) =  \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i,
-+
+\end{equation}
-+
+and $\Xi_f(t)$ is an approximate ($3 \times 3$) resistance tensor that
-+
+depends on the geometry and surface area of facet $f$ and the
-+
+viscosity of the bath.  The resistance tensor is related to the
-+
+fluctuations of the random force, $\mathbf{R}(t)$, by the
-+
+fluctuation-dissipation theorem,
-+
+\begin{eqnarray}
-+
+\left< {\mathbf R}_f(t) \right> & = & 0 \\
-+
+\left<{\mathbf R}_f(t) {\mathbf R}_f^T(t^\prime)\right> & = & 2 k_B T\
-+
+\Xi_f(t)\delta(t-t^\prime).
-+
+\label{eq:randomForce}
-+
+\end{eqnarray}
-+
-+
+Once the resistance tensor is known for a given facet, a stochastic
-+
+vector that has the properties in Eq. (\ref{eq:randomForce}) can be
-+
+calculated efficiently by carrying out a Cholesky decomposition to
-+
+obtain the square root matrix of the resistance tensor,
-+
+\begin{equation}
-+
+\Xi_f = {\bf S} {\bf S}^{T},
-+
+\label{eq:Cholesky}
-+
+\end{equation}
-+
+where ${\bf S}$ is a lower triangular matrix.\cite{Schlick2002} A
-+
+vector with the statistics required for the random force can then be
-+
+obtained by multiplying ${\bf S}$ onto a random 3-vector ${\bf Z}$ which
-+
+has elements chosen from a Gaussian distribution, such that:
-+
+\begin{equation}
-+
+\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot
-+
+{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij},
-+
+\end{equation}
-+
+where $\delta t$ is the timestep in use during the simulation. The
-+
+random force, ${\bf R}_{f} = {\bf S} {\bf Z}$, can be shown to
-+
+have the correct properties required by Eq. (\ref{eq:randomForce}).
-+
-+
+Our treatment of the resistance tensor is approximate.  $\Xi_f$ for a
-+
+rigid triangular plate would normally be treated as a $6 \times 6$
-+
+tensor that includes translational and rotational drag as well as
-+
+translational-rotational coupling. The computation of resistance
-+
+tensors for rigid bodies has been detailed
-+
+elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun:2008fk}
-+
+but the standard approach involving bead approximations would be
-+
+prohibitively expensive if it were recomputed at each step in a
-+
+molecular dynamics simulation.
-+
-+
+Instead, we are utilizing an approximate resistance tensor obtained by
-+
+first constructing the Oseen tensor for the interaction of the
-+
+centroid of the facet ($f$) with each of the subfacets $\ell=1,2,3$,
-+
+\begin{equation}
-+
+T_{\ell f}=\frac{A_\ell}{8\pi\eta R_{\ell f}}\left(I +
-+
+  \frac{\mathbf{R}_{\ell f}\mathbf{R}_{\ell f}^T}{R_{\ell f}^2}\right)
-+
+\end{equation}
-+
+Here, $A_\ell$ is the area of subfacet $\ell$ which is a triangle
-+
+containing two of the vertices of the facet along with the centroid.
-+
+$\mathbf{R}_{\ell f}$ is the vector between the centroid of facet $f$
-+
+and the centroid of sub-facet $\ell$, and $I$ is the ($3 \times 3$)
-+
+identity matrix.  $\eta$ is the viscosity of the external bath.
-+
-+
+The tensors for each of the sub-facets are added together, and the
-+
+resulting matrix is inverted to give a $3 \times 3$ resistance tensor
-+
+for translations of the triangular facet,
-+
+\begin{equation}
-+
+\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1}.
-+
+\end{equation}
-+
+Note that this treatment ignores rotations (and
-+
+translational-rotational coupling) of the facet.  In compact systems,
-+
+the facets stay relatively fixed in orientation between
-+
+configurations, so this appears to be a reasonably good approximation.
-+
-+
+At each
-+
+molecular dynamics time step, the following process is carried out:
-+
+\begin{enumerate}
-+
+\item The standard inter-atomic forces ($\nabla_iU$) are computed.
-+
+\item Delaunay triangulation is carried out using the current atomic
-+
+  configuration.
-+
+\item The convex hull is computed and facets are identified.
-+
+\item For each facet:
-+
+\begin{itemize}
-+
+\item[a.] The force from the pressure bath ($-\hat{n}_fPA_f$) is
-+
+  computed.
-+
+\item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the
-+
+  viscosity ($\eta$) of the bath.
-+
+\item[c.] Facet drag ($-\Xi_f(t) \mathbf{v}_f(t)$) forces are
-+
+  computed.
-+
+\item[d.] Random forces ($\mathbf{R}_f(t)$) are computed using the
-+
+  resistance tensor and the temperature ($T$) of the bath.
-+
+\end{itemize}
-+
+\item The facet forces are divided equally among the vertex atoms.
-+
+\item Atomic positions and velocities are propagated.
-+
+\end{enumerate}
-+
+The Delaunay triangulation and computation of the convex hull are done
-+
+using calls to the qhull library.\cite{Qhull} There is a minimal
-+
+penalty for computing the convex hull and resistance tensors at each
-+
+step in the molecular dynamics simulation (roughly 0.02 $\times$ cost
-+
+of a single force evaluation), and the convex hull is remarkably easy
-+
+to parallelize on distributed memory machines.
-+
-+
-+
+\begin{longtable}[c]{GBF}
-+
+\caption{Meta-data Keywords: Required parameters for the Langevin Hull integrator}
-+
+\\
-+
+{\bf keyword} & {\bf units} & {\bf use}  \\ \hline
-+
+\endhead
-+
+\hline
-+
+\endfoot
-+
+{\tt viscosity} & poise & Sets the value of viscosity of the implicit
-+
+solven . \\
-+
+{\tt targetTemp} & K & Sets the target temperature of the system.
-+
+This parameter must be specified to use Langevin Hull dynamics. \\
-+
+{\tt targetPressure} & atm & Sets the target pressure of the system.
-+
+This parameter must be specified to use Langevin Hull dynamics. \\
-+
+{\tt usePeriodicBoundaryConditions = false} & logical & Turns off periodic boundary conditions.
-+
+This parameter must be set to \tt false \\
-+
+\label{table:lhullParameters}
-+
+\end{longtable}
-+
-+
+\section{\label{sec:constraints}Constraint Methods}
+\subsection{\label{section:rattle}The {\sc rattle} Method for Bond
+\end{center}
+For example, the phrase {\tt select mass > 16.0 and charge < -2}
-<
+wouldselect StuntDoubles which have mass greater than 16.0 and charges
->
+would select StuntDoubles which have mass greater than 16.0 and charges
+less than -2.
+\subsection{\label{section:within}Within expressions}

Diff Legend

-–
+Removed lines
-+
+Added lines
-<
+Changed lines
->
+Changed lines

Comparing trunk/openmdDocs/openmdDoc.tex (file contents): Revision 3707 by gezelter, Wed Jun 16 15:38:45 2010 UTC vs. Revision 3708 by kstocke1, Mon Nov 22 22:34:45 2010 UTC

Diff Legend

Comparing trunk/openmdDocs/openmdDoc.tex (file contents):
Revision 3707 by gezelter, Wed Jun 16 15:38:45 2010 UTC vs.
Revision 3708 by kstocke1, Mon Nov 22 22:34:45 2010 UTC