2006-09-22 00:22:34 +08:00
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<HTML>
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<CENTER><A HREF = "http://lammps.sandia.gov">LAMMPS WWW Site</A> - <A HREF = "Manual.html">LAMMPS Documentation</A> - <A HREF = "Section_commands.html#comm">LAMMPS Commands</A>
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<HR>
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2010-05-02 08:59:55 +08:00
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<H3>fix rigid command
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</H3>
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<H3>fix rigid/nve command
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</H3>
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<H3>fix rigid/nvt command
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2006-09-22 00:22:34 +08:00
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</H3>
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<P><B>Syntax:</B>
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</P>
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2010-05-02 08:59:55 +08:00
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<PRE>fix ID group-ID style bodystyle args keyword values ...
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2006-09-22 00:22:34 +08:00
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</PRE>
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<UL><LI>ID, group-ID are documented in <A HREF = "fix.html">fix</A> command
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2010-05-02 08:59:55 +08:00
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<LI>style = <I>rigid</I> or <I>rigid/nve</I> or <I>rigid/nvt</I>
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2006-09-22 00:22:34 +08:00
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2009-01-13 22:38:26 +08:00
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<LI>bodystyle = <I>single</I> or <I>molecule</I> or <I>group</I>
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2006-09-22 00:22:34 +08:00
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2009-01-13 22:38:26 +08:00
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<PRE> <I>single</I> args = none
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<I>molecule</I> args = none
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<I>group</I> args = N groupID1 groupID2 ...
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N = # of groups
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groupID1, groupID2, ... = list of N group IDs
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</PRE>
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<LI>zero or more keyword/value pairs may be appended
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2011-04-30 00:27:56 +08:00
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<LI>keyword = <I>langevin</I> or <I>temp</I> or <I>tparam</I> or <I>force</I> or <I>torque</I>
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2009-01-13 22:38:26 +08:00
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2011-04-30 00:27:56 +08:00
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<PRE> <I>langevin</I> values = Tstart Tstop Tperiod seed
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Tstart,Tstop = desired temperature at start/stop of run (temperature units)
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Tdamp = temperature damping parameter (time units)
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seed = random number seed to use for white noise (positive integer)
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<I>temp</I> values = Tstart Tstop Tdamp
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2010-05-02 08:59:55 +08:00
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Tstart,Tstop = desired temperature at start/stop of run (temperature units)
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Tdamp = temperature damping parameter (time units)
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<I>tparam</I> values = Tchain Titer Torder
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Tchain = length of Nose/Hoover thermostat chain
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Titer = number of thermostat iterations performed
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Torder = 3 or 5 = Yoshida-Suzuki integration parameters
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<I>force</I> values = M xflag yflag zflag
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2009-01-13 22:38:26 +08:00
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M = which rigid body from 1-Nbody (see asterisk form below)
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xflag,yflag,zflag = off/on if component of center-of-mass force is active
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<I>torque</I> values = M xflag yflag zflag
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M = which rigid body from 1-Nbody (see asterisk form below)
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xflag,yflag,zflag = off/on if component of center-of-mass torque is active
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2006-09-22 00:22:34 +08:00
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</PRE>
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</UL>
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<P><B>Examples:</B>
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</P>
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<PRE>fix 1 clump rigid single
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2011-04-30 00:27:56 +08:00
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fix 1 clump rigid single force 1 off off on langevin 1.0 1.0 1.0 428984
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2010-05-02 08:59:55 +08:00
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fix 1 polychains rigid/nvt molecule temp 1.0 1.0 5.0
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2009-01-13 22:38:26 +08:00
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fix 1 polychains rigid molecule force 1*5 off off off force 6*10 off off on
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fix 2 fluid rigid group 3 clump1 clump2 clump3 torque * off off off
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2006-09-22 00:22:34 +08:00
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</PRE>
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<P><B>Description:</B>
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</P>
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2009-07-01 01:19:42 +08:00
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<P>Treat one or more sets of atoms as independent rigid bodies. This
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2006-09-22 00:22:34 +08:00
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means that each timestep the total force and torque on each rigid body
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2009-07-01 01:19:42 +08:00
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is computed as the sum of the forces and torques on its constituent
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particles and the coordinates, velocities, and orientations of the
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atoms in each body are updated so that the body moves and rotates as a
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single entity.
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</P>
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<P>Examples of large rigid bodies are a large colloidal particle, or
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portions of a large biomolecule such as a protein.
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</P>
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<P>Example of small rigid bodies are patchy nanoparticles, such as those
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2009-07-01 01:27:43 +08:00
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modeled in <A HREF = "#Zhang">this paper</A> by Sharon Glotzer's group, clumps of
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granular particles, lipid molecules consiting of one or more point
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dipoles connected to other spheroids or ellipsoids, and coarse-grain
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models of nano or colloidal particles consisting of a small number of
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constituent particles. Note that the <A HREF = "fix_shake.html">fix shake</A>
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command can also be used to rigidify small molecules of 2, 3, or 4
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atoms, e.g. water molecules. That fix treats the constituent atoms as
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point masses.
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2009-07-01 01:19:42 +08:00
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</P>
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2010-05-02 08:59:55 +08:00
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<P>These fixes also update the positions and velocities of the atoms in
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each rigid body via time integration. The <I>rigid</I> and <I>rigid/nve</I>
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styles do this via constant NVE integration. The only difference is
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that the <I>rigid</I> style uses an integration technique based on
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Richardson iterations. The <I>rigid/nve</I> style uses the methods
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described in the paper by <A HREF = "#Miller">Miller</A>, which are thought to
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provide better energy conservation than an iterative approach.
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</P>
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<P>The <I>rigid/nvt</I> style performs constant NVT integration using a
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Nose/Hoover thermostat with chains as described originally in
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<A HREF = "#Hoover">(Hoover)</A> and <A HREF = "#Martyna">(Martyna)</A>, which thermostats both
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the translational and rotational degrees of freedom of the rigid
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bodies. The rigid-body algorithm used by <I>rigid/nvt</I> is described in
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the paper by <A HREF = "#Kamberaj">Kamberaj</A>.
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</P>
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<P>IMPORTANT NOTE: You should not update the atoms in rigid bodies via
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other time-integration fixes (e.g. nve, nvt, npt), or you will be
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integrating their motion more than once each timestep.
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</P>
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<P>IMPORTANT NOTE: These fixes are overkill if you simply want to hold a
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collection of atoms stationary or have them move with a constant
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velocity. A simpler way to hold atoms stationary is to not include
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those atoms in your time integration fix. E.g. use "fix 1 mobile nve"
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instead of "fix 1 all nve", where "mobile" is the group of atoms that
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you want to move. You can move atoms with a constant velocity by
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assigning them an initial velocity (via the <A HREF = "velocity.html">velocity</A>
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command), setting the force on them to 0.0 (via the <A HREF = "fix_setforce.html">fix
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setforce</A> command), and integrating them as usual
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(e.g. via the <A HREF = "fix_nve.html">fix nve</A> command).
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</P>
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<HR>
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2009-07-01 01:19:42 +08:00
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<P>The constituent particles within a rigid body can be point particles
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2011-04-14 05:39:34 +08:00
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(the default in LAMMPS) or finite-size particles, such as spheres and
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ellipsoids. See the <A HREF = "atom_style.html">atom_style sphere and ellipsoid</A>
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commands for more details on these kinds of particles. Finite-size
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particles contribute differently to the moment of inertia of a rigid
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body than do point particles. Finite-size particles can also
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experience torque (e.g. due to <A HREF = "pair_gran.html">frictional granular
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2009-07-01 01:19:42 +08:00
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interactions</A>) and have an orientation. These
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2010-05-02 08:59:55 +08:00
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contributions are accounted for by these fixes.
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2009-07-01 01:19:42 +08:00
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</P>
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<P>Forces between particles within a body do not contribute to the
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external force or torque on the body. Thus for computational
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efficiency, you may wish to turn off pairwise and bond interactions
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between particles within each rigid body. The <A HREF = "neigh_modify.html">neigh_modify
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exclude</A> and <A HREF = "delete_bonds.html">delete_bonds</A>
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commands are used to do this. For finite-size particles this also
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means the particles can be highly overlapped when creating the rigid
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body.
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</P>
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<HR>
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2009-01-13 22:38:26 +08:00
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<P>Each body must have two or more atoms. An atom can belong to at most
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one rigid body. Which atoms are in which bodies can be defined via
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several options.
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2006-09-22 00:22:34 +08:00
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</P>
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2009-01-13 22:38:26 +08:00
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<P>For bodystyle <I>single</I> the entire fix group of atoms is treated as one
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rigid body.
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2006-09-22 00:22:34 +08:00
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</P>
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2009-01-13 22:38:26 +08:00
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<P>For bodystyle <I>molecule</I>, each set of atoms in the fix group with a
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different molecule ID is treated as a rigid body.
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2006-09-22 00:22:34 +08:00
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</P>
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2009-01-13 22:38:26 +08:00
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<P>For bodystyle <I>group</I>, each of the listed groups is treated as a
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separate rigid body. Only atoms that are also in the fix group are
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2006-09-22 00:22:34 +08:00
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included in each rigid body.
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</P>
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2011-02-16 05:14:11 +08:00
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<P>IMPORTANT NOTE: To compute the initial center-of-mass position and
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other properties of each rigid body, the image flags for each atom in
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the body are used to "unwrap" the atom coordinates. Thus you must
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insure that these image flags are consistent so that the unwrapping
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creates a valid rigid body (one where the atoms are close together),
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particularly if the atoms in a single rigid body straddle a periodic
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boundary. This means the input data file or restart file must define
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the image flags for each atom consistently or that you have used the
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<A HREF = "set.html">set</A> command to specify them correctly. If a dimension is
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non-periodic then the image flag of each atom must be 0 in that
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dimension, else an error is generated.
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</P>
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2009-10-30 00:10:55 +08:00
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<P>By default, each rigid body is acted on by other atoms which induce an
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external force and torque on its center of mass, causing it to
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translate and rotate. Components of the external center-of-mass force
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and torque can be turned off by the <I>force</I> and <I>torque</I> keywords.
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This may be useful if you wish a body to rotate but not translate, or
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vice versa, or if you wish it to rotate or translate continuously
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unaffected by interactions with other particles. Note that if you
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expect a rigid body not to move or rotate by using these keywords, you
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must insure its initial center-of-mass translational or angular
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velocity is 0.0. Otherwise the initial translational or angular
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momentum the body has will persist.
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2009-01-13 22:38:26 +08:00
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</P>
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<P>An xflag, yflag, or zflag set to <I>off</I> means turn off the component of
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force of torque in that dimension. A setting of <I>on</I> means turn on
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the component, which is the default. Which rigid body(s) the settings
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apply to is determined by the first argument of the <I>force</I> and
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<I>torque</I> keywords. It can be an integer M from 1 to Nbody, where
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Nbody is the number of rigid bodies defined. A wild-card asterisk can
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be used in place of, or in conjunction with, the M argument to set the
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flags for multiple rigid bodies. This takes the form "*" or "*n" or
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"n*" or "m*n". If N = the number of rigid bodies, then an asterisk
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with no numeric values means all bodies from 1 to N. A leading
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asterisk means all bodies from 1 to n (inclusive). A trailing
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asterisk means all bodies from n to N (inclusive). A middle asterisk
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means all types from m to n (inclusive). Note that you can use the
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<I>force</I> or <I>torque</I> keywords as many times as you like. If a
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particular rigid body has its component flags set multiple times, the
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settings from the final keyword are used.
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2009-01-13 22:38:26 +08:00
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</P>
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<P>For computational efficiency, you may wish to turn off pairwise and
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2006-09-22 00:22:34 +08:00
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bond interactions within each rigid body, as they no longer contribute
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to the motion. The <A HREF = "neigh_modify.html">neigh_modify exclude</A> and
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<A HREF = "delete_bonds.html">delete_bonds</A> commands are used to do this.
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</P>
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2010-05-02 08:59:55 +08:00
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<P>For computational efficiency, you should typically define one fix
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rigid which includes all the desired rigid bodies. LAMMPS will allow
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multiple rigid fixes to be defined, but it is more expensive.
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2006-09-22 00:22:34 +08:00
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</P>
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2010-05-02 08:59:55 +08:00
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<HR>
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2011-04-30 00:27:56 +08:00
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<P>The keyword/value option pairs are used in the following ways.
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</P>
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<P>The <I>langevin</I> and <I>temp</I> and <I>tparam</I> keywords perform thermostatting
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of the rigid bodies, altering both their translational and rotational
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degrees of freedom. What is meant by "temperature" of a collection of
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rigid bodies and how it can be monitored via the fix output is
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discussed below.
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</P>
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<P>The <I>langevin</I> keyword applies a Langevin thermostat to the constant
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NVE time integration performed by either the <I>rigid</I> or <I>rigid/nve</I>
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styles. It cannot be used with the <I>rigid/nvt</I> style. The desired
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temperature at each timestep is a ramped value during the run from
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<I>Tstart</I> to <I>Tstop</I>. The <I>Tdamp</I> parameter is specified in time units
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and determines how rapidly the temperature is relaxed. For example, a
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value of 100.0 means to relax the temperature in a timespan of
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(roughly) 100 time units (tau or fmsec or psec - see the
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<A HREF = "units.html">units</A> command). The random # <I>seed</I> must be a positive
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integer. The way the Langevin thermostatting operates is explained on
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the <A HREF = "fix_langevin.html">fix langevin</A> doc page.
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</P>
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<P>The <I>temp</I> and <I>tparam</I> keywords apply a Nose/Hoover thermostat to the
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NVT time integration performed by the <I>rigid/nvt</I> style. They cannot
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be used with the <I>rigid</I> or <I>rigid/nve</I> styles. The desired
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temperature at each timestep is a ramped value during the run from
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<I>Tstart</I> to <I>Tstop</I>. The <I>Tdamp</I> parameter is specified in time units
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and determines how rapidly the temperature is relaxed. For example, a
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value of 100.0 means to relax the temperature in a timespan of
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(roughly) 100 time units (tau or fmsec or psec - see the
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<A HREF = "units.html">units</A> command).
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2010-05-02 08:59:55 +08:00
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</P>
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<P>Nose/Hoover chains are used in conjunction with this thermostat. The
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<I>tparam</I> keyword can optionally be used to change the chain settings
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used. <I>Tchain</I> is the number of thermostats in the Nose Hoover chain.
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This value, along with <I>Tdamp</I> can be varied to dampen undesirable
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oscillations in temperature that can occur in a simulation. As a rule
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of thumb, increasing the chain length should lead to smaller
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oscillations.
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</P>
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2011-04-30 00:27:56 +08:00
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<P>IMPORTANT NOTE: There are alternate ways to thermostat a system of
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rigid bodies. You can use <A HREF = "fix_langevin.html">fix langevin</A> to treat
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|
the individual particles in the rigid bodies as effectively immersed
|
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in an implicit solvent, e.g. a Brownian dynamics model. For hybrid
|
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systems with both rigid bodies and solvent particles, you can
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thermostat only the solvent particles that surround one or more rigid
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bodies by appropriate choice of groups in the compute and fix commands
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for temperature and thermostatting. The solvent interactions with the
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rigid bodies should then effectively thermostat the rigid body
|
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|
temperature as well without use of the Langevin or Nose/Hoover options
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|
associated with the fix rigid commands.
|
2009-07-02 22:39:37 +08:00
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</P>
|
2010-05-02 08:59:55 +08:00
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<HR>
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|
2011-04-30 00:27:56 +08:00
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<P>The keyword/value option pairs are used in the following ways.
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</P>
|
2010-05-02 08:59:55 +08:00
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<P>If you use a <A HREF = "compute.html">temperature compute</A> with a group that
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includes particles in rigid bodies, the degrees-of-freedom removed by
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each rigid body are accounted for in the temperature (and pressure)
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computation, but only if the temperature group includes all the
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particles in a particular rigid body.
|
2009-07-02 22:39:37 +08:00
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</P>
|
2009-07-02 23:09:14 +08:00
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<P>A 3d rigid body has 6 degrees of freedom (3 translational, 3
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rotational), except for a collection of point particles lying on a
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straight line, which has only 5, e.g a dimer. A 2d rigid body has 3
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degrees of freedom (2 translational, 1 rotational).
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</P>
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<P>IMPORTANT NOTE: You may wish to explicitly subtract additional
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degrees-of-freedom if you use the <I>force</I> and <I>torque</I> keywords to
|
2010-05-02 08:59:55 +08:00
|
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|
eliminate certain motions of one or more rigid bodies. LAMMPS does
|
2009-07-02 23:09:14 +08:00
|
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not do this automatically.
|
2009-01-23 02:56:18 +08:00
|
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</P>
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|
<P>The rigid body contribution to the pressure of the system (virial) is
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also accounted for by this fix.
|
2006-09-22 00:22:34 +08:00
|
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</P>
|
2007-08-29 22:13:51 +08:00
|
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|
<P>IMPORTANT NOTE: The periodic image flags of atoms in rigid bodies are
|
2010-10-07 22:59:11 +08:00
|
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|
altered so that the rigid body can be reconstructed correctly when it
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|
straddles periodic boundaries. The atom image flags are not
|
2010-10-07 23:10:14 +08:00
|
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|
incremented/decremented as they would be for non-rigid atoms as the
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|
rigid body crosses periodic boundaries. This means you cannot
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|
|
interpret them as you normally would. For example, the image flag
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values written to a <A HREF = "dump.html">dump file</A> will be different than they
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|
would be if the atoms were not in a rigid body. Likewise the <A HREF = "compute_msd.html">compute
|
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|
|
msd</A> will not compute the expected mean-squared
|
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|
displacement for such atoms if the body moves across periodic
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|
boundaries. It also means that if you have bonds between a pair of
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|
rigid bodies and the bond straddles a periodic boundary, you cannot
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use the <A HREF = "replicate.html">replicate</A> command to increase the system
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|
|
size. Note that this fix does define image flags for each rigid body,
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|
|
which are incremented when the rigid body crosses a periodic boundary
|
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|
in the usual way. These image flags have the same meaning as atom
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|
images (see the "dump" command) and can be accessed and output as
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|
|
described below.
|
2007-08-29 22:13:51 +08:00
|
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|
</P>
|
2010-05-02 08:59:55 +08:00
|
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|
<HR>
|
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|
2007-10-11 06:28:11 +08:00
|
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|
<P><B>Restart, fix_modify, output, run start/stop, minimize info:</B>
|
2007-06-26 08:03:39 +08:00
|
|
|
</P>
|
2010-05-02 08:59:55 +08:00
|
|
|
<P>No information about the <I>rigid</I> and <I>rigid/nve</I> fixes are written to
|
|
|
|
|
<A HREF = "restart.html">binary restart files</A>. For style <I>rigid/nvt</I> the state
|
|
|
|
|
of the Nose/Hoover thermostat is written to <A HREF = "restart.html">binary restart
|
|
|
|
|
files</A>. See the <A HREF = "read_restart.html">read_restart</A> command
|
|
|
|
|
for info on how to re-specify a fix in an input script that reads a
|
|
|
|
|
restart file, so that the operation of the fix continues in an
|
|
|
|
|
uninterrupted fashion.
|
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|
|
|
</P>
|
2010-11-24 23:04:48 +08:00
|
|
|
<P>The <A HREF = "fix_modify.html">fix_modify</A> <I>energy</I> option is supported by the
|
|
|
|
|
rigid/nvt fix to add the energy change induced by the thermostatting
|
|
|
|
|
to the system's potential energy as part of <A HREF = "thermo_style.html">thermodynamic
|
|
|
|
|
output</A>.
|
|
|
|
|
</P>
|
2011-04-30 00:27:56 +08:00
|
|
|
<P>The rigid and rigid/nve fixes computes a global scalar which can be
|
2011-08-26 01:01:01 +08:00
|
|
|
accessed by various <A HREF = "Section_howto.html#howto_15">output commands</A>.
|
|
|
|
|
The scalar value calculated by these fixes is "intensive". The scalar
|
|
|
|
|
is the current temperature of the collection of rigid bodies. This is
|
2011-04-30 00:27:56 +08:00
|
|
|
averaged over all rigid bodies and their translational and rotational
|
|
|
|
|
degrees of freedom. The translational energy of a rigid body is 1/2 m
|
|
|
|
|
v^2, where m = total mass of the body and v = the velocity of its
|
|
|
|
|
center of mass. The rotational energy of a rigid body is 1/2 I w^2,
|
|
|
|
|
where I = the moment of inertia tensor of the body and w = its angular
|
|
|
|
|
velocity. Degrees of freedom constrained by the <I>force</I> and <I>torque</I>
|
|
|
|
|
keywords are removed from this calculation.
|
|
|
|
|
</P>
|
2010-11-24 23:04:48 +08:00
|
|
|
<P>The rigid/nvt fix computes a global scalar which can be accessed by
|
2011-08-26 01:01:01 +08:00
|
|
|
various <A HREF = "Section_howto.html#howto_15">output commands</A>. The scalar
|
|
|
|
|
value calculated by the rigid/nvt fix is "extensive". The scalar is
|
|
|
|
|
the cumulative energy change due to the thermostatting the fix
|
|
|
|
|
performs.
|
2010-11-24 23:04:48 +08:00
|
|
|
</P>
|
|
|
|
|
<P>All of these fixes compute a global array of values which can be
|
2011-08-26 01:01:01 +08:00
|
|
|
accessed by various <A HREF = "Section_howto.html#howto_15">output commands</A>.
|
|
|
|
|
The number of rows in the array is equal to the number of rigid
|
|
|
|
|
bodies. The number of columns is 15. Thus for each rigid body, 15
|
|
|
|
|
values are stored: the xyz coords of the center of mass (COM), the xyz
|
|
|
|
|
components of the COM velocity, the xyz components of the force acting
|
|
|
|
|
on the COM, the xyz components of the torque acting on the COM, and
|
|
|
|
|
the xyz image flags of the COM, which have the same meaning as image
|
|
|
|
|
flags for atom positions (see the "dump" command). The force and
|
|
|
|
|
torque values in the array are not affected by the <I>force</I> and
|
|
|
|
|
<I>torque</I> keywords in the fix rigid command; they reflect values before
|
|
|
|
|
any changes are made by those keywords.
|
2009-12-19 01:25:39 +08:00
|
|
|
</P>
|
|
|
|
|
<P>The ordering of the rigid bodies (by row in the array) is as follows.
|
|
|
|
|
For the <I>single</I> keyword there is just one rigid body. For the
|
|
|
|
|
<I>molecule</I> keyword, the bodies are ordered by ascending molecule ID.
|
|
|
|
|
For the <I>group</I> keyword, the list of group IDs determines the ordering
|
|
|
|
|
of bodies.
|
|
|
|
|
</P>
|
2010-05-02 08:59:55 +08:00
|
|
|
<P>The array values calculated by these fixes are "intensive", meaning
|
|
|
|
|
they are independent of the number of atoms in the simulation.
|
2009-01-13 22:38:26 +08:00
|
|
|
</P>
|
2010-05-02 08:59:55 +08:00
|
|
|
<P>No parameter of these fixes can be used with the <I>start/stop</I> keywords
|
|
|
|
|
of the <A HREF = "run.html">run</A> command. These fixes are not invoked during
|
|
|
|
|
<A HREF = "minimize.html">energy minimization</A>.
|
2007-06-26 08:03:39 +08:00
|
|
|
</P>
|
2006-09-22 00:22:34 +08:00
|
|
|
<P><B>Restrictions:</B>
|
|
|
|
|
</P>
|
2010-05-02 08:59:55 +08:00
|
|
|
<P>These fixes performs an MPI_Allreduce each timestep that is
|
|
|
|
|
proportional in length to the number of rigid bodies. Hence they will
|
|
|
|
|
not scale well in parallel if large numbers of rigid bodies are
|
|
|
|
|
simulated.
|
2006-09-22 00:22:34 +08:00
|
|
|
</P>
|
|
|
|
|
<P><B>Related commands:</B>
|
|
|
|
|
</P>
|
|
|
|
|
<P><A HREF = "delete_bonds.html">delete_bonds</A>, <A HREF = "neigh_modify.html">neigh_modify</A>
|
|
|
|
|
exclude
|
|
|
|
|
</P>
|
2009-01-13 22:38:26 +08:00
|
|
|
<P><B>Default:</B>
|
|
|
|
|
</P>
|
2010-05-02 08:59:55 +08:00
|
|
|
<P>The option defaults are force * on on on and torque * on on on,
|
|
|
|
|
meaning all rigid bodies are acted on by center-of-mass force and
|
|
|
|
|
torque. Also Tchain = 10, Titer = 1, Torder = 3.
|
2006-09-22 00:22:34 +08:00
|
|
|
</P>
|
2009-07-01 01:19:42 +08:00
|
|
|
<HR>
|
|
|
|
|
|
2010-05-02 08:59:55 +08:00
|
|
|
<A NAME = "Hoover"></A>
|
|
|
|
|
|
|
|
|
|
<P><B>(Hoover)</B> Hoover, Phys Rev A, 31, 1695 (1985).
|
|
|
|
|
</P>
|
|
|
|
|
<A NAME = "Kamberaj"></A>
|
|
|
|
|
|
|
|
|
|
<P><B>(Kamberaj)</B> Kamberaj, Low, Neal, J Chem Phys, 122, 224114 (2005).
|
|
|
|
|
</P>
|
|
|
|
|
<A NAME = "Martyna"></A>
|
|
|
|
|
|
|
|
|
|
<P><B>(Martyna)</B> Martyna, Klein, Tuckerman, J Chem Phys, 97, 2635 (1992);
|
|
|
|
|
Martyna, Tuckerman, Tobias, Klein, Mol Phys, 87, 1117.
|
|
|
|
|
</P>
|
|
|
|
|
<A NAME = "Miller"></A>
|
|
|
|
|
|
|
|
|
|
<P><B>(Miller)</B> Miller, Eleftheriou, Pattnaik, Ndirango, and Newns,
|
|
|
|
|
J Chem Phys, 116, 8649 (2002).
|
|
|
|
|
</P>
|
2009-07-01 01:27:43 +08:00
|
|
|
<A NAME = "Zhang"></A>
|
2009-07-01 01:19:42 +08:00
|
|
|
|
|
|
|
|
<P><B>(Zhang)</B> Zhang, Glotzer, Nanoletters, 4, 1407-1413 (2004).
|
|
|
|
|
</P>
|
2006-09-22 00:22:34 +08:00
|
|
|
</HTML>
|
