415 lines
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HTML
415 lines
30 KiB
HTML
<!DOCTYPE html>
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<title>replicate command — LAMMPS documentation</title>
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<div class="lammps_version">Version: <b>19 Nov 2024</b></div>
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<p class="caption" role="heading"><span class="caption-text">User Guide</span></p>
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<ul>
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<li class="toctree-l1"><a class="reference internal" href="Intro.html">1. Introduction</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Install.html">2. Install LAMMPS</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Build.html">3. Build LAMMPS</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Run_head.html">4. Run LAMMPS</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Commands.html">5. Commands</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Packages.html">6. Optional packages</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Speed.html">7. Accelerate performance</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Howto.html">8. Howto discussions</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Examples.html">9. Example scripts</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Tools.html">10. Auxiliary tools</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Errors.html">11. Errors</a></li>
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<p class="caption" role="heading"><span class="caption-text">Programmer Guide</span></p>
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<ul>
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<li class="toctree-l1"><a class="reference internal" href="Library.html">1. LAMMPS Library Interfaces</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Python_head.html">2. Use Python with LAMMPS</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Modify.html">3. Modifying & extending LAMMPS</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Developer.html">4. Information for Developers</a></li>
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</ul>
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<p class="caption" role="heading"><span class="caption-text">Command Reference</span></p>
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<ul class="current">
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<li class="toctree-l1 current"><a class="reference internal" href="commands_list.html">Commands</a><ul class="current">
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<li class="toctree-l2"><a class="reference internal" href="angle_coeff.html">angle_coeff command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="angle_style.html">angle_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="angle_write.html">angle_write command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="atom_modify.html">atom_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="atom_style.html">atom_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="balance.html">balance command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="bond_coeff.html">bond_coeff command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="bond_style.html">bond_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="bond_write.html">bond_write command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="boundary.html">boundary command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="change_box.html">change_box command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="clear.html">clear command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="comm_modify.html">comm_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="comm_style.html">comm_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="compute.html">compute command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="compute_modify.html">compute_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="create_atoms.html">create_atoms command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="create_bonds.html">create_bonds command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="create_box.html">create_box command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="delete_atoms.html">delete_atoms command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="delete_bonds.html">delete_bonds command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="dielectric.html">dielectric command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="dihedral_coeff.html">dihedral_coeff command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="dihedral_style.html">dihedral_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="dihedral_write.html">dihedral_write command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="dimension.html">dimension command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="displace_atoms.html">displace_atoms command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="dynamical_matrix.html">dynamical_matrix command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="echo.html">echo command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="fix.html">fix command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="fix_modify.html">fix_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="fitpod_command.html">fitpod command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="geturl.html">geturl command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="group.html">group command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="group2ndx.html">group2ndx command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="group2ndx.html#ndx2group-command">ndx2group command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="hyper.html">hyper command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="if.html">if command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="improper_coeff.html">improper_coeff command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="improper_style.html">improper_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="include.html">include command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="info.html">info command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="jump.html">jump command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="kim_commands.html">kim command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="kspace_modify.html">kspace_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="kspace_style.html">kspace_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="label.html">label command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="labelmap.html">labelmap command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="lattice.html">lattice command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="log.html">log command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="mass.html">mass command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="mdi.html">mdi command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_modify.html">min_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_spin.html">min_style spin command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_spin.html#min-style-spin-cg-command">min_style spin/cg command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_spin.html#min-style-spin-lbfgs-command">min_style spin/lbfgs command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html">min_style cg command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html#min-style-hftn-command">min_style hftn command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html#min-style-sd-command">min_style sd command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html#min-style-quickmin-command">min_style quickmin command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html#min-style-fire-command">min_style fire command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html#min-style-spin-command"><span class="xref std std-doc">min_style spin</span> command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html#min-style-spin-cg-command"><span class="xref std std-doc">min_style spin/cg</span> command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="min_style.html#min-style-spin-lbfgs-command"><span class="xref std std-doc">min_style spin/lbfgs</span> command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="minimize.html">minimize command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="molecule.html">molecule command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="neb.html">neb command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="neb_spin.html">neb/spin command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="neigh_modify.html">neigh_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="neighbor.html">neighbor command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="newton.html">newton command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="next.html">next command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="package.html">package command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="pair_coeff.html">pair_coeff command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="pair_modify.html">pair_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="pair_style.html">pair_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="pair_write.html">pair_write command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="partition.html">partition command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="plugin.html">plugin command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="prd.html">prd command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="print.html">print command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="processors.html">processors command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="python.html">python command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="quit.html">quit command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="read_data.html">read_data command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="read_dump.html">read_dump command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="read_restart.html">read_restart command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="region.html">region command</a></li>
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<li class="toctree-l2 current"><a class="current reference internal" href="#">replicate command</a><ul>
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<li class="toctree-l3"><a class="reference internal" href="#syntax">Syntax</a></li>
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<li class="toctree-l3"><a class="reference internal" href="#examples">Examples</a></li>
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<li class="toctree-l3"><a class="reference internal" href="#description">Description</a></li>
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<li class="toctree-l3"><a class="reference internal" href="#restrictions">Restrictions</a></li>
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<li class="toctree-l3"><a class="reference internal" href="#related-commands">Related commands</a></li>
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<li class="toctree-l3"><a class="reference internal" href="#default">Default</a></li>
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</ul>
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</li>
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<li class="toctree-l2"><a class="reference internal" href="rerun.html">rerun command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="reset_atoms.html">reset_atoms command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="reset_timestep.html">reset_timestep command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="restart.html">restart command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="run.html">run command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="run_style.html">run_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="set.html">set command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="shell.html">shell command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="special_bonds.html">special_bonds command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="suffix.html">suffix command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="tad.html">tad command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="temper.html">temper command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="temper_grem.html">temper/grem command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="temper_npt.html">temper/npt command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="thermo.html">thermo command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="thermo_modify.html">thermo_modify command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="thermo_style.html">thermo_style command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="third_order.html">third_order command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="timer.html">timer command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="timestep.html">timestep command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="uncompute.html">uncompute command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="undump.html">undump command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="unfix.html">unfix command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="units.html">units command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="variable.html">variable command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="velocity.html">velocity command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="write_coeff.html">write_coeff command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="write_data.html">write_data command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="write_dump.html">write_dump command</a></li>
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<li class="toctree-l2"><a class="reference internal" href="write_restart.html">write_restart command</a></li>
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</ul>
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</li>
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<li class="toctree-l1"><a class="reference internal" href="fixes.html">Fix Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="computes.html">Compute Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="pairs.html">Pair Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="bonds.html">Bond Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="angles.html">Angle Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="dihedrals.html">Dihedral Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="impropers.html">Improper Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="dumps.html">Dump Styles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="fix_modify_atc_commands.html">fix_modify AtC commands</a></li>
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<li class="toctree-l1"><a class="reference internal" href="Bibliography.html">Bibliography</a></li>
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</ul>
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</div>
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</div>
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<li class="breadcrumb-item active">replicate command</li>
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<a href="https://www.lammps.org"><img src="_static/lammps-logo.png" width="64" height="16" alt="LAMMPS Homepage"></a> | <a href="Commands_all.html">Commands</a>
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<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
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<div itemprop="articleBody">
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||
<p><span class="math notranslate nohighlight">\(\renewcommand{\AA}{\text{Å}}\)</span></p>
|
||
<section id="replicate-command">
|
||
<span id="index-0"></span><h1>replicate command<a class="headerlink" href="#replicate-command" title="Link to this heading"></a></h1>
|
||
<section id="syntax">
|
||
<h2>Syntax<a class="headerlink" href="#syntax" title="Link to this heading"></a></h2>
|
||
<div class="highlight-LAMMPS notranslate"><div class="highlight"><pre><span></span><span class="k">replicate</span><span class="w"> </span><span class="n">nx</span><span class="w"> </span><span class="n">ny</span><span class="w"> </span><span class="n">nz</span><span class="w"> </span><span class="n">keyword</span><span class="w"> </span><span class="n">...</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>nx,ny,nz = replication factors in each dimension</p>
|
||
<ul>
|
||
<li><p>zero or more keywords may be appended</p></li>
|
||
<li><p>keyword = <em>bbox</em> or <em>bond/periodic</em></p>
|
||
<pre class="literal-block"><em>bbox</em> = use a bounding-box algorithm which is faster for large proc counts
|
||
<em>bond/periodic</em> = use an algorithm that correctly replicates periodic bond loops</pre>
|
||
</li>
|
||
</ul>
|
||
</section>
|
||
<section id="examples">
|
||
<h2>Examples<a class="headerlink" href="#examples" title="Link to this heading"></a></h2>
|
||
<p>For examples of replicating simple linear polymer chains (periodic or
|
||
non-periodic) or periodic carbon nanotubes, see examples/replicate.</p>
|
||
<div class="highlight-LAMMPS notranslate"><div class="highlight"><pre><span></span><span class="k">replicate</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="m">3</span><span class="w"> </span><span class="m">2</span>
|
||
<span class="k">replicate</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="m">3</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="n">bbox</span>
|
||
<span class="k">replicate</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="m">3</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="n">bond</span><span class="o">/</span><span class="n">periodic</span>
|
||
</pre></div>
|
||
</div>
|
||
</section>
|
||
<section id="description">
|
||
<h2>Description<a class="headerlink" href="#description" title="Link to this heading"></a></h2>
|
||
<p>Replicate the current system one or more times in each dimension. For
|
||
example, replication factors of 2,2,2 will create a simulation with 8x
|
||
as many atoms by doubling the size of the simulation box in each
|
||
dimension. A replication factor of 1 leaves the simulation domain
|
||
unchanged in that dimension.</p>
|
||
<p>When the new simulation box is created it is partitioned into a
|
||
regular 3d grid of rectangular bricks, one per processor, based on the
|
||
number of processors being used and the settings of the
|
||
<a class="reference internal" href="processors.html"><span class="doc">processors</span></a> command. The partitioning can be
|
||
changed by subsequent <a class="reference internal" href="balance.html"><span class="doc">balance</span></a> or <a class="reference internal" href="fix_balance.html"><span class="doc">fix balance</span></a> commands.</p>
|
||
<p>All properties of each atom are replicated (except per-atom fix data,
|
||
see the Restrictions section below). This includes their velocities,
|
||
which may or may not be desirable. New atom IDs are assigned to new
|
||
atoms, as are new molecule IDs. Bonds and other topology interactions
|
||
are created between pairs of new atoms as well as between old and new
|
||
atoms.</p>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>The bond discussion which follows only refers to models with
|
||
permanent covalent bonds typically defined in LAMMPS via a data
|
||
file. It is not relevant to systems modeled with many-body
|
||
potentials which can define bonds on-the-fly, based on the current
|
||
positions of nearby atoms, e.g. models using the <a class="reference internal" href="pair_airebo.html"><span class="doc">AIREBO</span></a> or <a class="reference internal" href="pair_reaxff.html"><span class="doc">ReaxFF</span></a> potentials.</p>
|
||
</div>
|
||
<p>If the <em>bond/periodic</em> keyword is not specified, bond replication is
|
||
done by using the image flag for each atom to “unwrap” it out of the
|
||
periodic box before replicating it. After replication is performed,
|
||
atoms outside the new periodic box are wrapped back into it. This
|
||
assigns correct images flags to all atoms in the system. For this to
|
||
work, all original atoms in the original simulation box must have
|
||
consistent image flags. This means that if two atoms have a bond
|
||
between them which crosses a periodic boundary, their respective image
|
||
flags will differ by 1 in that dimension.</p>
|
||
<p>Image flag consistency is not possible if a system has a periodic bond
|
||
loop, meaning there is a chain of bonds which crosses an entire
|
||
dimension and re-connects to itself across a periodic boundary. In
|
||
this case you MUST use the <em>bond/periodic</em> keyword to correctly
|
||
replicate the system. This option zeroes the image flags for all
|
||
atoms and uses a different algorithm to find new (nearby) bond
|
||
neighbors in the replicated system. In the final replicated system
|
||
all image flags are zero (in each dimension).</p>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>LAMMPS does not check for image flag consistency before performing
|
||
the replication (it does issue a warning about this before a
|
||
simulation is run). If the original image flags are inconsistent,
|
||
the replicated system will also have inconsistent image flags, but
|
||
will otherwise be correctly replicated. This is NOT the case if
|
||
there is a periodic bond loop. See the next note.</p>
|
||
</div>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>LAMMPS does not check for periodic bond loops. If you use the
|
||
<em>bond/periodic</em> keyword for a system without periodic bond loops,
|
||
the system will be correctly replicated, but image flag information
|
||
will be lost (which may or may not be important to your model). If
|
||
you do not use the <em>bond/periodic</em> keyword for a system with
|
||
periodic bond loops, the replicated system will have invalid bonds
|
||
(typically very long), resulting in bad dynamics.</p>
|
||
</div>
|
||
<p>If possible, the <em>bbox</em> keyword should be used when running on a large
|
||
number of processors, as it can result in a substantial speed-up for
|
||
the replication operation. It uses a bounding box to only check atoms
|
||
in replicas that overlap with each processor’s new subdomain when
|
||
assigning atoms to processors. It also preserves image flag
|
||
information. The only drawback to the <em>bbox</em> option is that it
|
||
requires a temporary use of more memory. Each processor must be able
|
||
to store all atoms (and their per-atom data) in the original system,
|
||
before it is replicated.</p>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>The algorithm used by the <em>bond/periodic</em> keyword builds on the
|
||
algorithm used by the <em>bbox</em> keyword and thus has the same memory
|
||
requirements. If you specify only the <em>bond/peridoic</em> keyword it
|
||
will internally set the <em>bbox</em> keyword as well.</p>
|
||
</div>
|
||
</section>
|
||
<hr class="docutils" />
|
||
<section id="restrictions">
|
||
<h2>Restrictions<a class="headerlink" href="#restrictions" title="Link to this heading"></a></h2>
|
||
<p>A 2d simulation cannot be replicated in the z dimension.</p>
|
||
<p>If a simulation is non-periodic in a dimension, care should be used
|
||
when replicating it in that dimension, as it may generate atoms nearly
|
||
on top of each other.</p>
|
||
<p>If the current simulation was read in from a restart file (before a
|
||
run is performed), there must not be any fix information stored in the
|
||
file for individual atoms. Similarly, no fixes can be defined at the
|
||
time the replicate command is used that require vectors of atom
|
||
information to be stored. This is because the replicate command does
|
||
not know how to replicate that information for new atoms it creates.</p>
|
||
<p>To work around this restriction two options are possible. (1) Fixes
|
||
which use the stored data in the restart file can be defined before
|
||
replication and then deleted via the <a class="reference internal" href="unfix.html"><span class="doc">unfix</span></a> command and
|
||
re-defined after it. Or (2) the restart file can be converted to a
|
||
data file (which deletes the stored fix information) and fixes defined
|
||
after the replicate command. In both these scenarios, the per-atom
|
||
fix information in the restart file is lost.</p>
|
||
</section>
|
||
<section id="related-commands">
|
||
<h2>Related commands<a class="headerlink" href="#related-commands" title="Link to this heading"></a></h2>
|
||
<p>none</p>
|
||
</section>
|
||
<section id="default">
|
||
<h2>Default<a class="headerlink" href="#default" title="Link to this heading"></a></h2>
|
||
<p>No settings for using the <em>bbox</em> or <em>bond/periodic</em> algorithms.</p>
|
||
</section>
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||
</section>
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