.. index:: pair_style multi/lucy

pair_style multi/lucy command
=============================

Syntax
""""""

.. code-block:: LAMMPS

   pair_style multi/lucy style N keyword ...

* style = *lookup* or *linear* = method of interpolation
* N = use N values in *lookup*\ , *linear* tables

Examples
""""""""

.. code-block:: LAMMPS

   pair_style multi/lucy linear 1000
   pair_coeff * * multibody.table ENTRY1 7.0

Description
"""""""""""

Style *multi/lucy* computes a density-dependent force following from
the many-body form described in :ref:`(Moore) <Moore1>` and
:ref:`(Warren) <Warren1>` as

.. math::

   F_{i}^{DD}(\rho_i,\rho_j,r_{ij}) = \frac{1}{2} \omega_{DD}\left(r_{ij}\right)
   \left[A\left(\rho_i\right) + A\left(\rho_j\right)\right]e_{ij}

which consists of a density-dependent function, :math:`A(\rho)`, and a
radial-dependent weight function, :math:`\omega_{DD}(r_{ij})`.  The
radial-dependent weight function, :math:`\omega_{DD}(r_{ij})`, is taken
as the Lucy function:

.. math::

   \omega_{DD}\left(r_{ij}\right) = \left(1+\frac{3r_{ij}}{r_{cut}}\right)\left(1+\frac{r_{ij}}{r_{cut}}\right)^3

The density-dependent energy for a given particle is given by:

.. math::

   u_{i}^{DD}\left(\rho_{i}\right) = \frac{\pi r_{cut}^4}{84} \int_{\rho_0}^{\rho_i} A\left(\rho'\right) d\rho'

See the supporting information of :ref:`(Brennan) <Brennan1>` or the
publication by :ref:`(Moore) <Moore1>` for more details on the functional
form.

An interpolation table is used to evaluate the density-dependent energy
(:math:`\int A(\rho') d\rho'`) and force (:math:`A(\rho')`).  Note that
the pre-factor to the energy is computed after the interpolation, thus
the :math:`\int A(\rho') d \rho'` will have units of energy / length\^4.

The interpolation table is created as a pre-computation by fitting
cubic splines to the file values and interpolating the
density-dependent energy and force at each of *N* densities.  During a
simulation, the tables are used to interpolate the density-dependent
energy and force as needed for each pair of particles separated by a
distance *R*\ .  The interpolation is done in one of 2 styles: *lookup*
and *linear*\ .

For the *lookup* style, the density is used to find the nearest table
entry, which is the density-dependent energy and force.

For the *linear* style, the density is used to find the 2 surrounding
table values from which the density-dependent energy and force are
computed by linear interpolation.

The following coefficients must be defined for each pair of atoms
types via the :doc:`pair_coeff <pair_coeff>` command as in the examples
above.

* filename
* keyword
* cutoff (distance units)

The filename specifies a file containing the tabulated
density-dependent energy and force.  The keyword specifies a section
of the file.  The cutoff is an optional coefficient.  If not
specified, the outer cutoff in the table itself (see below) will be
used to build an interpolation table that extend to the largest
tabulated distance.  If specified, only file values up to the cutoff
are used to create the interpolation table.  The format of this file
is described below.

----------

The format of a tabulated file is a series of one or more sections,
defined as follows (without the parenthesized comments):

.. parsed-literal::

   # Density-dependent function (one or more comment or blank lines)

   DD-FUNCTION                (keyword is first text on line)
   N 500 R 1.0 10.0           (N, R, RSQ parameters)
                              (blank)
   1 1.0 25.5 102.34          (index, density, energy/r\^4, force)
   2 1.02 23.4 98.5
   ...
   500 10.0 0.001 0.003

A section begins with a non-blank line whose first character is not a
"#"; blank lines or lines starting with "#" can be used as comments
between sections.  The first line begins with a keyword which
identifies the section.  The line can contain additional text, but the
initial text must match the argument specified in the pair_coeff
command.  The next line lists (in any order) one or more parameters
for the table.  Each parameter is a keyword followed by one or more
numeric values.

The parameter "N" is required and its value is the number of table
entries that follow.  Note that this may be different than the *N*
specified in the :doc:`pair_style multi/lucy <pair_multi_lucy>` command.
Let Ntable = *N* in the pair_style command, and Nfile = "N" in the
tabulated file.  What LAMMPS does is a preliminary interpolation by
creating splines using the Nfile tabulated values as nodal points.  It
uses these to interpolate the density-dependent energy and force at
Ntable different points.  The resulting tables of length Ntable are
then used as described above, when computing the density-dependent
energy and force.  This means that if you want the interpolation
tables of length Ntable to match exactly what is in the tabulated file
(with effectively no preliminary interpolation), you should set Ntable
= Nfile, and use the "RSQ" parameter.  This is because the internal
table abscissa is always RSQ (separation distance squared), for
efficient lookup.

All other parameters are optional.  If "R" or "RSQ" does
not appear, then the distances in each line of the table are used
as-is to perform spline interpolation.  In this case, the table values
can be spaced in *density* uniformly or however you wish to position table
values in regions of large gradients.

If used, the parameters "R" or "RSQ" are followed by 2 values *rlo* and
*rhi*\ .  If specified, the density associated with each
density-dependent energy and force value is computed from these 2 values
(at high accuracy), rather than using the (low-accuracy) value listed in
each line of the table.  The density values in the table file are
ignored in this case.  For "R", distances uniformly spaced between *rlo*
and *rhi* are computed; for "RSQ", squared distances uniformly spaced
between *rlo\*rlo* and *rhi\*rhi* are computed.

.. note::

   If you use "R" or "RSQ", the tabulated distance values in the file
   are effectively ignored, and replaced by new values as described in
   the previous paragraph.  If the density value in the table is not
   very close to the new value (i.e. round-off difference), then you
   will be assigning density-dependent energy and force values to a
   different density, which is probably not what you want.  LAMMPS will
   warn if this is occurring.

Following a blank line, the next N lines list the tabulated values.
On each line, the first value is the index from 1 to N, the second value is
r (in density units), the third value is the density-dependent function value
(in energy units / length\^4), and the fourth is the force (in force units).  The
density values must increase from one line to the next.

Note that one file can contain many sections, each with a tabulated
potential.  LAMMPS reads the file section by section until it finds
one that matches the specified keyword.

----------

**Mixing, shift, table, tail correction, restart, rRESPA info**\ :

This pair style does not support mixing.  Thus, coefficients for all
I,J pairs must be specified explicitly.

The :doc:`pair_modify <pair_modify>` shift, table, and tail options are
not relevant for this pair style.

This pair style writes the settings for the "pair_style multi/lucy" command
to :doc:`binary restart files <restart>`, so a pair_style command does
not need to specified in an input script that reads a restart file.
However, the coefficient information is not stored in the restart
file, since it is tabulated in the potential files.  Thus, pair_coeff
commands do need to be specified in the restart input script.

This pair style can only be used via the *pair* keyword of the
:doc:`run_style respa <run_style>` command.  It does not support the
*inner*\ , *middle*\ , *outer* keywords.

----------

Restrictions
""""""""""""

This command is part of the USER-DPD package.  It is only enabled if
LAMMPS was built with that package.  See the :doc:`Build package <Build_package>` doc page for more info.

Related commands
""""""""""""""""

:doc:`pair_coeff <pair_coeff>`

**Default:** none

----------

.. _Warren1:

**(Warren)** Warren, Phys Rev E, 68, 066702 (2003).

.. _Brennan1:

**(Brennan)** Brennan, J Chem Phys Lett, 5, 2144-2149 (2014).

.. _Moore1:

**(Moore)** Moore, J Chem Phys, 144, 104501 (2016).
