lammps/doc/variable.txt

"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c

:link(lws,http://lammps.sandia.gov)
:link(ld,Manual.html)
:link(lc,Section_commands.html#comm)

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variable command :h3

[Syntax:]

variable name style args ... :pre

name = name of variable to define :ulb,l
style = {delete} or {index} or {loop} or {world} or {universe} or {uloop} or {string} or {equal} or {atom} :l
  {delete} = no args
  {index} args = one or more strings
  {loop} args = N
    N = integer size of loop, loop from 1 to N inclusive
  {loop} args = N pad
    N = integer size of loop, loop from 1 to N inclusive
    pad = all values will be same length, e.g. 001, 002, ..., 100
  {loop} args = N1 N2
    N1,N2 = loop from N1 to N2 inclusive
  {loop} args = N1 N2 pad
    N1,N2 = loop from N1 to N2 inclusive
    pad = all values will be same length, e.g. 050, 051, ..., 100
  {world} args = one string for each partition of processors
  {universe} args = one or more strings
  {uloop} args = N
    N = integer size of loop
  {uloop} args = N pad
    N = integer size of loop
    pad = all values will be same length, e.g. 001, 002, ..., 100
  {string} arg = one string
  {equal} or {atom} args = one formula containing numbers, thermo keywords, math operations, group functions, atom values and vectors, compute/fix/variable references
    numbers = 0.0, 100, -5.4, 2.8e-4, etc
    constants = PI
    thermo keywords = vol, ke, press, etc from "thermo_style"_thermo_style.html
    math operators = (), -x, x+y, x-y, x*y, x/y, x^y, 
                     x==y, x!=y, x<y, x<=y, x>y, x>=y, x&&y, x||y, !x
    math functions = sqrt(x), exp(x), ln(x), log(x),
                     sin(x), cos(x), tan(x), asin(x), acos(x), atan(x), atan2(y,x),
                     random(x,y,z), normal(x,y,z), ceil(x), floor(x), round(x)
                     ramp(x,y), stagger(x,y), logfreq(x,y,z), stride(x,y,z), vdisplace(x,y), swiggle(x,y,z), cwiggle(x,y,z)
    group functions = count(group), mass(group), charge(group),
		      xcm(group,dim), vcm(group,dim), fcm(group,dim),
		      bound(group,xmin), gyration(group), ke(group),
		      angmom(group,dim), torque(group,dim),
                      inertia(group,dimdim), omega(group,dim)
    region functions = count(group,region), mass(group,region), charge(group,region),
		      xcm(group,dim,region), vcm(group,dim,region), fcm(group,dim,region),
		      bound(group,xmin,region), gyration(group,region), ke(group,reigon),
		      angmom(group,dim,region), torque(group,dim,region), 
		      inertia(group,dimdim,region), omega(group,dim,region)
    special functions = sum(x), min(x), max(x), ave(x), trap(x), gmask(x), rmask(x), grmask(x,y)
    atom value = mass\[i\], type\[i\], x\[i\], y\[i\], z\[i\], vx\[i\], vy\[i\], vz\[i\], fx\[i\], fy\[i\], fz\[i\]
    atom vector = mass, type, x, y, z, vx, vy, vz, fx, fy, fz
    compute references = c_ID, c_ID\[i\], c_ID\[i\]\[j\]
    fix references = f_ID, f_ID\[i\], f_ID\[i\]\[j\]
    variable references = v_name, v_name\[i\] :pre
:ule

[Examples:]

variable x index run1 run2 run3 run4 run5 run6 run7 run8
variable LoopVar loop $n
variable beta equal temp/3.0
variable b1 equal x\[234\]+0.5*vol
variable b1 equal "x\[234\] + 0.5*vol"
variable b equal xcm(mol1,x)/2.0
variable b equal c_myTemp
variable b atom x*y/vol
variable foo string myfile
variable temp world 300.0 310.0 320.0 $\{Tfinal\}
variable x universe 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
variable x uloop 15 pad
variable x delete :pre

[Description:]

This command assigns one or more strings to a variable name for
evaluation later in the input script or during a simulation.

Variables can be used in several ways in LAMMPS.  A variable can be
referenced elsewhere in an input script to become part of a new input
command.  For variable styles that store multiple strings, the
"next"_next.html command can be used to increment which string is
assigned to the variable.  Variables of style {equal} store a formula
which when evaluated produces a single numeric value which can be
output either directly (see the "print"_print.html, "fix
print"_fix_print.html, and "run every"_run.html commands) or as part
of thermodynamic output (see the "thermo_style"_thermo_style.html
command), or used as input to an averaging fix (see the "fix
ave/time"_fix_ave_time.html command).  Variables of style {atom} store
a formula which when evaluated produces one numeric value per atom
which can be output to a dump file (see the "dump custom"_dump.html
command) or used as input to an averaging fix (see the "fix
ave/spatial"_fix_ave_spatial.html and "fix ave/atom"_fix_ave_atom.html
commands).

In the discussion that follows, the "name" of the variable is the
arbitrary string that is the 1st argument in the variable command.
This name can only contain alphanumeric characters and underscores.
The "string" is one or more of the subsequent arguments.  The "string"
can be simple text as in the 1st example above, it can contain other
variables as in the 2nd example, or it can be a formula as in the 3rd
example.  The "value" is the numeric quantity resulting from
evaluation of the string.  Note that the same string can generate
different values when it is evaluated at different times during a
simulation.

IMPORTANT NOTE: When the input script line that defines a variable of
style {equal} or {atom} that contain a formula is encountered, the
formula is NOT immediately evaluated and the result stored.  See the
discussion below about "Immediate Evaluation of Variables" if you want
to do this.

IMPORTANT NOTE: When a variable command is encountered in the input
script and the variable name has already been specified, the command
is ignored.  This means variables can NOT be re-defined in an input
script (with 2 exceptions, read further).  This is to allow an input
script to be processed multiple times without resetting the variables;
see the "jump"_jump.html or "include"_include.html commands.  It also
means that using the "command-line switch"_Section_start.html#start_7
-var will override a corresponding index variable setting in the input
script.

There are two exceptions to this rule.  First, variables of style
{string} and {equal} and {atom} ARE redefined each time the command is
encountered.  This allows these style of variables to be redefined
multiple times in an input script.  In a loop, this means the formula
associated with an {equal} or {atom} style variable can change if it
contains a substitution for another variable, e.g. $x.

Second, as described below, if a variable is iterated on to the end of
its list of strings via the "next"_next.html command, it is removed
from the list of active variables, and is thus available to be
re-defined in a subsequent variable command.  The {delete} style does
the same thing.

:line

"This section"_Section_commands.html#cmd_2 of the manual explains how
occurrences of a variable name in an input script line are replaced by
the variable's string.  The variable name can be referenced as $x if
the name "x" is a single character, or as $\{LoopVar\} if the name
"LoopVar" is one or more characters.

As described below, for variable styles {index}, {loop}, {universe},
and {uloop}, which string is assigned to a variable can be incremented
via the "next"_next.html command.  When there are no more strings to
assign, the variable is exhausted and a flag is set that causes the
next "jump"_jump.html command encountered in the input script to be
skipped.  This enables the construction of simple loops in the input
script that are iterated over and then exited from.

As explained above, an exhausted variable can be re-used in an input
script.  The {delete} style also removes the variable, the same as if
it were exhausted, allowing it to be redefined later in the input
script or when the input script is looped over.  This can be useful
when breaking out of a loop via the "if"_if.html and "jump"_jump.html
commands before the variable would become exhausted.  For example,

label	    loop
variable    a loop 5
print	    "A = $a"
if	    "$a > 2" then "jump in.script break"
next	    a
jump	    in.script loop
label	    break
variable    a delete :pre

:line

For the {index} style, one or more strings are specified.  Initially,
the 1st string is assigned to the variable.  Each time a
"next"_next.html command is used with the variable name, the next
string is assigned.  All processors assign the same string to the
variable.

{Index} style variables with a single string value can also be set by
using the command-line switch -var; see "this
section"_Section_start.html#start_7 for details.

The {loop} style is identical to the {index} style except that the
strings are the integers from 1 to N inclusive, if only one argument N
is specified.  This allows generation of a long list of runs
(e.g. 1000) without having to list N strings in the input script.
Initially, the string "1" is assigned to the variable.  Each time a
"next"_next.html command is used with the variable name, the next
string ("2", "3", etc) is assigned.  All processors assign the same
string to the variable.  The {loop} style can also be specified with
two arguments N1 and N2.  In this case the loop runs from N1 to N2
inclusive, and the string N1 is initially assigned to the variable.
N1 <= N2 and N2 >= 0 is required.

For the {world} style, one or more strings are specified.  There must
be one string for each processor partition or "world".  See "this
section"_Section_start.html#start_7 of the manual for information on
running LAMMPS with multiple partitions via the "-partition"
command-line switch.  This variable command assigns one string to each
world.  All processors in the world are assigned the same string.  The
next command cannot be used with {equal} style variables, since there
is only one value per world.  This style of variable is useful when
you wish to run different simulations on different partitions, or when
performing a parallel tempering simulation (see the
"temper"_temper.html command), to assign different temperatures to
different partitions.

For the {universe} style, one or more strings are specified.  There
must be at least as many strings as there are processor partitions or
"worlds".  See "this page"_Section_start.html#start_7 for information
on running LAMMPS with multiple partitions via the "-partition"
command-line switch.  This variable command initially assigns one
string to each world.  When a "next"_next.html command is encountered
using this variable, the first processor partition to encounter it, is
assigned the next available string.  This continues until all the
variable strings are consumed.  Thus, this command can be used to run
50 simulations on 8 processor partitions.  The simulations will be run
one after the other on whatever partition becomes available, until
they are all finished.  {Universe} style variables are incremented
using the files "tmp.lammps.variable" and "tmp.lammps.variable.lock"
which you will see in your directory during such a LAMMPS run.

The {uloop} style is identical to the {universe} style except that the
strings are the integers from 1 to N.  This allows generation of long
list of runs (e.g. 1000) without having to list N strings in the input
script.

:line

For the {equal} and {atom} styles, a single string is specified which
represents a formula that will be evaluated afresh each time the
variable is used.  If you want spaces in the string, enclose it in
double quotes so the parser will treat it as a single argument.  For
{equal} style variables the formula computes a scalar quantity, which
becomes the value of the variable whenever it is evaluated.  For
{atom} style variables the formula computes one quantity for each
atom whenever it is evaluated.

Note that {equal} and {atom} variables can produce different values at
different stages of the input script or at different times during a
run.  For example, if an {equal} variable is used in a "fix
print"_fix_print.html command, different values could be printed each
timestep it was invoked.  If you want a variable to be evaluated
immediately, so that the result is stored by the variable instead of
the string, see the section below on "Immediate Evaluation of
Variables".

The next command cannot be used with {equal} or {atom} style
variables, since there is only one string.

The formula for an {equal} or {atom} variable can contain a variety
of quantities.  The syntax for each kind of quantity is simple, but
multiple quantities can be nested and combined in various ways to
build up formulas of arbitrary complexity.  For example, this is a
valid (though strange) variable formula:

variable x equal "pe + c_MyTemp / vol^(1/3)" :pre

Specifically, an formula can contain numbers, thermo keywords, math
operators, math functions, group functions, region functions, atom
values, atom vectors, compute references, fix references, and
references to other variables.

Number: 0.2, 100, 1.0e20, -15.4, etc
Constant: PI
Thermo keywords: vol, pe, ebond, etc
Math operators: (), -x, x+y, x-y, x*y, x/y, x^y, x==y, x!=y, x<y, x<=y, x>y, x>=y, x&&y, x||y, !x
Math functions: sqrt(x), exp(x), ln(x), log(x), sin(x), cos(x), tan(x), asin(x), acos(x), atan(x), atan2(y,x), random(x,y,z), normal(x,y,z), ceil(x), floor(x), round(x), ramp(x,y), stagger(x,y), logfreq(x,y,z), stride(x,y,z), vdisplace(x,y), swiggle(x,y,z), cwiggle(x,y,z)
Group functions: count(ID), mass(ID), charge(ID), xcm(ID,dim), \
                 vcm(ID,dim), fcm(ID,dim), bound(ID,dir), \
		 gyration(ID), ke(ID), angmom(ID,dim), torque(ID,dim), \
		 inertia(ID,dimdim), omega(ID,dim)
Region functions: count(ID,IDR), mass(ID,IDR), charge(ID,IDR), \
		  xcm(ID,dim,IDR), vcm(ID,dim,IDR), fcm(ID,dim,IDR), \
		  bound(ID,dir,IDR), gyration(ID,IDR), ke(ID,IDR), \
		  angmom(ID,dim,IDR), torque(ID,dim,IDR), \
                  inertia(ID,dimdim,IDR), omega(ID,dim,IDR)
Special functions: sum(x), min(x), max(x), ave(x), trap(x), gmask(x), rmask(x), grmask(x,y)
Atom values: mass\[i\], type\[i\], x\[i\], y\[i\], z\[i\], \
             vx\[i\], vy\[i\], vz\[i\], fx\[i\], fy\[i\], fz\[i\]
Atom vectors: mass, type, x, y, z, vx, vy, vz, fx, fy, fz
Compute references: c_ID, c_ID\[i\], c_ID\[i\]\[j\]
Fix references: f_ID, f_ID\[i\], f_ID\[i\]\[j\]
Other variables: v_name, v_name\[i\] :tb(s=:)

:line

Most of the formula elements produce a scalar value.  A few produce a
per-atom vector of values.  These are the atom vectors, compute
references that represent a per-atom vector, fix references that
represent a per-atom vector, and variables that are atom-style
variables.  Math functions that operate on scalar values produce a
scalar value; math function that operate on per-atom vectors do so
element-by-element and produce a per-atom vector.

A formula for equal-style variables cannot use any formula element
that produces a per-atom vector.  A formula for an atom-style variable
can use formula elements that produce either a scalar value or a
per-atom vector.  Atom-style variables are evaluated by other commands
that define a "group"_group.html on which they operate, e.g. a
"dump"_dump.html or "compute"_compute.html or "fix"_fix.html command.
When they invoke the atom-style variable, only atoms in the group are
inlcuded in the formula evaluation.  The variable evaluates to 0.0 for
atoms not in the group.

The thermo keywords allowed in a formula are those defined by the
"thermo_style custom"_thermo_style.html command.  Thermo keywords that
require a "compute"_compute.html to calculate their values such as
"temp" or "press", use computes stored and invoked by the
"thermo_style"_thermo_style.html command.  This means that you can
only use those keywords in a variable if the style you are using with
the thermo_style command (and the thermo keywords associated with that
style) also define and use the needed compute.  Note that some thermo
keywords use a compute indirectly to calculate their value (e.g. the
enthalpy keyword uses temp, pe, and pressure).  If a variable is
evaluated directly in an input script (not during a run), then the
values accessed by the thermo keyword must be current.  See the
discussion below about "Variable Accuracy".

:line

Math Operators :h4

Math operators are written in the usual way, where the "x" and "y" in
the examples can themselves be arbitrarily complex formulas, as in the
examples above.  In this syntax, "x" and "y" can be scalar values or
per-atom vectors.  For example, "ke/natoms" is the division of two
scalars, where "vy+vz" is the element-by-element sum of two per-atom
vectors of y and z velocities.

Operators are evaluated left to right and have the usual C-style
precedence: unary minus and unary logical NOT operator "!" have the
highest precedence, exponentiation "^" is next; multiplication and
division are next; addition and subtraction are next; the 4 relational
operators "<", "<=", ">", and ">=" are next; the two remaining
relational operators "==" and "!=" are next; then the logical AND
operator "&&"; and finally the logical OR operator "||" has the lowest
precedence.  Parenthesis can be used to group one or more portions of
a formula and/or enforce a different order of evaluation than what
would occur with the default precedence.

The 6 relational operators return either a 1.0 or 0.0 depending on
whether the relationship between x and y is TRUE or FALSE.  For
example the expression x<10.0 in an atom-style variable formula will
return 1.0 for all atoms whose x-coordinate is less than 10.0, and 0.0
for the others.  The logical AND operator will return 1.0 if both its
arguments are non-zero, else it returns 0.0.  The logical OR operator
will return 1.0 if either of its arguments is non-zero, else it
returns 0.0.  The logical NOT operator returns 1.0 if its argument is
0.0, else it returns 0.0.

These relational and logical operators can be used as a masking or
selection operation in a formula.  For example, the number of atoms
whose properties satifsy one or more criteria could be calculated by
taking the returned per-atom vector of ones and zeroes and passing it
to the "compute reduce"_compute_reduce.html command.

:line

Math Functions :h4

Math functions are specified as keywords followed by one or more
parenthesized arguments "x", "y", "z", each of which can themselves be
arbitrarily complex formulas.  In this syntax, the arguments can
represent scalar values or per-atom vectors.  In the latter case, the
math operation is performed on each element of the vector.  For
example, "sqrt(natoms)" is the sqrt() of a scalar, where "sqrt(y*z)"
yields a per-atom vector with each element being the sqrt() of the
product of one atom's y and z coordinates.

Most of the math functions perform obvious operations.  The ln() is
the natural log; log() is the base 10 log. 

The random(x,y,z) function takes 3 arguments: x = lo, y = hi, and z =
seed.  It generates a uniform random number between lo and hi.  The
normal(x,y,z) function also takes 3 arguments: x = mu, y = sigma, and
z = seed.  It generates a Gaussian variate centered on mu with
variance sigma^2.  In both cases the seed is used the first time the
internal random number generator is invoked, to initialize it.  For
equal-style variables, every processor uses the same seed so that they
each generate the same sequence of random numbers.  For atom-style
variables, a unique seed is created for each processor, based on the
specified seed.  This effectively generates a different random number
for each atom being looped over in the atom-style variable.

IMPORTANT NOTE: Internally, there is just one random number generator
for all equal-style variables and one for all atom-style variables.
If you define multiple variables (of each style) which use the
random() or normal() math functions, then the internal random number
generators will only be initialized once, which means only one of the
specified seeds will determine the sequence of generated random
numbers.

The ceil(), floor(), and round() functions are those in the C math
library.  Ceil() is the smallest integer not less than its argument.
Floor() if the largest integer not greater than its argument.  Round()
is the nearest integer to its argument.

The ramp(x,y) function uses the current timestep to generate a value
linearly intepolated between the specified x,y values over the course
of a run, according to this formula:

value = x + (y-x) * (timestep-startstep) / (stopstep-startstep) :pre

The run begins on startstep and ends on stopstep.  Startstep and
stopstep can span multiple runs, using the {start} and {stop} keywords
of the "run"_run.html command.  See the "run"_run.html command for
details of how to do this.

The stagger(x,y) function uses the current timestep to generate a new
timestep.  X,y > 0 and x > y are required.  The generated timesteps
increase in a staggered fashion, as the sequence
x,x+y,2x,2x+y,3x,3x+y,etc.  For any current timestep, the next
timestep in the sequence is returned.  Thus if stagger(1000,100) is
used in a variable by the "dump_modify every"_dump_modify.html
command, it will generate the sequence of output timesteps:

100,1000,1100,2000,2100,3000,etc :pre

The logfreq(x,y,z) function uses the current timestep to generate a
new timestep.  X,y,z > 0 and y < z are required.  The generated
timesteps increase in a logarithmic fashion, as the sequence
x,2x,3x,...y*x,z*x,2*z*x,3*z*x,...y*z*x,z*z*x,2*z*x*x,etc.  For any
current timestep, the next timestep in the sequence is returned.  Thus
if logfreq(100,4,10) is used in a variable by the "dump_modify
every"_dump_modify.html command, it will generate the sequence of
output timesteps:

100,200,300,400,1000,2000,3000,4000,10000,20000,etc :pre

The stride(x,y,z) function uses the current timestep to generate a new
timestep.  X,y >= 0 and z > 0 and x <= y are required.  The generated
timesteps increase in increments of z, from x to y, I.e. it generates
the sequece x,x+z,x+2z,...,y.  If y-x is not a multiple of z, then
similar to the way a for loop operates, the last value will be one
that does not exceed y.  For any current timestep, the next timestep
in the sequence is returned.  Thus if stagger(1000,2000,100) is used
in a variable by the "dump_modify every"_dump_modify.html command, it
will generate the sequence of output timesteps:

1000,1100,1200, ... ,1900,2000 :pre

The vdisplace(x,y) function takes 2 arguments: x = value0 and y =
velocity, and uses the elapsed time to change the value by a linear
displacement due to the applied velocity over the course of a run,
according to this formula:

value = value0 + velocity*(timestep-startstep)*dt :pre

where dt = the timestep size.

The run begins on startstep.  Startstep can span multiple runs, using
the {start} keyword of the "run"_run.html command.  See the
"run"_run.html command for details of how to do this.  Note that the
"thermo_style"_thermo_style.html keyword elaplong =
timestep-startstep.

The swiggle(x,y,z) and cwiggle(x,y,z) functions each take 3 arguments:
x = value0, y = amplitude, z = period.  They use the elapsed time to
oscillate the value by a sin() or cos() function over the course of a
run, according to one of these formulas, where omega = 2 PI / period:

value = value0 + Amplitude * sin(omega*(timestep-startstep)*dt)
value = value0 + Amplitude * (1 - cos(omega*(timestep-startstep)*dt)) :pre

where dt = the timestep size.

The run begins on startstep.  Startstep can span multiple runs, using
the {start} keyword of the "run"_run.html command.  See the
"run"_run.html command for details of how to do this.  Note that the
"thermo_style"_thermo_style.html keyword elaplong =
timestep-startstep.

:line

Group and Region Functions :h4

Group functions are specified as keywords followed by one or two
parenthesized arguments.  The first argument is the group-ID.  The
{dim} argument, if it exists, is {x} or {y} or {z}.  The {dir}
argument, if it exists, is {xmin}, {xmax}, {ymin}, {ymax}, {zmin}, or
{zmax}.  The {dimdim} argument, if it exists, is {xx} or {yy} or {zz}
or {xy} or {yz} or {xz}.

The group function count() is the number of atoms in the group.  The
group functions mass() and charge() are the total mass and charge of
the group.  Xcm() and vcm() return components of the position and
velocity of the center of mass of the group.  Fcm() returns a
component of the total force on the group of atoms.  Bound() returns
the min/max of a particular coordinate for all atoms in the group.
Gyration() computes the radius-of-gyration of the group of atoms.  See
the "compute gyration"_compute_gyration.html command for a definition
of the formula.  Angmom() returns components of the angular momentum
of the group of atoms around its center of mass.  Torque() returns
components of the torque on the group of atoms around its center of
mass, based on current forces on the atoms.  Inertia() returns one of
6 components of the inertia tensor of the group of atoms around its
center of mass.  Omega() returns components of the angular velocity of
the group of atoms around its center of mass.

Region functions are specified exactly the same way as group functions
except they take an extra argument which is the region ID.  The
function is computed for all atoms that are in both the group and the
region.  If the group is "all", then the only criteria for atom
inclusion is that it be in the region.

:line

Special Functions :h4

Special functions take specific kinds of arguments, meaning their
arguments cannot be formulas themselves.

The sum(x), min(x), max(x), ave(x), and trap(x) functions each take 1
argument which is of the form "c_ID" or "c_ID\[N\]" or "f_ID" or
"f_ID\[N\]".  The first two are computes and the second two are fixes;
the ID in the reference should be replaced by the ID of a compute or
fix defined elsewhere in the input script.  The compute or fix must
produce either a global vector or array.  If it produces a global
vector, then the notation without "\[N\]" should be used.  If it
produces a global array, then the notation with "\[N\]" should be
used, when N is an integer, to specify which column of the global
array is being referenced.

These functions operate on the global vector of inputs and reduce it
to a single scalar value.  This is analagous to the operation of the
"compute reduce"_compute_reduce.html command, which invokes the same
functions on per-atom and local vectors.

The sum() function calculates the sum of all the vector elements.  The
min() and max() functions find the minimum and maximum element
respectively.  The ave() function is the same as sum() except that it
divides the result by the length of the vector.  The trap() function
is the same as sum() except the first and last elements are multiplied
by a weighting factor of 1/2 when performing the sum.  This
effectively implements an integratiion via the trapezoidal rule on the
global vector of data.  I.e. consider a set of points, equally spaced
by 1 in their x coordinate: (1,V1), (2,V2), ..., (N,VN), where the Vi
are the values in the global vector of length N.  The integral from 1
to N of these points is trap().  When appropriately normalized by the
timestep size, this function is useful for calculating integrals of
time-series data, like that generated by the "fix
ave/correlate"_fix_ave_correlate.html command.

The gmask(x) function takes 1 argument which is a group ID.  It
can only be used in atom-style variables.  It returns a 1 for
atoms that are in the group, and a 0 for atoms that are not.

The rmask(x) function takes 1 argument which is a region ID.  It can
only be used in atom-style variables.  It returns a 1 for atoms that
are in the geometric region, and a 0 for atoms that are not.

The grmask(x,y) function takes 2 arguments.  The first is a group ID,
and the second is a region ID.  It can only be used in atom-style
variables.  It returns a 1 for atoms that are in both the group and
region, and a 0 for atoms that are not in both.

:line

Atom Values and Vectors :h4

Atom values take a single integer argument I from 1 to N, where I is
the an atom-ID, e.g. x\[243\], which means use the x coordinate of the
atom with ID = 243.

Atom vectors generate one value per atom, so that a reference like
"vx" means the x-component of each atom's velocity will be used when
evaluating the variable.  Note that other atom attributes can be used
as inputs to a variable by using the "compute
property/atom"_compute_property_atom.html command and then specifying
a quantity from that compute.

:line

Compute References :h4

Compute references access quantities calculated by a
"compute"_compute.html.  The ID in the reference should be replaced by
the ID of a compute defined elsewhere in the input script.  As
discussed in the doc page for the "compute"_compute.html command,
computes can produce global, per-atom, or local values.  Only global
and per-atom values can be used in a variable.  Computes can also
produce a scalar, vector, or array.  An equal-style variable can only
use scalar values, which means a global scalar, or an element of a
global or per-atom vector or array.  Atom-style variables can use the
same scalar values.  They can also use per-atom vector values.  A
vector value can be a per-atom vector itself, or a column of an
per-atom array.  See the doc pages for individual computes to see what
kind of values they produce.

Examples of different kinds of compute references are as follows.
There is no ambiguity as to what a reference means, since computes
only produce global or per-atom quantities, never both.

c_ID: global scalar, or per-atom vector
c_ID\[I\]: Ith element of global vector, or atom I's value in per-atom vector, or Ith column from per-atom array
c_ID\[I\]\[J\]: I,J element of global array, or atom I's Jth value in per-atom array :tb(s=:)

If a variable containing a compute is evaluated
directly in an input script (not during a run), then the values
accessed by the compute must be current.  See the discussion below
about "Variable Accuracy".

:line

Fix References :h4

Fix references access quantities calculated by a "fix"_compute.html.
The ID in the reference should be replaced by the ID of a fix defined
elsewhere in the input script.  As discussed in the doc page for the
"fix"_fix.html command, fixes can produce global, per-atom, or local
values.  Only global and per-atom values can be used in a variable.
Fixes can also produce a scalar, vector, or array.  An equal-style
variable can only use scalar values, which means a global scalar, or
an element of a global or per-atom vector or array.  Atom-style
variables can use the same scalar values.  They can also use per-atom
vector values.  A vector value can be a per-atom vector itself, or a
column of an per-atom array.  See the doc pages for individual fixes
to see what kind of values they produce.

The different kinds of fix references are exactly the same as the
compute references listed in the above table, where "c_" is replaced
by "f_".  Again, there is no ambiguity as to what a reference means,
since fixes only produce global or per-atom quantities, never both.

f_ID: global scalar, or per-atom vector
f_ID\[I\]: Ith element of global vector, or atom I's value in per-atom vector, or Ith column from per-atom array
f_ID\[I\]\[J\]: I,J element of global array, or atom I's Jth value in per-atom array :tb(s=:)

If a variable containing a fix is evaluated directly in an input
script (not during a run), then the values accessed by the fix should
be current.  See the discussion below about "Variable Accuracy".

Note that some fixes only generate quantities on certain timesteps.
If a variable attempts to access the fix on non-allowed timesteps, an
error is generated.  For example, the "fix ave/time"_fix_ave_time.html
command may only generate averaged quantities every 100 steps.  See
the doc pages for individual fix commands for details.

:line

Variable References :h4

Variable references access quantities calulated by other variables,
which will cause those variables to be evaluated.  The name in the
reference should be replaced by the name of a variable defined
elsewhere in the input script.  As discussed on this doc page,
atom-style variables generate a per-atom vector of values; all other
variable styles generate a global scalar value.  An equal-style
variable can only use scalar values, which means another equal-style
variable or an element of an atom-style variable.  Atom-style
variables can use the same scalar values.  They can also use other
atom-style variables.

Examples of different kinds of variable references are as follows.
There is no ambiguity as to what a reference means, since variables
produce only a global scalar or a per-atom vectors, never both.

v_name: scalar, or per-atom vector
v_name\[I\]: atom I's value in per-atom vector :tb(s=:)

:line

[Immediate Evaluation of Variables:]

There is a difference between referencing a variable with a leading $
sign (e.g. $x or $\{abc\}) versus with a leading "v_" (e.g. v_x or
v_abc).  The former can be used in any command, including a variable
command, to force the immediate evaluation of the referenced variable
and the substitution of its value into the command.  The latter is a
required kind of argument to some commands (e.g. the "fix
ave/spatial"_fix_ave_spatial.html or "dump custom"_dump.html or
"thermo_style"_thermo_style.html commands) if you wish it to evaluate
a variable periodically during a run.  It can also be used in a
variable formula if you wish to reference a second variable.  The
second variable will be evaluated whenever the first variable is
evaluated.

As an example, suppose you use this command in your input script to
define the variable "v" as

variable v equal vol :pre

before a run where the simulation box size changes.  You might think
this will assign the initial volume to the variable "v".  That is not
the case.  Rather it assigns a formula which evaluates the volume
(using the thermo_style keyword "vol") to the variable "v".  If you
use the variable "v" in some other command like "fix
ave/time"_fix_ave_time.html then the current volume of the box will be
evaluated continuously during the run.

If you want to store the initial volume of the system, you can do it
this way:

variable v equal vol
variable v0 equal $v :pre

The second command will force "v" to be evaluated (yielding the
initial volume) and assign that value to the variable "v0".  Thus the
command

thermo_style custom step v_v v_v0 :pre

would print out both the current and initial volume periodically
during the run.

Note that it is a mistake to enclose a variable formula in double
quotes if it contains variables preceeded by $ signs.  For example,

variable vratio equal "$\{vfinal\}/$\{v0\}" :pre

This is because the quotes prevent variable substitution (see "this
section"_Section_commands.html#cmd_2 on parsing input script
commands), and thus an error will occur when the formula for "vratio"
is evaluated later.

:line

[Variable Accuracy:]

Obviously, LAMMPS attempts to evaluate variables containing formulas
({equal} and {atom} style variables) accurately whenever the
evaluation is performed.  Depending on what is included in the
formula, this may require invoking a "compute"_compute.html, either
directly or indirectly via a thermo keyword, or accessing a value
previously calculated by a compute, or accessing a value calculated
and stored by a "fix"_fix.html.  If the compute is one that calculates
the pressure or energy of the system, then these quantities need to be
tallied during the evaluation of the interatomic potentials (pair,
bond, etc) on timesteps that the variable will need the values.

LAMMPS keeps track of all of this during a "run"_run.html or "energy
minimization"_minimize.html.  An error will be generated if you
attempt to evaluate a variable on timesteps when it cannot produce
accurate values.  For example, if a "thermo_style
custom"_thermo_style.html command prints a variable which accesses
values stored by a "fix ave/time"_fix_ave_time.html command and the
timesteps on which thermo output is generated are not multiples of the
averaging frequency used in the fix command, then an error will occur.

An input script can also request variables be evaluated before or
after or in between runs, e.g. by including them in a
"print"_print.html command.  In this case, if a compute is needed to
evaluate a variable (either directly or indirectly), LAMMPS will not
invoke the compute, but it will use a value previously calculated by
the compute, and can do this only if it is current.  Fixes will always
provide a quantity needed by a variable, but the quantity may or may
not be current.  This leads to one of three kinds of behavior:

(1) The variable may be evaluated accurately.  If it contains
references to a compute or fix, and these values were calculated on
the last timestep of a preceeding run, then they will be accessed and
used by the variable and the result will be accurate.

(2) LAMMPS may not be able to evaluate the variable and will generate
an error message stating so.  For example, if the variable requires a
quantity from a "compute"_compute.html that is not current, LAMMPS
will generate an error.  This means, for example, that such a variable
cannot be evaluated before the first run has occurred.  Likewise, in
between runs, such a variable cannot be accessed unless it was
evaluated on the last timestep of the preceding run, e.g. by
thermodynamic output.

One way to get around this problem is to perform a 0-timestep run
before using the variable.  For example, these commands

variable t equal temp
print "Initial temperature = $t"
run 1000 :pre

will generate an error if the run is the first run specified in the
input script, because generating a value for the "t" variable requires
a compute for calculating the temperature to be invoked.

However, this sequence of commands would be fine:

run 0
variable t equal temp
print "Initial temperature = $t"
run 1000 :pre

The 0-timestep run initializes and invokes various computes, including
the one for temperature, so that the value it stores is current and
can be accessed by the variable "t" after the run has completed.  Note
that a 0-timestep run does not alter the state of the system, so it
does not change the input state for the 1000-timestep run that
follows.  Also note that the 0-timestep run must actually use and
invoke the compute in question (e.g. via "thermo"_thermo_style.html or
"dump"_dump.html output) in order for it to enable the compute to be
used in a variable after the run.  Thus if you are trying to print a
variable that uses a compute you have defined, you could insure it was
invoked on the last timestep of the preceding run by including it in
thermodynamic output.

Unlike computes, "fixes"_fix.html will never generate an error if
their values are accessed by a variable in between runs.  They always
return some value to the variable.  However, the value may not be what
you expect if the fix has not yet calculated the quantity of interest
or it is not current.  For example, the "fix indent"_fix_indent.html
command stores the force on the indenter.  But this is not computed
until a run is performed.  Thus if a variable attempts to print this
value before the first run, zeroes will be output.  Again, performing
a 0-timestep run before printing the variable has the desired effect.

(3) The variable may be evaluated incorrectly and LAMMPS may have no
way to detect this has occurred.  Consider the following sequence of
commands:

pair_coeff 1 1 1.0 1.0
run 1000
pair_coeff 1 1 1.5 1.0
variable e equal pe
print "Final potential energy = $e" :pre

The first run is performed using one setting for the pairwise
potential defined by the "pair_style"_pair_style.html and
"pair_coeff"_pair_coeff.html commands.  The potential energy is
evaluated on the final timestep and stored by the "compute
pe"_compute_pe.html compute (this is done by the
"thermo_style"_thermo_style.html command).  Then a pair coefficient is
changed, altering the potential energy of the system.  When the
potential energy is printed via the "e" variable, LAMMPS will use the
potential energy value stored by the "compute pe"_compute_pe.html
compute, thinking it is current.  There are many other commands which
could alter the state of the system between runs, causing a variable
to evaluate incorrectly.

The solution to this issue is the same as for case (2) above, namely
perform a 0-timestep run before the variable is evaluated to insure
the system is up-to-date.  For example, this sequence of commands
would print a potential energy that reflected the changed pairwise
coefficient:

pair_coeff 1 1 1.0 1.0
run 1000
pair_coeff 1 1 1.5 1.0
run 0
variable e equal pe
print "Final potential energy = $e" :pre

:line

[Restrictions:]

Indexing any formula element by global atom ID, such as an atom value,
requires the atom style to use a global mapping in order to look up
the vector indices.  By default, only atom styles with molecular
information create global maps.  The "atom_modify
map"_atom_modify.html command can override the default.

All {universe}- and {uloop}-style variables defined in an input script
must have the same number of values.

[Related commands:]

"next"_next.html, "jump"_jump.html, "include"_include.html,
"temper"_temper.html, "fix print"_fix_print.html, "print"_print.html

[Default:] none