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			<H3>variable command
			</H3>
			<P><B>Syntax:</B>
			</P>
			<PRE>variable name style args ...
			</PRE>
			<UL><LI>name = name of variable to define

			<LI>style = <I>delete</I> or <I>index</I> or <I>loop</I> or <I>world</I> or <I>universe</I> or
			<I>uloop</I> or <I>string</I> or <I>format</I> or <I>getenv</I> or <I>file</I> or <I>atomfile</I> or <I>python</I> or <I>equal</I> or <I>atom</I>

			<PRE> <I>delete</I> = no args
			<I>index</I> args = one or more strings
			<I>loop</I> args = N
			N = integer size of loop, loop from 1 to N inclusive
			<I>loop</I> 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
			<I>loop</I> args = N1 N2
			N1,N2 = loop from N1 to N2 inclusive
			<I>loop</I> args = N1 N2 pad
			N1,N2 = loop from N1 to N2 inclusive
			pad = all values will be same length, e.g. 050, 051, ..., 100
			<I>world</I> args = one string for each partition of processors
			<I>universe</I> args = one or more strings
			<I>uloop</I> args = N
			N = integer size of loop
			<I>uloop</I> args = N pad
			N = integer size of loop
			pad = all values will be same length, e.g. 001, 002, ..., 100
			<I>string</I> arg = one string
			<I>format</I> args = vname fstr
			vname = name of equal-style variable to evaluate
			fstr = C-style format string
			<I>getenv</I> arg = one string
			<I>file</I> arg = filename
			<I>atomfile</I> arg = filename
			<I>python</I> arg = function
			<I>equal</I> or <I>atom</I> 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 <A HREF = "thermo_style.html">thermo_style</A>
			math operators = (), -x, x+y, x-y, x*y, x/y, x^y, x%y,
			x == y, x != y, x &lt y, x &lt= y, x &gt y, x &gt= y, x && y, x \|\| y, !x
			math functions = sqrt(x), exp(x), ln(x), log(x), abs(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), logfreq2(x,y,z),
			stride(x,y,z), stride2(x,y,z,a,b,c),
			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,dir), 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,dir,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), slope(x), gmask(x), rmask(x), grmask(x,y), next(x)
			atom value = id[i], mass[i], type[i], mol[i], x[i], y[i], z[i], vx[i], vy[i], vz[i], fx[i], fy[i], fz[i], q[i]
			atom vector = id, mass, type, mol, x, y, z, vx, vy, vz, fx, fy, fz, q
			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>

			</UL>
			<P><B>Examples:</B>
			</P>
			<PRE>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 myPy python increase
			variable f file values.txt
			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 str format x %.6g
			variable x delete
			</PRE>
			<P><B>Description:</B>
			</P>
			<P>This command assigns one or more strings to a variable name for
			evaluation later in the input script or during a simulation.
			</P>
			<P>Variables can thus be useful in several contexts. A variable can be
			defined and then referenced elsewhere in an input script to become
			part of a new input command. For variable styles that store multiple
			strings, the <A HREF = "next.html">next</A> command can be used to increment which
			string is assigned to the variable. Variables of style <I>equal</I> store
			a formula which when evaluated produces a single numeric value which
			can be output either directly (see the <A HREF = "print.html">print</A>, <A HREF = "fix_print.html">fix
			print</A>, and <A HREF = "run.html">run every</A> commands) or as part
			of thermodynamic output (see the <A HREF = "thermo_style.html">thermo_style</A>
			command), or used as input to an averaging fix (see the <A HREF = "fix_ave_time.html">fix
			ave/time</A> command). Variables of style <I>atom</I> store
			a formula which when evaluated produces one numeric value per atom
			which can be output to a dump file (see the <A HREF = "dump.html">dump custom</A>
			command) or used as input to an averaging fix (see the <A HREF = "fix_ave_spatial.html">fix
			ave/spatial</A> and <A HREF = "fix_ave_atom.html">fix ave/atom</A>
			commands). Variables of style <I>atomfile</I> can be used anywhere in an
			input script that atom-style variables are used; they get their
			per-atom values from a file rather than from a formula. Variables can
			be hooked to Python functions using code you provide, so that the
			variable gets its value from the evaluation of the Python code.
			</P>
			<P>IMPORTANT NOTE: As discussed in <A HREF = "Section_commands.html#cmd_2">Section 3.2</A>
			of the manual, an input script can use "immediate" variables, specified
			as $(formula) with parenthesis, where the formula has the same syntax
			as equal-style variables described on this page. This is a convenient
			way to evaluate a formula immediately without using the variable command
			to define a named variable and then evaluate that variable. See below
			for a more detailed discussion of this feature.
			</P>
			<P>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.
			</P>
			<P>IMPORTANT NOTE: When the input script line is encountered that defines
			a variable of style <I>equal</I> or <I>atom</I> or <I>python</I> that contains a
			formula or Python code, the formula is NOT immediately evaluated.
			It will be evaluated every time when the variable is <B>used</B> instead.
			If you simply want to evaluate a formula in place you can use as
			so-called. See the section below about "Immediate Evaluation
			of Variables" for more details on the topic. This is also true of
			a <I>format</I> style variable since it evaluates another variable when
			it is invoked.
			</P>
			<P>IMPORTANT NOTE: Variables of style <I>equal</I> and <I>atom</I> can be used as
			inputs to various other LAMMPS commands which evaluate their formulas
			as needed, e.g. at different timesteps during a <A HREF = "run.html">run</A>.
			Variables of style <I>python</I> can be used in place of an equal-style
			variable so long as the associated Python function, as defined by the
			<A HREF = "python.html">python</A> command, returns a numeric value. Thus any
			command that states it can use an equal-style variable as an argument,
			can also use such a python-style variable. This means that when the
			LAMMPS command evaluates the variable, the Python function will be
			executed.
			</P>
			<P>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 two exceptions, read further). This is to allow an input
			script to be processed multiple times without resetting the variables;
			see the <A HREF = "jump.html">jump</A> or <A HREF = "include.html">include</A> commands. It also
			means that using the <A HREF = "Section_start.html#start_7">command-line switch</A>
			-var will override a corresponding index variable setting in the input
			script.
			</P>
			<P>There are two exceptions to this rule. First, variables of style
			<I>string</I>, <I>getenv</I>, <I>equal</I>, <I>atom</I>, and <I>python</I> 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 <I>equal</I> or <I>atom</I> style variable
			can change if it contains a substitution for another variable, e.g. $x
			or v_x.
			</P>
			<P>Second, as described below, if a variable is iterated on to the end of
			its list of strings via the <A HREF = "next.html">next</A> command, it is removed
			from the list of active variables, and is thus available to be
			re-defined in a subsequent variable command. The <I>delete</I> style does
			the same thing.
			</P>
			<HR>

			<P><A HREF = "Section_commands.html#cmd_2">This section</A> 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.
			</P>
			<P>As described below, for variable styles <I>index</I>, <I>loop</I>, <I>file</I>,
			<I>universe</I>, and <I>uloop</I>, which string is assigned to a variable can be
			incremented via the <A HREF = "next.html">next</A> command. When there are no more
			strings to assign, the variable is exhausted and a flag is set that
			causes the next <A HREF = "jump.html">jump</A> 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.
			</P>
			<P>As explained above, an exhausted variable can be re-used in an input
			script. The <I>delete</I> 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 <A HREF = "if.html">if</A> and <A HREF = "jump.html">jump</A>
			commands before the variable would become exhausted. For example,
			</P>
			<PRE>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>
			<HR>

			<P>This section describes how all the various variable styles are defined
			and what they store. Except for the <I>equal</I> and <I>atom</I> styles,
			which are explaine in the next section.
			</P>
			<P>Many of the styles store one or more strings. Note that a single
			string can contain spaces (multiple words), if it is enclosed in
			quotes in the variable command. When the variable is substituted for
			in another input script command, its returned string will then be
			interpreted as multiple arguments in the expanded command.
			</P>
			<P>For the <I>index</I> style, one or more strings are specified. Initially,
			the 1st string is assigned to the variable. Each time a
			<A HREF = "next.html">next</A> command is used with the variable name, the next
			string is assigned. All processors assign the same string to the
			variable.
			</P>
			<P><I>Index</I> style variables with a single string value can also be set by
			using the command-line switch -var; see <A HREF = "Section_start.html#start_7">this
			section</A> for details.
			</P>
			<P>The <I>loop</I> style is identical to the <I>index</I> 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
			<A HREF = "next.html">next</A> 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 <I>loop</I> 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.
			</P>
			<P>For the <I>world</I> style, one or more strings are specified. There must
			be one string for each processor partition or "world". See <A HREF = "Section_start.html#start_7">this
			section</A> 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 <I>equal</I> 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
			<A HREF = "temper.html">temper</A> command), to assign different temperatures to
			different partitions.
			</P>
			<P>For the <I>universe</I> style, one or more strings are specified. There
			must be at least as many strings as there are processor partitions or
			"worlds". See <A HREF = "Section_start.html#start_7">this page</A> 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 <A HREF = "next.html">next</A> 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. <I>Universe</I> 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.
			</P>
			<P>The <I>uloop</I> style is identical to the <I>universe</I> 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.
			</P>
			<P>For the <I>string</I> style, a single string is assigned to the variable.
			The only difference between this and using the <I>index</I> style with a
			single string is that a variable with <I>string</I> style can be redefined.
			E.g. by another command later in the input script, or if the script is
			read again in a loop.
			</P>
			<P>For the <I>format</I> style, an equal-style variable is specified along
			with a C-style format string, e.g. "%f" or "%.10g", which must be
			appropriate for formatting a double-precision floating-point value.
			This allows an equal-style variable to be formatted specifically for
			output as a string, e.g. by the <A HREF = "print.html">print</A> command, if the
			default format "%.15g" has too much precision.
			</P>
			<P>For the <I>getenv</I> style, a single string is assigned to the variable
			which should be the name of an environment variable. When the
			variable is evaluated, it returns the value of the environment
			variable, or an empty string if it not defined. This style of
			variable can be used to adapt the behavior of LAMMPS input scripts via
			environment variable settings, or to retrieve information that has
			been previously stored with the <A HREF = "shell.html">shell putenv</A> command.
			Note that because environment variable settings are stored by the
			operating systems, they persist beyond a <A HREF = "clear.html">clear</A> command.
			</P>
			<P>For the <I>file</I> style, a filename is provided which contains a list of
			strings to assign to the variable, one per line. The strings can be
			numeric values if desired. See the discussion of the next() function
			below for equal-style variables, which will convert the string of a
			file-style variable into a numeric value in a formula.
			</P>
			<P>When a file-style variable is defined, the file is opened and the
			string on the first line is read and stored with the variable. This
			means the variable can then be evaluated as many times as desired and
			will return that string. There are two ways to cause the next string
			from the file to be read: use the <A HREF = "next.html">next</A> command or the
			next() function in an equal- or atom-style variable, as discussed
			below.
			</P>
			<P>The rules for formatting the file are as follows. A comment character
			"#" can be used anywhere on a line; text starting with the comment
			character is stripped. Blank lines are skipped. The first "word" of
			a non-blank line, delimited by white space, is the "string" assigned
			to the variable.
			</P>
			<P>For the <I>atomfile</I> style, a filename is provided which contains one or
			more sets of values, to assign on a per-atom basis to the variable.
			The format of the file is described below.
			</P>
			<P>When an atomfile-style variable is defined, the file is opened and the
			first set of per-atom values are read and stored with the variable.
			This means the variable can then be evaluated as many times as desired
			and will return those values. There are two ways to cause the next
			set of per-atom values from the file to be read: use the
			<A HREF = "next.html">next</A> command or the next() function in an atom-style
			variable, as discussed below.
			</P>
			<P>The rules for formatting the file are as follows. Each time a set of
			per-atom values is read, a non-blank line is searched for in the file.
			A comment character "#" can be used anywhere on a line; text starting
			with the comment character is stripped. Blank lines are skipped. The
			first "word" of a non-blank line, delimited by white space, is read as
			the count N of per-atom lines to immediately follow. N can be be the
			total number of atoms in the system, or only a subset. The next N
			lines have the following format
			</P>
			<PRE>ID value
			</PRE>
			<P>where ID is an atom ID and value is the per-atom numeric value that
			will be assigned to that atom. IDs can be listed in any order.
			</P>
			<P>IMPORTANT NOTE: Every time a set of per-atom lines is read, the value
			for all atoms is first set to 0.0. Thus values for atoms whose ID
			does not appear in the set, will remain 0.0.
			</P>
			<P>For the <I>python</I> style a Python function name is provided. This needs
			to match a function name specified in a <A HREF = "python.html">python</A> command
			which returns a value to this variable as defined by its <I>return</I>
			keyword. For exampe these two commands would be self-consistent:
			</P>
			<PRE>variable foo python myMultiply
			python myMultiply return v_foo format f file funcs.py
			</PRE>
			<P>The two commands can appear in either order so long as both are
			specified before the Python function is invoked for the first time.
			</P>
			<P>Each time the variable is evaluated, the associated Python function is
			invoked, and the value it returns is also returned by the variable.
			Since the Python function can use other LAMMPS variables as input, or
			query interal LAMMPS quantities to perform its computation, this means
			the variable can return a different value each time it is evaluated.
			</P>
			<P>The type of value stored in the variable is determined by the <I>format</I>
			keyword of the <A HREF = "python.html">python</A> command. It can be an integer
			(i), floating point (f), or string (s) value. As mentioned above, if
			it is a numeric value (integer or floating point), then the
			python-style variable can be used in place of an equal-style variable
			anywhere in an input script, e.g. as an argument to another command
			that allows for equal-style variables.
			</P>
			<HR>

			<P>For the <I>equal</I> and <I>atom</I> 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
			<I>equal</I> style variables the formula computes a scalar quantity, which
			becomes the value of the variable whenever it is evaluated. For
			<I>atom</I> style variables the formula computes one quantity for each
			atom whenever it is evaluated.
			</P>
			<P>Note that <I>equal</I> and <I>atom</I> variables can produce different values at
			different stages of the input script or at different times during a
			run. For example, if an <I>equal</I> variable is used in a <A HREF = "fix_print.html">fix
			print</A> 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".
			</P>
			<P>The next command cannot be used with <I>equal</I> or <I>atom</I> style
			variables, since there is only one string.
			</P>
			<P>The formula for an <I>equal</I> or <I>atom</I> 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:
			</P>
			<PRE>variable x equal "pe + c_MyTemp / vol^(1/3)"
			</PRE>
			<P>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.
			</P>
			<DIV ALIGN=center><TABLE BORDER=1 >
			<TR><TD >Number</TD><TD > 0.2, 100, 1.0e20, -15.4, etc</TD></TR>
			<TR><TD >Constant</TD><TD > PI</TD></TR>
			<TR><TD >Thermo keywords</TD><TD > vol, pe, ebond, etc</TD></TR>
			<TR><TD >Math operators</TD><TD > (), -x, x+y, x-y, x*y, x/y, x^y, x%y, </TD></TR>
			<TR><TD >Math operators</TD><TD > (), -x, x+y, x-y, x*y, x/y, x^y, x%y, x == y, x != y, x &lt y, x &lt= y, x &gt y, x &gt= y, x && y, x \|\| y, !x</TD></TR>
			<TR><TD >Math functions</TD><TD > sqrt(x), exp(x), ln(x), log(x), abs(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), logfreq2(x,y,z), stride(x,y,z), stride2(x,y,z,a,b,c), vdisplace(x,y), swiggle(x,y,z), cwiggle(x,y,z)</TD></TR>
			<TR><TD >Group functions</TD><TD > 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)</TD></TR>
			<TR><TD >Region functions</TD><TD > 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)</TD></TR>
			<TR><TD >Special functions</TD><TD > sum(x), min(x), max(x), ave(x), trap(x), slope(x), gmask(x), rmask(x), grmask(x,y), next(x)</TD></TR>
			<TR><TD >Atom values</TD><TD > id[i], mass[i], type[i], mol[i], x[i], y[i], z[i], vx[i], vy[i], vz[i], fx[i], fy[i], fz[i], q[i]</TD></TR>
			<TR><TD >Atom vectors</TD><TD > id, mass, type, mol, x, y, z, vx, vy, vz, fx, fy, fz, q</TD></TR>
			<TR><TD >Compute references</TD><TD > c_ID, c_ID[i], c_ID[i][j]</TD></TR>
			<TR><TD >Fix references</TD><TD > f_ID, f_ID[i], f_ID[i][j]</TD></TR>
			<TR><TD >Other variables</TD><TD > v_name, v_name[i]
			</TD></TR></TABLE></DIV>

			<HR>

			<P>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.
			</P>
			<P>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 <A HREF = "group.html">group</A> on which they operate, e.g. a
			<A HREF = "dump.html">dump</A> or <A HREF = "compute.html">compute</A> or <A HREF = "fix.html">fix</A> 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.
			</P>
			<P>The thermo keywords allowed in a formula are those defined by the
			<A HREF = "thermo_style.html">thermo_style custom</A> command. Thermo keywords that
			require a <A HREF = "compute.html">compute</A> to calculate their values such as
			"temp" or "press", use computes stored and invoked by the
			<A HREF = "thermo_style.html">thermo_style</A> 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".
			</P>
			<HR>

			<H4>Math Operators
			</H4>
			<P>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.
			</P>
			<P>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 and the modulo operator "%" 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.
			</P>
			<P>IMPORTANT NOTE: Because a unary minus is higher precedence than
			exponentiation, the formula "-2^2" will evaluate to 4, not -4. This
			convention is compatible with some programming languages, but not
			others. As mentioned, this behavior can be easily overridden with
			parenthesis; the formula "-(2^2)" will evaluate to -4.
			</P>
			<P>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.
			</P>
			<P>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 <A HREF = "compute_reduce.html">compute reduce</A> command.
			</P>
			<HR>

			<H4>Math Functions
			</H4>
			<P>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.
			</P>
			<P>Most of the math functions perform obvious operations. The ln() is
			the natural log; log() is the base 10 log.
			</P>
			<P>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.
			</P>
			<P>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.
			</P>
			<P>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.
			</P>
			<P>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:
			</P>
			<PRE>value = x + (y-x) * (timestep-startstep) / (stopstep-startstep)
			</PRE>
			<P>The run begins on startstep and ends on stopstep. Startstep and
			stopstep can span multiple runs, using the <I>start</I> and <I>stop</I> keywords
			of the <A HREF = "run.html">run</A> command. See the <A HREF = "run.html">run</A> command for
			details of how to do this.
			</P>
			<P>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 <A HREF = "dump_modify.html">dump_modify every</A>
			command, it will generate the sequence of output timesteps:
			</P>
			<PRE>100,1000,1100,2000,2100,3000,etc
			</PRE>
			<P>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 are on a base-z logarithmic scale, starting with x, and the
			y value is how many of the z-1 possible timesteps within one
			logarithmic interval are generated. I.e. the timesteps follow the
			sequence x,2x,3x,...yx,xz,2xz,3xz,...yxz,xz^2,2xz^2,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 <A HREF = "dump_modify.html">dump_modify
			every</A> command, it will generate this sequence of
			output timesteps:
			</P>
			<PRE>100,200,300,400,1000,2000,3000,4000,10000,20000,etc
			</PRE>
			<P>The logfreq2(x,y,z) function is similar to logfreq, except a single
			logarithmic interval is divided into y equally-spaced timesteps and
			all of them are output. Y < z is not required. Thus, if
			logfreq2(100,18,10) is used in a variable by the <A HREF = "dump_modify.html">dump_modify
			every</A> command, then the interval between 100 and
			1000 is divided as 900/18 = 50 steps, and it will generate the
			sequence of output timesteps:
			</P>
			<PRE>100,150,200,...950,1000,1500,2000,...9500,10000,15000,etc
			</PRE>
			<P>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 stride(1000,2000,100) is used
			in a variable by the <A HREF = "dump_modify.html">dump_modify every</A> command, it
			will generate the sequence of output timesteps:
			</P>
			<PRE>1000,1100,1200, ... ,1900,2000
			</PRE>
			<P>The stride2(x,y,z,a,b,c) function is similar to the stride() function
			except it generates two sets of strided timesteps, one at a coarser
			level and one at a finer level. Thus it is useful for debugging,
			e.g. to produce output every timestep at the point in simulation when
			a problem occurs. X,y >= 0 and z > 0 and x <= y are required, as are
			a,b >= 0 and c > 0 and a < b. Also, a >= x and b <= y are required so
			that the second stride is inside the first. The generated timesteps
			increase in increments of z, starting at x, until a is reached. At
			that point the timestep increases in increments of c, from a to b,
			then after b, increments by z are resumed until y is reached. For any
			current timestep, the next timestep in the sequence is returned. Thus
			if stride(1000,2000,100,1350,1360,1) is used in a variable by the
			<A HREF = "dump_modify.html">dump_modify every</A> command, it will generate the
			sequence of output timesteps:
			</P>
			<PRE>1000,1100,1200,1300,1350,1351,1352, ... 1359,1360,1400,1500, ... ,2000
			</PRE>
			<P>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:
			</P>
			<PRE>value = value0 + velocity(timestep-startstep)dt
			</PRE>
			<P>where dt = the timestep size.
			</P>
			<P>The run begins on startstep. Startstep can span multiple runs, using
			the <I>start</I> keyword of the <A HREF = "run.html">run</A> command. See the
			<A HREF = "run.html">run</A> command for details of how to do this. Note that the
			<A HREF = "thermo_style.html">thermo_style</A> keyword elaplong =
			timestep-startstep.
			</P>
			<P>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:
			</P>
			<PRE>value = value0 + Amplitude * sin(omega(timestep-startstep)dt)
			value = value0 + Amplitude * (1 - cos(omega(timestep-startstep)dt))
			</PRE>
			<P>where dt = the timestep size.
			</P>
			<P>The run begins on startstep. Startstep can span multiple runs, using
			the <I>start</I> keyword of the <A HREF = "run.html">run</A> command. See the
			<A HREF = "run.html">run</A> command for details of how to do this. Note that the
			<A HREF = "thermo_style.html">thermo_style</A> keyword elaplong =
			timestep-startstep.
			</P>
			<HR>

			<H4>Group and Region Functions
			</H4>
			<P>Group functions are specified as keywords followed by one or two
			parenthesized arguments. The first argument <I>ID</I> is the group-ID.
			The <I>dim</I> argument, if it exists, is <I>x</I> or <I>y</I> or <I>z</I>. The <I>dir</I>
			argument, if it exists, is <I>xmin</I>, <I>xmax</I>, <I>ymin</I>, <I>ymax</I>, <I>zmin</I>, or
			<I>zmax</I>. The <I>dimdim</I> argument, if it exists, is <I>xx</I> or <I>yy</I> or <I>zz</I>
			or <I>xy</I> or <I>yz</I> or <I>xz</I>.
			</P>
			<P>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 <A HREF = "compute_gyration.html">compute gyration</A> 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 symmetric inertia tensor of the group of atoms
			around its center of mass, ordered as Ixx,Iyy,Izz,Ixy,Iyz,Ixz.
			Omega() returns components of the angular velocity of the group of
			atoms around its center of mass.
			</P>
			<P>Region functions are specified exactly the same way as group functions
			except they take an extra final argument <I>IDR</I> 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.
			</P>
			<HR>

			<H4>Special Functions
			</H4>
			<P>Special functions take specific kinds of arguments, meaning their
			arguments cannot be formulas themselves.
			</P>
			<P>The sum(x), min(x), max(x), ave(x), trap(x), and slope(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.
			</P>
			<P>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
			<A HREF = "compute_reduce.html">compute reduce</A> command, which invokes the same
			functions on per-atom and local vectors.
			</P>
			<P>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.
			</P>
			<P>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 integration 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 <A HREF = "fix_ave_correlate.html">fix ave/correlate</A> command.
			</P>
			<P>The slope() function uses linear regression to fit a line to the 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 returned value is the slope of the line. If the line
			has a single point or is vertical, it returns 1.0e20.
			</P>
			<P>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.
			</P>
			<P>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.
			</P>
			<P>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.
			</P>
			<P>The next(x) function takes 1 argument which is a variable ID (not
			"v_foo", just "foo"). It must be for a file-style or atomfile-style
			variable. Each time the next() function is invoked (i.e. each time
			the equal-style or atom-style variable is evaluated), the following
			steps occur.
			</P>
			<P>For file-style variables, the current string value stored by the
			file-style variable is converted to a numeric value and returned by
			the function. And the next string value in the file is read and
			stored. Note that if the line previously read from the file was not a
			numeric string, then it will typically evaluate to 0.0, which is
			likely not what you want.
			</P>
			<P>For atomfile-style variables, the current per-atom values stored by
			the atomfile-style variable are returned by the function. And the
			next set of per-atom values in the file is read and stored.
			</P>
			<P>Since file-style and atomfile-style variables read and store the first
			line of the file or first set of per-atoms values when they are
			defined in the input script, these are the value(s) that will be
			returned the first time the next() function is invoked. If next() is
			invoked more times than there are lines or sets of lines in the file,
			the variable is deleted, similar to how the <A HREF = "next.html">next</A> command
			operates.
			</P>
			<HR>

			<H4>Atom Values and Vectors
			</H4>
			<P>Atom values take an integer argument I from 1 to N, where I is the
			atom-ID, e.g. x[243], which means use the x coordinate of the atom
			with ID = 243. Or they can take a variable name, specified as v_name,
			where name is the name of the variable, like x[v_myIndex]. The
			variable can be of any style except atom or atom-file variables. The
			variable is evaluated and the result is expected to be numeric and is
			cast to an integer (i.e. 3.4 becomes 3), to use an an index, which
			must be a value from 1 to N. Note that a "formula" cannot be used as
			the argument between the brackets, e.g. x[243+10] or
			x[v_myIndex+1] are not allowed. To do this a single variable can be
			defined that contains the needed formula.
			</P>
			<P>Note that the 0 < atom-ID <= N, where N is the largest atom ID
			in the system. If an ID is specified for an atom that does not
			currently exist, then the generated value is 0.0.
			</P>
			<P>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.
			</P>
			<P>The meaning of the different atom values and vectors is mostly
			self-explanatory. Mol refers to the molecule ID of an atom, and is
			only defined if an <A HREF = "atom_style.html">atom_style</A> is being used that
			defines molecule IDs.
			</P>
			<P>Note that many other atom attributes can be used as inputs to a
			variable by using the <A HREF = "compute_property_atom.html">compute
			property/atom</A> command and then specifying
			a quantity from that compute.
			</P>
			<HR>

			<H4>Compute References
			</H4>
			<P>Compute references access quantities calculated by a
			<A HREF = "compute.html">compute</A>. 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 <A HREF = "compute.html">compute</A> 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.
			</P>
			<P>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.
			</P>
			<DIV ALIGN=center><TABLE BORDER=1 >
			<TR><TD >c_ID</TD><TD > global scalar, or per-atom vector</TD></TR>
			<TR><TD >c_ID[I]</TD><TD > Ith element of global vector, or atom I's value in per-atom vector, or Ith column from per-atom array</TD></TR>
			<TR><TD >c_ID[I][J]</TD><TD > I,J element of global array, or atom I's Jth value in per-atom array
			</TD></TR></TABLE></DIV>

			<P>For I and J, integers can be specified or a variable name, specified
			as v_name, where name is the name of the variable. The rules for this
			syntax are the same as for the "Atom Values and Vectors" discussion
			above.
			</P>
			<P>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".
			</P>
			<HR>

			<H4>Fix References
			</H4>
			<P>Fix references access quantities calculated by a <A HREF = "compute.html">fix</A>.
			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
			<A HREF = "fix.html">fix</A> 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.
			</P>
			<P>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.
			</P>
			<DIV ALIGN=center><TABLE BORDER=1 >
			<TR><TD >f_ID</TD><TD > global scalar, or per-atom vector</TD></TR>
			<TR><TD >f_ID[I]</TD><TD > Ith element of global vector, or atom I's value in per-atom vector, or Ith column from per-atom array</TD></TR>
			<TR><TD >f_ID[I][J]</TD><TD > I,J element of global array, or atom I's Jth value in per-atom array
			</TD></TR></TABLE></DIV>

			<P>For I and J, integers can be specified or a variable name, specified
			as v_name, where name is the name of the variable. The rules for this
			syntax are the same as for the "Atom Values and Vectors" discussion
			above.
			</P>
			<P>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".
			</P>
			<P>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 <A HREF = "fix_ave_time.html">fix ave/time</A>
			command may only generate averaged quantities every 100 steps. See
			the doc pages for individual fix commands for details.
			</P>
			<HR>

			<H4>Variable References
			</H4>
			<P>Variable references access quantities stored or calculated 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.
			</P>
			<P>As discussed on this doc page, equal-style variables generate a global
			scalar numeric value; atom-style and atomfile-style variables generate
			a per-atom vector of numeric values; all other variables store a
			string. The formula for an equal-style variable can use any style of
			variable except an atom-style or atomfile-style (unless only a single
			value from the variable is accessed via a subscript). If a
			string-storing variable is used, the string is converted to a numeric
			value. Note that this will typically produce a 0.0 if the string is
			not a numeric string, which is likely not what you want. The formula
			for an atom-style variable can use any style of variable, including
			other atom-style or atomfile-style variables.
			</P>
			<P>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 vector, never both.
			</P>
			<DIV ALIGN=center><TABLE BORDER=1 >
			<TR><TD >v_name</TD><TD > scalar, or per-atom vector</TD></TR>
			<TR><TD >v_name[I]</TD><TD > atom I's value in per-atom vector
			</TD></TR></TABLE></DIV>

			<P>For I, an integer can be specified or a variable name, specified as
			v_name, where name is the name of the variable. The rules for this
			syntax are the same as for the "Atom Values and Vectors" discussion
			above.
			</P>
			<HR>

			<P><B>Immediate Evaluation of Variables:</B>
			</P>
			<P>If you want an equal-style variable to be evaluated immediately, it
			may be the case that you do not need to define a variable at all. See
			<A HREF = "Section_commands.html#cmd_2">Section 3.2</A> of the manual, which
			describes the use of "immediate" variables in an input script,
			specified as $(formula) with parenthesis, where the formula has the
			same syntax as equal-style variables described on this page. This
			effectively evaluates a formula immediately without using the variable
			command to define a named variable.
			</P>
			<P>More generally, 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 input script
			command, including a variable command. The input script parser
			evaluates the reference variable immediately and substitutes its value
			into the command. As explained in <A HREF = "Section_commands.html#3_2">Section commands
			3.2</A> for "Parsing rules", you can also use
			un-named "immediate" variables for this purpose. For example, a
			string like this $((xlo+xhi)/2+sqrt(v_area)) in an input script
			command evaluates the string between the parenthesis as an equal-style
			variable formula.
			</P>
			<P>Referencing a variable with a leading "v_" is an optional or required
			kind of argument for some commands (e.g. the <A HREF = "fix_ave_spatial.html">fix
			ave/spatial</A> or <A HREF = "dump.html">dump custom</A> or
			<A HREF = "thermo_style.html">thermo_style</A> 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.
			</P>
			<P>As an example, suppose you use this command in your input script to
			define the variable "v" as
			</P>
			<PRE>variable v equal vol
			</PRE>
			<P>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 <A HREF = "fix_ave_time.html">fix
			ave/time</A> then the current volume of the box will be
			evaluated continuously during the run.
			</P>
			<P>If you want to store the initial volume of the system, you can do it
			this way:
			</P>
			<PRE>variable v equal vol
			variable v0 equal $v
			</PRE>
			<P>The second command will force "v" to be evaluated (yielding the
			initial volume) and assign that value to the variable "v0". Thus the
			command
			</P>
			<PRE>thermo_style custom step v_v v_v0
			</PRE>
			<P>would print out both the current and initial volume periodically
			during the run.
			</P>
			<P>Note that it is a mistake to enclose a variable formula in double
			quotes if it contains variables preceeded by $ signs. For example,
			</P>
			<PRE>variable vratio equal "${vfinal}/${v0}"
			</PRE>
			<P>This is because the quotes prevent variable substitution (see <A HREF = "Section_commands.html#cmd_2">this
			section</A> on parsing input script
			commands), and thus an error will occur when the formula for "vratio"
			is evaluated later.
			</P>
			<HR>

			<P><B>Variable Accuracy:</B>
			</P>
			<P>Obviously, LAMMPS attempts to evaluate variables containing formulas
			(<I>equal</I> and <I>atom</I> style variables) accurately whenever the
			evaluation is performed. Depending on what is included in the
			formula, this may require invoking a <A HREF = "compute.html">compute</A>, 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 <A HREF = "fix.html">fix</A>. 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.
			</P>
			<P>LAMMPS keeps track of all of this during a <A HREF = "run.html">run</A> or <A HREF = "minimize.html">energy
			minimization</A>. An error will be generated if you
			attempt to evaluate a variable on timesteps when it cannot produce
			accurate values. For example, if a <A HREF = "thermo_style.html">thermo_style
			custom</A> command prints a variable which accesses
			values stored by a <A HREF = "fix_ave_time.html">fix ave/time</A> 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.
			</P>
			<P>An input script can also request variables be evaluated before or
			after or in between runs, e.g. by including them in a
			<A HREF = "print.html">print</A> 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 was invoked on the current
			timestep. 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:
			</P>
			<P>(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.
			</P>
			<P>(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 <A HREF = "compute.html">compute</A> that has not been invoked on
			the current timestep, 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, a variable containing a
			compute cannot be evaluated unless the compute was invoked on the last
			timestep of the preceding run, e.g. by thermodynamic output.
			</P>
			<P>One way to get around this problem is to perform a 0-timestep run
			before using the variable. For example, these commands
			</P>
			<PRE>variable t equal temp
			print "Initial temperature = $t"
			run 1000
			</PRE>
			<P>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.
			</P>
			<P>However, this sequence of commands would be fine:
			</P>
			<PRE>run 0
			variable t equal temp
			print "Initial temperature = $t"
			run 1000
			</PRE>
			<P>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 <A HREF = "thermo_style.html">thermo</A> or
			<A HREF = "dump.html">dump</A> 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 can insure it is
			invoked on the last timestep of the preceding run by including it in
			thermodynamic output.
			</P>
			<P>Unlike computes, <A HREF = "fix.html">fixes</A> 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 <A HREF = "fix_indent.html">fix indent</A>
			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.
			</P>
			<P>(3) The variable may be evaluated incorrectly and LAMMPS may have no
			way to detect this has occurred. Consider the following sequence of
			commands:
			</P>
			<PRE>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>
			<P>The first run is performed using one setting for the pairwise
			potential defined by the <A HREF = "pair_style.html">pair_style</A> and
			<A HREF = "pair_coeff.html">pair_coeff</A> commands. The potential energy is
			evaluated on the final timestep and stored by the <A HREF = "compute_pe.html">compute
			pe</A> compute (this is done by the
			<A HREF = "thermo_style.html">thermo_style</A> 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 <A HREF = "compute_pe.html">compute pe</A>
			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.
			</P>
			<P>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:
			</P>
			<PRE>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>
			<HR>

			<P><B>Restrictions:</B>
			</P>
			<P>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 <A HREF = "atom_modify.html">atom_modify
			map</A> command can override the default.
			</P>
			<P>All <I>universe</I>- and <I>uloop</I>-style variables defined in an input script
			must have the same number of values.
			</P>
			<P><B>Related commands:</B>
			</P>
			<P><A HREF = "next.html">next</A>, <A HREF = "jump.html">jump</A>, <A HREF = "include.html">include</A>,
			<A HREF = "temper.html">temper</A>, <A HREF = "fix_print.html">fix print</A>, <A HREF = "print.html">print</A>
			</P>
			<P><B>Default:</B> none
			</P>
			</HTML>

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