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<CENTER><A HREF = "http://lammps.sandia.gov">LAMMPS WWW Site</A> - <A HREF = "Manual.html">LAMMPS Documentation</A> - <A HREF = "Section_commands.html#comm">LAMMPS Commands</A>
</CENTER>
<HR>
<H3>compute heat/flux command
</H3>
<P><B>Syntax:</B>
</P>
<PRE>compute ID group-ID heat/flux ke-ID pe-ID stress-ID
</PRE>
<UL><LI>ID, group-ID are documented in <A HREF = "compute.html">compute</A> command
<LI>heat/flux = style name of this compute command
<LI>ke-ID = ID of a compute that calculates per-atom kinetic energy
<LI>pe-ID = ID of a compute that calculates per-atom potential energy
<LI>stress-ID = ID of a compute that calculates per-atom stress
</UL>
<P><B>Examples:</B>
</P>
<PRE>compute myFlux all heat/flux myKE myPE myStress
</PRE>
<P><B>Description:</B>
</P>
<P>Define a computation that calculates the heat flux vector based on
contributions from atoms in the specified group. This can be used by
itself to measure the heat flux into or out of a reservoir of atoms,
or to calculate a thermal conductivity using the Green-Kubo formalism.
</P>
<P>See the <A HREF = "fix_thermal_conductivity.html">fix thermal/conductivity</A>
command for details on how to compute thermal conductivity in an
alternate way, via the Muller-Plathe method. See the <A HREF = "fix_heat.html">fix
heat</A> command for a way to control the heat added or
subtracted to a group of atoms.
</P>
<P>The compute takes three arguments which are IDs of other
<A HREF = "compute.html">computes</A>. One calculates per-atom kinetic energy
(<I>ke-ID</I>), one calculates per-atom potential energy (<I>pe-ID)</I>, and the
third calcualtes per-atom stress (<I>stress-ID</I>). These should be
defined for the same group used by compute heat/flux, though LAMMPS
does not check for this.
</P>
<P>The Green-Kubo formulas relate the ensemble average of the
auto-correlation of the heat flux J to the thermal conductivity kappa:
</P>
<CENTER><IMG SRC = "Eqs/heat_flux_J.jpg">
</CENTER>
<CENTER><IMG SRC = "Eqs/heat_flux_k.jpg">
</CENTER>
<P>Ei in the first term of the equation for J is the per-atom energy
(potential and kinetic). This is calculated by the computes <I>ke-ID</I>
and <I>pe-ID</I>. Si in the second term of the equation for J is the
per-atom stress tensor calculated by the compute <I>stress-ID</I>. The
tensor multiplies Vi as a 3x3 matrix-vector multiply to yield a
vector. Note that as discussed below, the 1/V scaling factor in the
equation for J is NOT included in the calculation performed by this
compute; you need to add it for a volume appropriate to the atoms
included in the calculation.
</P>
<P>IMPORTANT NOTE: The <A HREF = "compute_pe_atom.html">compute pe/atom</A> and
<A HREF = "compute_stress_atom.html">compute stress/atom</A> commands have options
for which terms to include in their calculation (pair, bond, etc).
The heat flux calculation will thus include exactly the same terms.
Normally you should use <A HREF = "compute_stress_atom.html">compute stress/atom
virial</A> so as not to include a kinetic energy
term in the heat flux.
</P>
<P>This compute calculates 6 quantities and stores them in a 6-component
vector. The first 3 components are the x, y, z components of the full
heat flux vector, i.e. (Jx, Jy, Jz). The next 3 components are the x,
y, z components of just the convective portion of the flux, i.e. the
first term in the equation for J above.
</P>
<HR>
<P>The heat flux can be output every so many timesteps (e.g. via the
<A HREF = "thermo_style.html">thermo_style custom</A> command). Then as a
post-processing operation, an autocorrelation can be performed, its
integral estimated, and the Green-Kubo formula above evaluated.
</P>
<P>The <A HREF = "fix_ave_correlate.html">fix ave/correlate</A> command can calclate
the autocorrelation. The trap() function in the
<A HREF = "variable.html">variable</A> command can calculate the integral.
</P>
<P>An example LAMMPS input script for solid Ar is appended below. The
result should be: average conductivity ~0.29 in W/mK.
</P>
<HR>
<P><B>Output info:</B>
</P>
<P>This compute calculates a global vector of length 6 (total heat flux
vector, followed by conductive heat flux vector), which can be
accessed by indices 1-6. These values can be used by any command that
uses global vector values from a compute as input. See <A HREF = "Section_howto.html#howto_15">this
section</A> for an overview of LAMMPS output
options.
</P>
<P>The vector values calculated by this compute are "extensive", meaning
they scale with the number of atoms in the simulation. They can be
divided by the appropriate volume to get a flux, which would then be
an "intensive" value, meaning independent of the number of atoms in
the simulation. Note that if the compute is "all", then the
appropriate volume to divide by is the simulation box volume.
However, if a sub-group is used, it should be the volume containing
those atoms.
</P>
<P>The vector values will be in energy*velocity <A HREF = "units.html">units</A>. Once
divided by a volume the units will be that of flux, namely
energy/area/time <A HREF = "units.html">units</A>
</P>
<P><B>Restrictions:</B> none
</P>
<P><B>Related commands:</B>
</P>
<P><A HREF = "fix_thermal_conductivity.html">fix thermal/conductivity</A>,
<A HREF = "fix_ave_correlate.html">fix ave/correlate</A>,
<A HREF = "variable.html">variable</A>
</P>
<P><B>Default:</B> none
</P>
<HR>
<PRE># Sample LAMMPS input script for thermal conductivity of solid Ar
</PRE>
<PRE>units real
variable T equal 70
variable V equal vol
variable dt equal 4.0
variable p equal 200 # correlation length
variable s equal 10 # sample interval
variable d equal $p*$s # dump interval
</PRE>
<PRE># convert from LAMMPS real units to SI
</PRE>
<PRE>variable kB equal 1.3806504e-23 # [J/K] Boltzmann
variable kCal2J equal 4186.0/6.02214e23
variable A2m equal 1.0e-10
variable fs2s equal 1.0e-15
variable convert equal ${kCal2J}*${kCal2J}/${fs2s}/${A2m}
</PRE>
<PRE># setup problem
</PRE>
<PRE>dimension 3
boundary p p p
lattice fcc 5.376 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
region box block 0 4 0 4 0 4
create_box 1 box
create_atoms 1 box
mass 1 39.948
pair_style lj/cut 13.0
pair_coeff * * 0.2381 3.405
timestep ${dt}
thermo $d
</PRE>
<PRE># equilibration and thermalization
</PRE>
<PRE>velocity all create $T 102486 mom yes rot yes dist gaussian
fix NVT all nvt temp $T $T 10 drag 0.2
run 8000
</PRE>
<PRE># thermal conductivity calculation, switch to NVE if desired
</PRE>
<PRE>#unfix NVT
#fix NVE all nve
</PRE>
<PRE>reset_timestep 0
compute myKE all ke/atom
compute myPE all pe/atom
compute myStress all stress/atom NULL virial
compute flux all heat/flux myKE myPE myStress
variable Jx equal c_flux[1]/vol
variable Jy equal c_flux[2]/vol
variable Jz equal c_flux[3]/vol
fix JJ all ave/correlate $s $p $d &
c_flux[1] c_flux[2] c_flux[3] type auto file J0Jt.dat ave running
variable scale equal ${convert}/${kB}/$T/$T/$V*$s*${dt}
variable k11 equal trap(f_JJ[3])*${scale}
variable k22 equal trap(f_JJ[4])*${scale}
variable k33 equal trap(f_JJ[5])*${scale}
thermo_style custom step temp v_Jx v_Jy v_Jz v_k11 v_k22 v_k33
run 100000
variable k equal (v_k11+v_k22+v_k33)/3.0
variable ndens equal count(all)/vol
print "average conductivity: $k[W/mK] @ $T K, ${ndens} /A^3"
</PRE>
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