# compute heat/flux command

## Syntax

``` LAMMPS
compute ID group-ID heat/flux ke-ID pe-ID stress-ID
```

-   ID, group-ID are documented in [compute](compute) command
-   heat/flux = style name of this compute command
-   ke-ID = ID of a compute that calculates per-atom kinetic energy
-   pe-ID = ID of a compute that calculates per-atom potential energy
-   stress-ID = ID of a compute that calculates per-atom stress

## Examples

``` LAMMPS
compute myFlux all heat/flux myKE myPE myStress
```

## Description

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 through a set of atoms (e.g., a region
between two thermostatted reservoirs held at different temperatures), or
to calculate a thermal conductivity using the equilibrium Green-Kubo
formalism.

For other non-equilibrium ways to compute a thermal conductivity, see
the [Howto kappa](Howto_kappa) doc page. These include use of the [fix
thermal/conductivity](fix_thermal_conductivity) command for the
Muller-Plathe method. Or the [fix heat](fix_heat) command which can add
or subtract heat from groups of atoms.

The compute takes three arguments which are IDs of other
[computes](compute). One calculates per-atom kinetic energy (\*ke-ID\*),
one calculates per-atom potential energy (\*pe-ID)\*, and the third
calculates per-atom stress (\*stress-ID\*).

:::: note
::: title
Note
:::

These other computes should provide values for all the atoms in the
group this compute specifies. That means the other computes could use
the same group as this compute, or they can just use group \"all\" (or
any group whose atoms are superset of the atoms in this compute\'s
group). LAMMPS does not check for this.
::::

In case of two-body interactions, the heat flux $\mathbf{J}$ is defined
as

$$\begin{aligned}
\mathbf{J} &= \frac{1}{V} \left[ \sum_i e_i \mathbf{v}_i - \sum_{i} \mathbf{S}_{i} \mathbf{v}_i \right] \\
&= \frac{1}{V} \left[ \sum_i e_i \mathbf{v}_i + \sum_{i<j} \left( \mathbf{F}_{ij} \cdot \mathbf{v}_j \right) \mathbf{r}_{ij} \right] \\
&= \frac{1}{V} \left[ \sum_i e_i \mathbf{v}_i + \frac{1}{2} \sum_{i<j} \bigl( \mathbf{F}_{ij} \cdot \left(\mathbf{v}_i + \mathbf{v}_j \right) \bigr) \mathbf{r}_{ij} \right]
\end{aligned}$$

$e_i$ in the first term of the equation is the per-atom energy
(potential and kinetic). This is calculated by the computes *ke-ID* and
*pe-ID*. $\mathbf{S}_i$ in the second term is the per-atom stress tensor
calculated by the compute *stress-ID*. See [compute
stress/atom](compute_stress_atom) and [compute
centroid/stress/atom](compute_stress_atom) for possible definitions of
atomic stress $\mathbf{S}_i$ in the case of bonded and many-body
interactions. The tensor multiplies $\mathbf{v}_i$ by a $3\times3$
matrix to yield a vector. Note that as discussed below, the $1/V$
scaling factor in the equation for $\mathbf{J}$ is **not** included in
the calculation performed by these computes; you need to add it for a
volume appropriate to the atoms included in the calculation.

:::: note
::: title
Note
:::

The [compute pe/atom](compute_pe_atom) and [compute
stress/atom](compute_stress_atom) 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 [compute stress/atom virial](compute_stress_atom) or [compute
centroid/stress/atom virial](compute_stress_atom) so as not to include a
kinetic energy term in the heat flux.
::::

:::: warning
::: title
Warning
:::

The compute *heat/flux* has been reported to produce unphysical values
for angle, dihedral, improper and constraint force contributions when
used with [compute stress/atom](compute_stress_atom), as discussed in
[(Surblys2019)](Surblys3), [(Boone)](Boone) and
[(Surblys2021)](Surblys4). You are strongly advised to use [compute
centroid/stress/atom](compute_stress_atom), which has been implemented
specifically for such cases.
::::

:::: warning
::: title
Warning
:::

Due to an implementation detail, the $y$ and $z$ components of heat flux
from [fix rigid](fix_rigid) contribution when computed via [compute
stress/atom](compute_stress_atom) are highly unphysical and should not
be used.
::::

The Green\--Kubo formulas relate the ensemble average of the
auto-correlation of the heat flux $\mathbf{J}$ to the thermal
conductivity $\kappa$:

$$\kappa  = \frac{V}{k_B T^2} \int_0^\infty \langle J_x(0)  J_x(t) \rangle \, \mathrm{d} t = \frac{V}{3 k_B T^2} \int_0^\infty \langle \mathbf{J}(0) \cdot  \mathbf{J}(t)  \rangle \, \mathrm{d}t$$

------------------------------------------------------------------------

The heat flux can be output every so many timesteps (e.g., via the
[thermo_style custom](thermo_style) command). Then as a post-processing
operation, an auto-correlation can be performed, its integral estimated,
and the Green\--Kubo formula above evaluated.

The [fix ave/correlate](fix_ave_correlate) command can calculate the
auto-correlation. The trap() function in the [variable](variable)
command can calculate the integral.

An example LAMMPS input script for solid argon is appended below. The
result should be an average conductivity
$\approx 0.29~\mathrm{W/m \cdot K}$.

------------------------------------------------------------------------

## Output info

This compute calculates a global vector of length 6. The first three
components are the $x$, $y$, and $z$ components of the full heat flux
vector (i.e., $J_x$, $J_y$, and $J_z$). The next three components are
the $x$, $y$, and $z$ components of just the convective portion of the
flux (i.e., the first term in the equation for $\mathbf{J}$). Each
component 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
the [Howto output](Howto_output) documentation for an overview of LAMMPS
output options.

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 group is \"all\", then the
appropriate volume to divide by is the simulation box volume. However,
if a group with a subset of atoms is used, it should be the volume
containing those atoms.

The vector values will be in energy\*velocity [units](units). Once
divided by a volume the units will be that of flux, namely
energy/area/time [units](units)

## Restrictions

> none

## Related commands

[fix thermal/conductivity](fix_thermal_conductivity), [fix
ave/correlate](fix_ave_correlate), [variable](variable)

## Default

none

------------------------------------------------------------------------

### Example Input File

``` LAMMPS
# Sample LAMMPS input script for thermal conductivity of solid Ar

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

# convert from LAMMPS real units to SI

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}

# setup problem

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

# equilibration and thermalization

velocity     all create $T 102486 mom yes rot yes dist gaussian
fix          NVT all nvt temp $T $T 10 drag 0.2
run          8000

# thermal conductivity calculation, switch to NVE if desired

#unfix       NVT
#fix         NVE all nve

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"
```

------------------------------------------------------------------------

::: {#Surblys3}
**(Surblys2019)** Surblys, Matsubara, Kikugawa, Ohara, Phys Rev E, 99,
051301(R) (2019).
:::

::: {#Boone}
**(Boone)** Boone, Babaei, Wilmer, J Chem Theory Comput, 15, 5579\--5587
(2019).
:::

::: {#Surblys4}
**(Surblys2021)** Surblys, Matsubara, Kikugawa, Ohara, J Appl Phys 130,
215104 (2021).
:::