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# compute heat/flux command

## Syntax

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

- ID, group-ID are documented in 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

    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 doc page. These include use of the fix thermal/conductivity command for the Muller-Plathe method. Or the fix heat command which can add or subtract heat from groups of atoms.

The compute takes three arguments which are IDs of other computes. 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

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

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

\(e_i\) 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 and compute centroid/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

The compute pe/atom and 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 or compute centroid/stress/atom virial so as not to include a kinetic energy term in the heat flux.

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, as discussed in (Surblys2019), (Boone) and (Surblys2021). You are strongly advised to use compute centroid/stress/atom, which has been implemented specifically for such cases.

Warning

Due to an implementation detail, the \(y\) and \(z\) components of heat flux from fix rigid contribution when computed via 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 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 command can calculate the auto-correlation. The trap() function in the 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 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. Once divided by a volume the units will be that of flux, namely energy/area/time units

## Restrictions

none

## Related commands

fix thermal/conductivity, fix ave/correlate, variable

## Default

none

### Example Input File

    # 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"

(Surblys2019) Surblys, Matsubara, Kikugawa, Ohara, Phys Rev E, 99, 051301(R) (2019).

(Boone) Boone, Babaei, Wilmer, J Chem Theory Comput, 15, 5579–5587 (2019).

(Surblys2021) Surblys, Matsubara, Kikugawa, Ohara, J Appl Phys 130, 215104 (2021).