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update compute heat/flux documentation

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Donatas Surblys 2019-11-12 22:43:52 +09:00
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doc/src/compute_heat_flux.rst
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@ -54,38 +54,58 @@ third calculates per-atom stress (\ *stress-ID*\ ).
(or any group whose atoms are superset of the atoms in this compute's
group). LAMMPS does not check for this.
The Green-Kubo formulas relate the ensemble average of the
auto-correlation of the heat flux J to the thermal conductivity kappa:
In case of pairwise interactions the heat flux is defined as:
.. image:: Eqs/heat_flux_J.jpg
:align: center
.. math::
\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} \left( \mathbf{F}_{ij} \cdot \left(\mathbf{v}_i + \mathbf{v}_j \right) \right) \mathbf{r}_{ij} \right]
.. image:: Eqs/heat_flux_k.jpg
:align: center
Ei in the first term of the equation for J is the per-atom energy
(potential and kinetic). This is calculated by the computes *ke-ID*
and *pe-ID*\ . Si in the second term of the equation for J is the
per-atom stress tensor calculated by the compute *stress-ID*\ . 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
:math:`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*. :math:`\mathbf{S}_i` in the second term is the
per-atom stress tensor calculated by the compute *stress-ID*.
See :doc:`compute stress/atom <compute_stress_atom>`
and :doc:`compute centroid/stress/atom <compute_stress_atom>`
for possible definitions of atomic stress :math:`\mathbf{S}_i`
in the case of thee-body and larger many-body interactions.
The tensor multiplies :math:`\mathbf{v}_i` as a 3x3 matrix-vector multiply
to yield a vector.
Note that as discussed below, the :math:`\frac{1}{V}` scaling factor in the
equation for :math:`\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 :doc:`compute pe/atom <compute_pe_atom>` and :doc:`compute stress/atom <compute_stress_atom>` commands have options for which
The :doc:`compute pe/atom <compute_pe_atom>` and
:doc:`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
flux calculation will thus include exactly the same terms. Normally
you should use :doc:`compute stress/atom virial <compute_stress_atom>`
or :doc:`compute centroid/stress/atom virial <compute_stress_atom>`
so as not to include a kinetic energy term in the heat flux.
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.
.. warning::
The compute *heat/flux* has been reported to produce unphysical
values for three and larger many-body interactions such
as angles, dihedrals and torsions,
when used with :doc:`compute stress/atom <compute_stress_atom>`,
as discussed in :ref:`(Surblys) <Surblys2>` and :ref:`(Boone) <Boone>`.
You are strongly advised to
use :doc:`compute centroid/stress/atom <compute_stress_atom>`,
which has been implemented specifically for such cases.
The Green-Kubo formulas relate the ensemble average of the
auto-correlation of the heat flux :math:`\mathbf{J}`
to the thermal conductivity :math:`\kappa`:
.. math::
\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
----------
@ -109,9 +129,15 @@ result should be: average conductivity ~0.29 in W/mK.
**Output info:**
This compute calculates a global vector of length 6 (total heat flux
vector, followed by convective heat flux vector), which can be
accessed by indices 1-6. These values can be used by any command that
This compute calculates a global vector of length 6.
The first 3 components are the :math:`x`, :math:`y`, :math:`z`
components of the full heat flux vector,
i.e. (:math:`J_x`, :math:`J_y`, :math:`J_z`).
The next 3 components are the :math:`x`, :math:`y`, :math:`z` components
of just the convective portion of the flux, i.e. the
first term in the equation for :math:`\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 :doc:`Howto output <Howto_output>` doc page for an overview of LAMMPS output
options.
@ -212,6 +238,22 @@ Related commands
print "average conductivity: $k[W/mK] @ $T K, ${ndens} /A\^3"
----------
.. _Surblys2:
**(Surblys)** Surblys, Matsubara, Kikugawa, Ohara, Phys Rev E, 99, 051301(R) (2019).
.. _Boone:
**(Boone)** Boone, Babaei, Wilmer, J Chem Theory Comput, 15, 5579--5587 (2019).
.. _lws: http://lammps.sandia.gov
.. _ld: Manual.html
.. _lc: Commands_all.html