# Calculate thermal conductivity

The thermal conductivity kappa of a material can be measured in at least
4 ways using various options in LAMMPS. See the examples/KAPPA directory
for scripts that implement the 4 methods discussed here for a simple
Lennard-Jones fluid model. Also, see the [Howto
viscosity](Howto_viscosity) page for an analogous discussion for
viscosity.

The thermal conductivity tensor kappa is a measure of the propensity of
a material to transmit heat energy in a diffusive manner as given by
Fourier\'s law

J = -kappa grad(T)

where J is the heat flux in units of energy per area per time and
grad(T) is the spatial gradient of temperature. The thermal conductivity
thus has units of energy per distance per time per degree K and is often
approximated as an isotropic quantity, i.e. as a scalar.

The first method is to setup two thermostatted regions at opposite ends
of a simulation box, or one in the middle and one at the end of a
periodic box. By holding the two regions at different temperatures with
a [thermostatting fix](Howto_thermostat), the energy added to the hot
region should equal the energy subtracted from the cold region and be
proportional to the heat flux moving between the regions. See the papers
by [Ikeshoji and Hafskjold](howto-Ikeshoji) and [Wirnsberger et
al](howto-Wirnsberger) for details of this idea. Note that
thermostatting fixes such as [fix nvt](fix_nh), [fix
langevin](fix_langevin), and [fix temp/rescale](fix_temp_rescale) store
the cumulative energy they add/subtract.

Alternatively, as a second method, the [fix heat](fix_heat) or [fix
ehex](fix_ehex) commands can be used in place of thermostats on each of
two regions to add/subtract specified amounts of energy to both regions.
In both cases, the resulting temperatures of the two regions can be
monitored with the \"compute temp/region\" command and the temperature
profile of the intermediate region can be monitored with the [fix
ave/chunk](fix_ave_chunk) and [compute ke/atom](compute_ke_atom)
commands.

The third method is to perform a reverse non-equilibrium MD simulation
using the [fix thermal/conductivity](fix_thermal_conductivity) command
which implements the rNEMD algorithm of Muller-Plathe. Kinetic energy is
swapped between atoms in two different layers of the simulation box.
This induces a temperature gradient between the two layers which can be
monitored with the [fix ave/chunk](fix_ave_chunk) and [compute
ke/atom](compute_ke_atom) commands. The fix tallies the cumulative
energy transfer that it performs. See the [fix
thermal/conductivity](fix_thermal_conductivity) command for details.

The fourth method is based on the Green-Kubo (GK) formula which relates
the ensemble average of the auto-correlation of the heat flux to kappa.
The heat flux can be calculated from the fluctuations of per-atom
potential and kinetic energies and per-atom stress tensor in a
steady-state equilibrated simulation. This is in contrast to the two
preceding non-equilibrium methods, where energy flows continuously
between hot and cold regions of the simulation box.

The [compute heat/flux](compute_heat_flux) command can calculate the
needed heat flux and describes how to implement the Green_Kubo formalism
using additional LAMMPS commands, such as the [fix
ave/correlate](fix_ave_correlate) command to calculate the needed
auto-correlation. See the page for the [compute
heat/flux](compute_heat_flux) command for an example input script that
calculates the thermal conductivity of solid Ar via the GK formalism.

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

::: {#howto-Ikeshoji}
**(Ikeshoji)** Ikeshoji and Hafskjold, Molecular Physics, 81, 251-261
(1994).
:::

::: {#howto-Wirnsberger}
**(Wirnsberger)** Wirnsberger, Frenkel, and Dellago, J Chem Phys, 143,
124104 (2015).
:::