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# ML LHEAT

ML_LHEAT = [logical]
Default: ML_LHEAT = .FALSE.

Description: This tag specifies whether the heat flux is calculated or not in the machine learning force field method.

The heat flux within machine learning force fields can is decomposed into atomic contributions written as

${\displaystyle \mathbf {q} (t)=\sum \limits _{i=1}^{N_{a}}{\frac {d}{dt}}\left(\mathbf {r} _{i}E_{i}\right),}$

${\displaystyle E_{i}={\frac {m_{i}\left|\mathbf {v} _{i}\right|^{2}}{2}}+U_{i}}$

where ${\displaystyle \mathbf {r} _{i}}$, ${\displaystyle \mathbf {v} _{i}}$ and ${\displaystyle E_{i}}$ denote the position vector, velocity and energy of atom ${\displaystyle i}$, respectively. The number of atoms in the system is denoted by ${\displaystyle N_{a}}$. The heat flux can be further rewritten as

${\displaystyle \mathbf {q} (t)=\sum \limits _{i=1}^{N_{a}}\mathbf {v} _{i}E_{i}+\sum \limits _{i=1}^{N_{a}}\mathbf {r} _{i}\left(m_{i}\mathbf {v} _{i}\cdot {\frac {d\mathbf {v} _{i}}{dt}}+\sum \limits _{j=1}^{N_{a}}\mathbf {v} _{j}\cdot \nabla _{j}U_{i}\right).}$

Using the equation of motions

${\displaystyle m_{i}{\frac {d\mathbf {v} _{i}}{dt}}=-\sum \limits _{j=1}^{N_{a}}\nabla _{i}U_{j}}$

the heat flux can be simplified to

${\displaystyle \mathbf {q} (t)=\sum \limits _{i=1}^{N_{a}}\mathbf {v} _{i}E_{i}-\sum \limits _{i=1}^{N_{a}}\sum \limits _{j=1}^{N_{a}}\mathbf {r} _{i}\left(\mathbf {v} _{i}\cdot \nabla _{i}U_{j}\right)+\sum \limits _{i=1}^{N_{a}}\sum \limits _{j=1}^{N_{a}}\mathbf {r} _{i}\left(\mathbf {v} _{j}\cdot \nabla _{j}U_{i}\right)=\sum \limits _{i=1}^{N_{a}}\mathbf {v} _{i}E_{i}+\sum \limits _{i=1}^{N_{a}}\sum \limits _{j=1}^{N_{a}}\left(\mathbf {r} _{i}-\mathbf {r} _{j}\right)\left(\mathbf {v} _{j}\cdot \nabla _{j}U_{i}\right).}$

Finally (in a post-processing step), the thermal conductivity at temperature ${\displaystyle T}$ in the Green-Kubo formalism can be calculated from the correlation of the heat flux ${\displaystyle \mathbf {q} }$ as

${\displaystyle \kappa ={\frac {1}{3Vk_{b}T^{2}}}\int \limits _{0}^{\infty }\langle \mathbf {q} (t)\cdot \mathbf {q} (0)\rangle dt,}$

where ${\displaystyle V}$ and ${\displaystyle k_{b}}$ denotes the volume of the system and the Boltzmann constant, respectively.

The heat flux is written to the file ML_HEAT.