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.

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

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

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

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

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

Using the equation of motions

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

the heat flux can be simplified to

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

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

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

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

The heat flux is written to the file ML_HEAT.

Related tags and articles

ML_LMLFF, ML_LEATOM

Examples that use this tag

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