ML LHEAT

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

Description: This flag 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 can 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 Sections

ML_FF_LMLFF, ML_FF_LEATOM_MB

Examples that use this tag

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