ML LHEAT: Difference between revisions

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<math>
<math>
m_{i} \frac{d \mathbf{v}_{i}}{dt} = - \sum\limits_{j=1}{N_{a}} \nabla_{i} U_{j}
m_{i} \frac{d \mathbf{v}_{i}}{dt} = - \sum\limits_{j=1}{N_{a}} \nabla_{i} U_{j}
</math>
the heat flux can be simplified to
<math>
\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) + um\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).
</math>
</math>



Revision as of 09:48, 12 June 2021

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.


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

where , and denote the position vector, velocity and energy of atom , respectively. The number of atoms in the system is denoted by . The heat flux can be further rewritten as

Using the equation of motions

the heat flux can be simplified to

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: \limits is allowed only on operators"): {\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)+um\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 in the Green-Kubo formalism can be calculated from the correlation of the heat flux as


where and 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