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

$\mathbf{𝐪}(t)=\sum _{i=1}^{N_a}\frac{d}{dt}\left({\mathbf{𝐫}}_i{E}_i\right),$

$E_i=\frac{m_i{\left|{\mathbf{𝐯}}_i\right|}^2}{2}+U_i$

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

$\mathbf{𝐪}(t)=\sum _{i=1}^{N_a}{\mathbf{𝐯}}_i{E}_i+\sum _{i=1}^{N_a}{\mathbf{𝐫}}_i(m_i{\mathbf{𝐯}}_i\cdot \frac{d{\mathbf{𝐯}}_i}{dt}+\sum _{j=1}^{N_a}{\mathbf{𝐯}}_j\cdot {\mathrm{\nabla }}_j{U}_i).$

Using the equation of motions

$m_i\frac{d{\mathbf{𝐯}}_i}{dt}=-\sum _{j=1}^{N_a}{\mathrm{\nabla }}_i{U}_j$

the heat flux can be simplified to

$\mathbf{𝐪}(t)=\sum _{i=1}^{N_a}{\mathbf{𝐯}}_i{E}_i-\sum _{i=1}^{N_a}\sum _{j=1}^{N_a}{\mathbf{𝐫}}_i({\mathbf{𝐯}}_i\cdot {\mathrm{\nabla }}_i{U}_j)+\sum _{i=1}^{N_a}\sum _{j=1}^{N_a}{\mathbf{𝐫}}_i({\mathbf{𝐯}}_j\cdot {\mathrm{\nabla }}_j{U}_i)=\sum _{i=1}^{N_a}{\mathbf{𝐯}}_i{E}_i+\sum _{i=1}^{N_a}\sum _{j=1}^{N_a}({\mathbf{𝐫}}_i-{\mathbf{𝐫}}_j)({\mathbf{𝐯}}_j\cdot {\mathrm{\nabla }}_j{U}_i).$

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

$\kappa =\frac{1}{3V{k}_b{T}^2}\underset{0}{\overset{\mathrm{\infty }}{\int }}⟨\mathbf{𝐪}(t)\cdot \mathbf{𝐪}(0)⟩dt,$

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_LMLFF, ML_LEATOM

Examples that use this tag

---