Heat current

Using the force expression, one can derive the following expression for the heat current for the whole system (\(E_i\) is the total energy of atom \(i\)) [Fan2015]:

\[\boldsymbol{J} = \boldsymbol{J}^{\text{pot}} + \boldsymbol{J}^{\text{kin}} = \sum_{i} \boldsymbol{J}^{\text{pot}}_i + \sum_{i} \boldsymbol{J}^{\text{kin}}_i;\]

where

\[\boldsymbol{J}^{\text{kin}}_i = \boldsymbol{v}_i E_i;\]

and

\[\boldsymbol{J}^{\text{pot}}_i = -\frac{1}{2}\sum_{j \neq i} \boldsymbol{r}_{ij} \left(\frac{\partial U_i}{\partial \boldsymbol{r}_{ij}} \cdot \boldsymbol{v}_j -\frac{\partial U_j}{\partial \boldsymbol{r}_{ji}} \cdot \boldsymbol{v}_i\right).\]

The potential part of the per-atom heat current can also be written in the following equivalent forms [Fan2015]:

\[\boldsymbol{J}^{\text{pot}}_i = -\sum_{j \neq i} \boldsymbol{r}_{ij} \left(\frac{\partial U_i}{\partial \boldsymbol{r}_{ij}} \cdot \boldsymbol{v}_j\right);\]

and

\[\boldsymbol{J}^{\text{pot}}_i = \sum_{j \neq i} \boldsymbol{r}_{ij} \left(\frac{ \partial U_j} {\partial \boldsymbol{r}_{ji}} \cdot \boldsymbol{v}_i\right).\]

Therefore, the per-atom heat current can also be expressed in terms of the per-atom virial [Gabourie2021]:

\[\boldsymbol{J}^{\text{pot}}_i = \mathbf{W}_i \cdot \boldsymbol{v}_i.\]

where the per-atom virial tensor cannot be assumed to be symmetric and the full tensor with 9 components should be used [Gabourie2021]. This result has actually been clear from the derivations in [Fan2015] but it was wrongly stated there that the potential part of the heat current cannot be expressed in terms of the per-atom virial.

One can also derive the following expression for the heat current from a subsystem \(A\) to a subsystem \(B\) [Fan2017]:

\[Q_{A \rightarrow B} = -\sum_{i \in A} \sum_{j \in B} \left(\frac{\partial U_i}{\partial \boldsymbol{r}_{ij}} \cdot \boldsymbol{v}_j -\frac{\partial U_j}{\partial \boldsymbol{r}_{ji}} \cdot \boldsymbol{v}_i\right).\]