Hello everyone,

I'm trying to understand the equation of heat flux which is being used by molecular dynamic calculations. In order to use it for multi component system, the equation has to be modified according to [1][2][3]. In the modified equation, they are subtracting the average enthalpy of the sub-species from the general heat flux as in attached EQ-1.png. Green outlined part is the stranded heat flux calculation in molecular dynamics and Red outlined part is the modification involving average enthalpy (halpha) of sub-species. Here halpha is defined as in attached EQ-2.png. First two terms in EQ-2 is straight forward, with the summation, it's the total kinetic and potential energy of species alpha. But I'm confusing about the third term here, which is the virial contribution. I don't understand why is there a 1/3 factor in front of it and rij*Fij, is this the dot product? If it is the dot product, then it would calculate a scalar value (rxFx+ryFy+rzFz) and equation would be pretty understandable. But standard virial calculation in MD codes gives a 9 component vector (rxFx ryFy rzFz rxFy rxFz ryFz ryFx rzFx rzFy) or basically with 6 as a symmetric tensor. But in EQ-2 only the diagonal elements of the virial tensor is used if we consider the dot product of r and F. Wish someone could help me to clear my foggy understanding. Thank you very much.

[1] - http://dx.doi.org/10.1063/1.4749265

[2] - http://dx.doi.org/10.1103/PhysRevLett.98.028302

[3] - https://doi.org/10.1088/1361-648X/aa5f08