**Methodology and meaning of computing heat flux via atomic stress in
systems with constraint dynamics**

D Surblys and H Matsubara and G Kikugawa and T Ohara, JOURNAL OF APPLIED PHYSICS, 130, 215104 (2021).

DOI: 10.1063/5.0070930

Reliably obtaining thermal properties of complex systems, which often
involves computing heat flux to obtain thermal conductivity via either
Fourier's law or the Green-Kubo relation, is an important task in modern
molecular dynamics simulations. In our previous work **Surblys et al.,
Phys. Rev. E 99, 051301(R) (2019)**, we have demonstrated that atomic
stress could be used to efficiently compute heat flux for molecules with
angle, dihedral, or improper many-body interactions, provided a newly
derived "centroid" form was used. This was later successfully
implemented in the LAMMPS simulation package. On the other hand, small
rigid molecules, like water and partial constraints in semi-flexible
molecules, are often implemented via constraint force algorithms. There
has been a lack of clarification if the constraint forces that maintain
geometric constraints and can also be considered as many-body forces
contribute to the overall heat flux and how to compute them correctly
and efficiently. To address this, we investigate how to apply the
centroid atomic stress form to reliably compute heat flux for systems
with constraint or rigid body dynamics. We successfully apply the
centroid atomic stress form to flexible, semi-flexible, and rigid water
models; decompose the computed thermal conductivity into separate
components; and demonstrate that the contribution from constraint forces
to the overall heat flux and thermal conductivity is small but non-
negligible. We also show that while the centroid formulation produces
correct heat flux values, the original "group" formulation produces
incorrect and sometimes unphysical results. Finally, we provide insight
into the meaning of constraint force contribution.(c) 2021 Author(s).
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