The quantum method of planes-local pressure definitions for machine learning potentials

Stress, or pressure, is a central quantity in engineering and remains vital in molecular modeling. However, the commonly used virial stress tensor is invalid for an inhomogeneous fluid, which is essential in fluid dynamics and non-equilibrium molecular dynamics (NEMD) simulations. This is solved by using the method of planes (MoP), a mechanical form of pressure, simply interpreted as the force divided by area, yet it is derived from the firm foundations of statistical mechanics. We present an extension of MoP stress [B. D. Todd et al., Phys. Rev. E 52, 1627-1638 (1995)] to the MACE potential, a particular form of machine learning (ML) potential allowing the incorporation of quantum mechanical physics into classical simulation. We present the derivation of this local stress for the MACE potential using the theoretical framework set out by Irving and Kirkwood [J. Chem. Phys. 18(6), 817-829 (1950)]. For the test case of an interface between water and zirconium oxide, we show that the MoP measures the correct force balance while the virial form fails. Furthermore, we demonstrate that this planar definition of stress is valid arbitrarily far from equilibrium, showing exact conservation every time step in a control volume bounded by MoP. This links the stress directly to the conservation equations and demonstrates the validity in non-equilibrium molecular dynamics systems. All code to reproduce these validations for any MACE system, together with ASE accelerated code to calculate the MoP, is provided as open source. This work helps build the foundation to extend the ML revolution in materials to NEMD and molecular fluid dynamics modeling.