Unbiased virial-based pressure estimator

Pressure is a mechanical but not an atomistic concept. It follows either from continuum mechanics or thermodynamics. Statistical mechanics allows us to introduce an atomistic definition of pressure. In statistical mechanics, two equivalent definitions of pressure are possible. The first follows from Green's approach, which relates pressure to the thermodynamic derivative of the Helmholtz free energy (the logarithm of the canonical partition function) with respect to volume, and the second from a formulation of the mechanical Clausius virial theorem for a closed-or open [J. A. Schweitz, J. Phys. A 10, 507 (1977)]-system. However, as we demonstrate in this paper, both approaches are incomplete, as they result in atomistic definitions of the pressure that are only asymptotically unbiased. Here, exploiting an idea advanced by D. H. Tsai [J. Phys. A 70, 1375 (1979)], we find a simple justification for an ingenious observation made by T. W. Lion and R. J. Allen [J. Phys. Condens. Matter 24, 284133 (2012)] based on the study of J. F. Lutsko [J. Appl. Phys. 64, 1152 (1988)] and J. Cormier et al. [J. Appl. Phys. 89, 99 (2001)]. We find that the virial by itself is not enough to properly define (if not asymptotically, in the thermodynamic limit) the atomistic estimator of the pressure. Rather, one needs a proper definition of what contributes to the pressure (the force exerted per unit area on a system in equilibrium) acting on a surface dividing the space into two parts. With that, and a properly generalized concept of additive cell virial, we provide a simple justification of an already known unbiased estimator for the pressure of an open atomistic system in a local cell in equilibrium.