The Isotropy of the Pressure in Anisotropic Systems

1. The Problem The virial equation gives the pressure tensor, p αβ , in terms of the equilibrium ensemble average of the microscopic stress tensor, σ αβ. Thus we have p αβ V = σ αβ = mv iα v iβ i ∑ + 1 2 r ij ,α i ≠ j ∑ f ij ,β (1) where V is the volume of the system, m is the particle mass, v i is the velocity of particle i, r ij is the centre of mass separation between particles i and j and f ij is the force exerted on particle i by particle j. The Greek subscripts indicate Cartesian tensor components. Eq. (1) presupposes that the system considered is classical and that the forces are pairwise additive. The pressure tensor is thus an average of a second rank tensor. Let us consider an arbitrary second rank tensor, A αβ , dependent on the molecular configuration. In the isotropic phase, it must be true that A αβ = aδ αβ (2) where a is a scalar constant. Thus it follows that the pressure tensor must also be