Microscopic pressure tensor for hard-sphere fluids

The microscopic pressure tensor, which is not uniquely defined, is analyzed for a uniform hard-sphere fluid in Cartesian and spherical coordinates. Two popular definitions, one due to Irving and Kirkwood (IK) [J. H. Irving and J. G. Kirkwood, J. Chem. Phys. 18, 817 (1950)] and the other due to Harasima (H) [A. Harasima, Adv. Chem. Phys. 1, 203 (1958)], were used in this work. The IK definition is found to give the same ensemble average of the local pressure in Cartesian and spherical coordinates for a homogeneous hard-sphere system. The pressure obtained from the H definition gives, on the other hand, different results in the two coordinate systems. In Cartesian coordinates, the H pressure is identical to the IK pressure, but in spherical coordinates, the pressure depends on R (the distance from the origin). Therefore the H definition does not give a proper pressure tensor.