The stress tensor of a molecular system: an exercise in statistical mechanics

We prove that conservation of the stress tensor is a consequence of the invariance of the partition function under canonical diffeomorphisms. From this observation a simple and general derivation of the formula which gives the local expression of the stress tensor of a molecular system in terms of its microscopic degrees of freedom readily follows. The derivation is valid in the canonical as well as the microcanonical ensemble. It works both in the classical and in the quantum mechanical settings and for arbitrary boundary conditions. In particular, if periodic boundary conditions are assigned to the system, the usual minimal-image prescription is naturally born out for mathematical consistency. An interesting outcome of our general analysis is that only in the case of a short-range interaction potential a truly local formula for the stress tensor can exist.