Calculating the pressure in simulations using periodic boundary conditions (Q1906419)

Because it is not immediately clear how to write down a proper Hamiltonian for a system in periodic boundary conditions, particularly with Coulomb interactions, we consider a large, finite array of copies of a basic simulation cell containing \(N\) particles with some interaction between them. We also put \(N\) independent copy particles in each of the copy cells of the array and write down a constrained Lagrangian for the whole system. We derive a Hamiltonian for the whole system with constraints and then derive the equations of motion and a virial expression for the pressure tensor in terms of the forces on the system.