Direct dynamical calculation of entropy and free energy by adiabatic switching

If an ergodic Hamiltonian is changing slowly in time, its energy shell is an adiabatic invariant: Even though the energy changes, an initial surface of constant energy is mapped into a continuous family of surfaces, each of which is also of constant energy. This observation allows efficient and direct dynamical calculation of the entropy of classical fluids, exemplified here by a simple model of liquid water. The results depend substantially on the form of the ``switching'' function, suggesting possible improvements in traditional thermodynamic switching processes. Application of a similar adiabaticity to Nos\'e dynamics allows dynamical computation of differences in the Helmholtz free energy.