AbstractThe thermodynamic irreversibility of molecular processes involving heat transfer is established by incorporating time‐reversible dynamical thermostats in the equations of motion. It is then possible to understand why nonequilibrium steady states must obey the Second Law of Thermodynamics, not just with high probability, but with probability one. This is because nonequilibrium states are so rare, relative to equilibrium ones, as to occupy a zero‐volume multifractal part of equilibrium phase space. These conclusions are generally valid. We illustrate them here by studying equilibrium manybody systems, both solid and fluid, as well as a simple one‐body chaotic nonequilibrium steady state. In all these cases we follow the motion of an orthonormal set of comoving vectors {δ}, determining the time‐averaged “Lyapunov Spectrum” of dynamic instability rates, as well as the vectors “Rotation Spectrum”, and relating these spectra to the irreversible nature of nonequilibrium flows described above.