Boltzmann's ergodic hypothesis

Summary Boltzmann's ergodic hypothesis is usually understood as the assumption that the trajectory of an isolated mechanical system runs through all states compatible with the total energy of the system. This understanding of Boltzmann stems from the Ehrenfests' review of the foundations of statistical mechanics in 1911. If Boltzmann's work is read with any attention, it becomes impossible to ascribe to him the claim that one single trajectory would fill the whole of state space. He admitted a continuous number of different possible mechanical trajectories. Ergodicity was formulated as the condition that only one integral of motion, the total energy, is preserved in time. The two reasons for this are external disturbing forces and collisions within the system. Boltzmann found it difficult to ascribe ergodic behavior to a single system where the theoretical dependence on initial conditions, though never observed, has to be admitted as possible. To circumvent the dependence, he invented the concept of a microcanonical ensemble.