text: This condition in its original form was shown to be impossible for systems with dimension greater than two in papers by Plancherel and Rosenthal.
full_question: This condition in its original form was shown to be impossible for systems with dimension greater than two in papers by Plancherel and Rosenthal. Birkhoff showed that this property is equivalent to a system having no invariant proper subspaces, the condition of metric transitivity. This condition is the weakest in a chain of systems with the Bernoulli property, K property, and mixing, all of which imply this condition. A computer simulation showed that this condition was not present in a 1D lattice of anharmonic oscillators, a result which surprised Fermi, Pasta, and Ulam. This condition does not exist on the "invariant KAM tori," and it is a requirement for the equipartition theorem to hold. A statement of this condition is that the time average of an observable is equal to its ensemble average. For 10 points, name this property of some types of dynamical systems which means that every microstate is equally likely.
first_sentence: This condition in its original form was shown to be impossible for systems with dimension greater than two in papers by Plancherel and Rosenthal.
raw_answer: ergodicity [or word forms, such as the ergodic hypothesis]