Ergodic theory has been approached from the side of the time averages with correlation functions from many-body models. The condition for ergodic behavior is formulated in terms of infinite products of certain numbers associated with time evolution. Physical properties that make a model ergodic are identified.

Khinchin's theorem of ergodicity is examined by means of linear response theory. The resulting ergodic condition shows that, contrary to the theorem, irreversibility is not a sufficient condition for ergodicity. By the recurrence relations method, we prove that irreversibility is broader in scope than ergodicity, showing why it can only be a necessary condition for ergodicity.

Some years ago the Heisenberg equation of motion was formally solved by the recurrence relations approach. It is shown here that the Langevin equation represents a structural property of the recurrence relations. The Langevin equation is useful for studying the time evolution of the current. The resulting current-current correlation function is compared with Luttinger's phenomenological theory. Geometric interpretations are made for the conductivity and the dielectric function.