 Malament and Zabell (1980) respond to this challenge by suggesting a way of explaining the success of equilibrium theory that still invokes ergodicity, but avoids appeal to time averages. This solves the above mentioned problems, but suffers from the difficulty that many systems that are successfully dealt with by the formalism of SM are not ergodic. To circumvent this difficulty Vranas (1998) suggested replacing ergodicity with what he calls \(\varepsilon\)-ergodicity. Intuitively a system is \(\varepsilon\)-ergodic if it is ergodic not on the entire phase space, but on a very large part of it (those parts on which it is not ergodic having measure \(\varepsilon\), where \(\varepsilon\) is very small). The leading idea behind his approach is to challenge the commonly held belief that even if a system is just a ‘little bit’ non-ergodic, then it behaves in a completely ‘un-ergodic’ way. Vranas points out that there is a middle ground and then argues that this middle ground actually provides us with everything we need. This is a promising proposal, but it faces three challenges. First, it needs to be shown that all relevant systems really are \(\varepsilon\)-ergodicity. Second, the argument so far has only been developed for the microcanonical ensemble, but one would like to know whether, and if so how, it works for the canonical and the grandcanonical ensembles. Third, it is still based on the assumption that equilibrium is characterised by a stationary distribution, which, as we will see below, is an obstacle when it comes to formulating a workable Gibbsian non-equilibrium theory.