The definition of a mixing system

While Boltzmann fixed his attention on the motion of the phase point for a single system and was led to the concept of ergodicity, Gibbs took another approach to the same problem. Since one never knows precisely what the initial phase point of a system is, Gibbs decided to consider the average behavior of a set of points on the constant-energy surface with more or less the same macroscopic initial state. Without worrying too much about how such a set might be defined precisely, let's consider an initial set of points A. As the set travels through Γ-space, it changes shape but its measure stays the same, μ{A) = μ(At). The set gets stretched and folded and may eventually appear on a coarse enough scale to fill the energy surface uniformly. However, the set At has the same topological structure as the set A and the initial set is not ‘forgotten’, in the sense that a time-reversal operation on the set At will produce the set A. There is a nice lecture-demonstration apparatus that illustrates this time-reversal operation: A drop of immiscible ink is added to a container of glycerine. If you stir the glycerine slowly, the drop will stretch and form a thin line. Eventually it seems to fill the whole space, but if the stirring is reversed, the initial configuration of the drop of ink surprisingly reappears.

Gibbs thought that the apparently uniform distribution of the set At on the energy surface was the key to understanding how mechanically reversible systems could approach an equilibrium state.