Space plasmas from the solar wind to planetary magnetospheres and the outer heliosphere are systems in stationary states out of equilibrium. Empirical kappa distributions successfully describe these space plasmas. Non-extensive Statistical Mechanics offers a solid statistical foundation and provides a set of proven tools for understanding these distributions. Here, by expressing the entropy in terms of the κ-index, we show the detailed paths by which the transition of stationary states evolves toward equilibrium. This procedure naturally provides four indices that are frequently observed in space plasmas: The extreme kappa values of infinity and ∼1.5 correspond to the stationary states at equilibrium and the furthest state from equilibrium, respectively; the value of κ∼1.63 is the fundamental state exhibiting the minimum entropy; and the value of κ∼2.45 is the escape state which separates the near-equilibrium states included in the monotonic region κ>2.45, from the far-equilibrium states included in the non-monotonic cavity of 2.45>κ>1.5. The fundamental state separates the entropy in two monotonic branches, κ>1.63 (A) and 1.63>κ (D). As the entropy increases, the value of κ-index increases along the A-branch, while it decreases along the D-branch, exhibiting a phenomenological Acceleration or Deceleration of particles, respectively. Starting from stationary states near the fundamental state, spontaneous procedures that increase entropy, move the system toward equilibrium either directly, along the A-branch, or indirectly, along the D-branch first, and then, along the A-branch, after isentropic switching between the branches. Finally, external factors that can decrease the entropy of the system, move it back into stationary states closer to the fundamental state. In the case of solar wind, newly formed pick-up ions play just such a role because their motion is highly ordered.