These are related to the physical conditions for the possible occurrence of the so-called property of macroscopic irreversibility (PMI) and the consequent one represented by the decay to kinetic equilibrium (DKE) which characterize the 1-body (kinetic) PDF in these N-body systems, i.e., when 1-body-factorized initial conditions are considered for the N-body Liouville equation .
In fact for the same equation it is not known in satisfactory generality whether smooth enough solutions of the same equation exist which satisfy the H-theorem inequality and decay asymptotically to kinetic equilibrium [23, 30].
(iv) MKI Prescription No.#4: fourth, in order to warrant the existence of DKE we shall require the functional [I.sub.M]([[rho].sup.(N).sub.1](t)) to be prescribed in such a way that at an arbitrary time t [member of] I, with t [??] [t.sub.o], the vanishing of both [I.sub.M]([[rho].sup.(N).sub.1](t)) and its time derivative ([partial derivative]/[partial derivative]t) [I.sub.M]([[rho].sup.(N).sub.1](t)) should occur if and only if the 1-body PDF solution of the Master kinetic equation coincides with kinetic equilibrium. As a consequence, for the functional [I.sub.M]([[rho].sup.(N).sub.1](t)) the following propositions should be equivalent:
with [[rho].sup.(N).sub.1M]([v.sub.1]) being a kinetic equilibrium PDF of the form (6).
(i) the time derivative inequality (18) and the conditions of existence of kinetic equilibrium (21);
Caption: Figure 2: Conventional kinetic equilibrium extraction of Clinacanthus nutans fresh and dried leaves (n = 3).
Caption: Figure 4: Peleg model kinetic equilibrium extraction of Clinacanthus nutans fresh and dried leaves (n = 3).
Caption: Figure 7: Peleg model kinetic equilibrium extraction of orientin and vitexin dried leaves extract (n = 3).
As it can be seen from Figure 1, the cassava husks powder presents a very rapid adsorption process, reaching kinetic equilibrium in less than 1 min.
The material reached kinetic equilibrium in less than 1 minute, and its optimum pH to perform Cd(II) uptake was found to be 5, what is very approximate to the material's point of zero charge (5.33).