  Since the foundations of Population Genetics the notion of genetic
equilibrium (in close analogy to Classical Mechanics) has been associated to
the Hardy-Weinberg (HW) Principle and the identification of equilibrium is
currently assumed by stating that the HW axioms are valid if appropriate values
of Chi-Square (p<0.05) are observed in experiments. Here we show by numerical
experiments with the genetic system of one locus/two alleles that considering
large ensembles of populations the Chi-Square test is not decisive and may lead
to false negatives in random mating populations and false positives in
nonrandom mating populations. As a result we confirm the logical statement that
statistical tests can not be used to deduce if the genetic population is under
the HW conditions. Furthermore, we show that under the HW conditions
populations of any finite size evolve in time according to what can be
identified as neutral dynamics to which the very notion of equilibrium is
unattainable for any practical purpose. Therefore, under the HW conditions
equilibrium properties are not observable. We also show that by relaxing the
condition of random mating the dynamics acquires all the characteristics of
asymptotic stable equilibrium. As a consequence our results show that the
question of equilibrium in genetic systems should be approached in close
analogy to non-equilibrium statistical physics and its observability should be
focused on dynamical quantities like the typical decay properties of the
allelic auto correlation function in time. In this perspective one should
abandon the classical notion of genetic equilibrium and its relation to the HW
proportions and open investigations in the direction of searching for unifying
general principles of population genetic transformations capable to take in
consideration these systems in their full complexity.
