Analytical time-dependent functions describing the change of the concentration of the solvent S(t) and the homeopathic active substance A(t) during decimal and centesimal dilution are derived. The function S(t) is a special case of the West–Brown–Enquist curve describing ontogenic growth, the increase in concentration of the solvent during potentization resembles the growth of biological systems. It is demonstrated that the macroscopic S(t) function is the ground state solution of the microscopic non-local Horodecki–Feinberg equation for the time-dependent Hulthèn potential at the critical screening. In consequence potentization belongs to the class of quasi-quantum phenomena playing an important role both in biological systems and homeopathy. A comparison of the results predicted by the model proposed with the results of experiments on delayed luminescence of a homeopathic medicine is made.