On the Growth of Chemical Diversity

In complex systems that host evolutionary processes, in which entirely new entities may enter the scene, some variables can sometimes show a “hockey-stick” behavior, that is a long period of slow growth followed by an “explosive” increase. The TAP equation was proposed with the aim of describing the growth of the number of different types of entities in systems where new entities (e.g., artifacts) can be created, supposing that they derive from transformations of pre-existing ones. It shows a very interesting divergence in finite times, different from the usual exponential growth where divergence takes place in the infinite time limit. The TAP equation does not deal with the growth of the number of actual types, but rather with the number of the possible ones (the members of the so-called set of Adjacent Possible), and it can therefore overestimate the actual rate of growth. In this paper, we introduce a model (called BPSM, focused on systems that may be relevant for the origin of life) that takes into account the difference between the Adjacent Possible and the set of types that are actually created. Using simulations, it has been observed that the growth of the number of chemical species in the system resembles that of the corresponding TAP equation. Since in this case only combinations of at most two entities can be considered at each time, the TAP equation can be analytically integrated. Its behavior can be then compared to the (necessarily finite) behavior of model simulations; their behaviors turn out to be quite similar, and proper tests are introduced, which show that they differ from the familiar exponential growth. Therefore, the BPSM model provides a description of the rapid increase of diversity which resembles TAP, while based upon the growth of the actual entities rather than on the Adjacent Possible.