Reproduction and ferromagnetism

I present a stochastic population model that combines cooperative interactions of the type often used in physics with the process of reproduction and death familiar to biology, and I refer to reasons why such interlocking may be of interest to both fields. Physics and biology have developed good frameworks to model the process of stochastic change in systems which experience the action of cooperation, and reproduction, respectively. These two frameworks overlap substantially, and it is surprising that neither the physical or the biological literature seem to consider models interlocking the two effects. Considering a system that exhibits both reproduction and cooperation is particularly significant in view of the fact that the conventional explanation of ferromagnetism as a cooperative effect, arising somehow out of quantum mechanical assumptions, has repeatedly been called into question [3, 8, 6]. Indeed, it has been closely argumented recently, on empirical grounds, that the ferromagnetic transition should be interpreted as one due to change through nucleation and growth in the material’s crystal structure, with cooperative magnetics effects playing no significant role [12, 13]. However, even though the assumptions of quantum mechanics don’t offer a good premise for the existence of molecular cooperation either from an empirical or a conceptual point of view [1], it is unsatisfactory to rule out the possibility of the cooperative effect altogether, especially in view of recent advances in biology. There are obvious analogies linking the growth processes of bacterial colonies and of crystals. On the other hand, bacterial species have been found to coordinate their activity through several types of cooperative interactions, that have been characterised in mechanistic detail [14]. Since cooperation is the traditional explanation for the ferromagnetic email: [email protected] 1 ar X iv :1 30 7. 29 11 v2 [ ph ys ic s. bi oph ] 2 0 Ju l 2 01 3 phenomenon, and since crystal nucleation and growth are currently being proposed as a radically alternative cause, it is sensible to consider that both effects may be involved in physical materials, in analogy to what is observed in biological systems. In other words, evidence suggests that both physical and biological processes may involve the intertwined action of reproduction and cooperation. In this paper I consider one of simplest ways to model this, which can be described in a nutshell as a superposition of Moran’s random-walk model of genetic drift through death and birth [11], and the Glauber dynamics of a mean field ferromagnet [10]. I will now introduce the process, and use the subsequent section to characterise the stochastic equilibrium that arises from it. The characterisation gives a full summary of the possible regimes, and allows some considerations regarding the significance of the model.