Phase Transitions in Social Sciences: Two-Population Mean Field Theory

Abstract

A new mean field statistical mechanics model of two interacting groups of spins is introduced and the phase transition studied in terms of their relative size. A jump of the average magnetization is found for large values of the mutual interaction when the relative percentage of the two populations crosses a critical threshold.It is shown how the critical percentage depends on internal interactions and on the initial magnetizations. The model is interpreted as a prototype of residentimmigrant cultural interaction and conclusions from the social sciences perspectives are drawn. IntroductionIn the last few decades the statistical mechanics approach has seen an impressive expansion in fields as diverse as combinatorial optimization, finance, biology and others. Its success relies upon the fact that every problem of many interacting parts may be studied by its methods and techniques. Our plan in this work is to introduce a statistical mechanics model with the purpose of describing two homogeneous groups of individuals whose interaction is imitative within the same group but may be imitative or counter-imitative between different groups. Such a model may represent a first attempt toward the construction of a statistical mechanics theory of the phenomenon of cultural interchange. When 1 two populations come in contact, like in the case of immigration but also in a more general context through the media, it is often seen how cultural traits are sometimes mixed, while some other times one of the two dominates and prevails on the other. Examples are found in linguistics, as well as in opinion forming dynamics ([1], [2], [3]). One interesting feature of those changes is that sometimes they appear to be smoothly varying with the relative proportions of the two groups, some other times the crossing of a critical value triggers a jump of the observed quantity [4]. What we are building here is the mean field theory of the two population problem i.e. we consider every individual interacting with every other with the same strength. In future works we plan to introduce a more realistic model by allowing randomness of the interaction strength like in the Sherrington-Kirkpatrick spin glass model ([5] [6]) and also a more structured network connection like, for instance, the one predicted by the small-world theory [7] [8]. We want to stress that, although our model is inspired by the Curie-Weiss theory of ferromagnetism, the problem we deal with here is quite different because we do not study the phase transition in terms of the temperature but in terms of the relative size of the two populations.The dictionary we plan to follow is easily explained by saying that a cultural trait is considered for simplicity as a dichotomic variable σ i = ±1. The interaction between particles is built up as a sum of pairwise interactions and plays the role of cultural interaction between two individuals i and j as described by a potential, or a cost function, which simply reflects the will to "agree" or "disagree" among the two. The two at...

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