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We have seen that one often encounters in may disciplines stochastic systems consisting of many autocatalytic elements (i.e described by autocatalysis equations 1-3). Some of the most striking examples affecting our daily life are in the financial field. For instance, the wealth of the individual traders [3], the market capitalization of each firm in the market [4] and the number of investors adopting a common investment strategy [5] are all stochastically autocatalytic in the sense that their jumps (elementary quanta of change) from one time instant to the next are typically proportional (via stochastic factors) to their value.

Even though such systems are rare in physics, advanced statistical mechanics techniques do apply to them and have turned out to have crucial relevance: while the usual partial differential equation treatment of these systems often predicts a 'dead', tradeless market, the renormalization group 'corrections' ensure the emergence of a macroscopic adaptive collective dynamics which allows the survival and development of a lively robust market [6].

Stochastic autocatalytic growth [7] generates, even in very non-stationary logistic systems, robust Pareto–Zipf power law wealth distributions [8]. It was shown [9] that it if the market is efficient, one can map it onto a statistical mechanics system in thermal equilibrium. Consequently, the Pareto law emerges in efficient markets as universally as the Boltzmann law holds universally for all systems in thermal equilibrium.

The study of markets as composed of interacting stochastic autocatalytic elements has led to many theoretical quantitative predictions, some of which have been brilliantly confirmed by financial measurements. Among the most surprising ones is the theoretically predicted equality between the exponent of the Pareto wealth distribution and the exponent governing the time interval dependence of the market returns distribution.

In fact in all the fields described in the preceding sections, one finds the recurring connection between the autocatalytic dynamics and the emergence of the power laws and fractal fluctuations.

In the presence of competition, the autocatalytic systems reduce to logistic systems which were long recognized as ubiquitous.

Thus the observation by Montroll:

all the social phenomena obey logistic growth

becomes the direct explanation of the older observation by Davis:

'No one, however, has yet exhibited a stable social order, ancient or modern, which has not followed the Pareto pattern'.

In fact, beyond Davis's statement, the Pareto law stability holds even for non-stable social order (booms, crashes, wars, revolutions, etc), provided the markets are efficient.

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