Phase Transitions and Complex Systems

PHASE TRANSmONS ANO ORDER PARAMETERS ministic or stochastic. {c) is a set of parameters and r' are a H ow to define complexity is not a trivial question. Several given set of nearest neighboring positions of r. 1'ypically, fue definitions have been proposed, and all of them share set of neighboring points, ?([r), is symmetrically distributed fue intuitive idea that complexity is neither complete around fue site r, and those CA able to simulate real systems arder nor complete disorder [1-5]. But this statement, though often verify some particular constraints. fairly intuitive, is far from satisfactory. A quantitative characIn this section, we summarize fue most relevant phase tranterization of complexity is necessary. Which "universal" feasitions (PHTs). To beginwith, weusetheapproachof(l), where tures share apparently different complex systems? The bespatial degrees of freedom and explicit local properties are havior ofphysical systems clase to criticalpoints mar answer neglected} We will assume that fuese are nonequilibrium this question. It is welI known, from fue theory of phase tranphase transitions, which are achieved by means of energy or sitions, that a given system (possibly made of manY submatter inputs into fue system. systems) can undergo strong qualitative changes in its macroscopic properties if a suitable control parameter is Flrs.t-Order Transitions adequately tuned and that clase to fuese critical points some Sudden changes in the behavior of nonlinear systems as a key characteristic constants (the so-called critical exponents) consequence of a continuous change in a given parameter are are fue salDe for very different systems [6]. At critical points, welI known [1,8]. An example is given by fue one-dimensional fractal structures, complex dynamical pattems and optimal nonlinear system information transfer appear in a spontaneous way. Observing such properties in those systems which we call "complex," dxl dt = fix) = ;x3+ .ux + p. (3) we can conjecture that complexity tends to appear clase to in-