It is suggested that the notion of equation-of-state serves as appropriate common basis for studying the macroscopic behavior of both traditional physical systems and complex systems. The reason is that while the equilibrium systems are characterized both by their energy function and the corresponding equation-of-state, the steady states of out-of-equilibrium systems are defined only by their dynamics, i.e. by their equations-of-state. It is demonstrated that there exists a common measure which generalizes the notion of Gibbs measure so that it acquires two-fold meaning: it appears both as local thermodynamical potential and as probability for robustness to environmental fluctuations. It is proven that the obtained Gibbs measure has very different meaning and role than its traditional counterpart. The first one is that it is derived without prerequisite requirement for simultaneous achieving of any extreme property of the system such as maximization of the entropy.
TopIntroduction
So far we have studied the grounding principles that underlie the functioning and organization of complex systems so that their evolution to be self-sustained in a stable way in a non-predetermined environment subject to the constraint of boundedness alone. This setting is fundamentally different from the traditional one which always pre-determines environment thus aiming to provide the conditions for reoccurrence of the event. Thereby, the predetermination of the environment is viewed as necessary prerequisite of a law: a law will state that when an event reoccurs the environment reoccurs too. The present book aims at developing a systematic alternative to this view. That is why it does not involve any pre-determination of the environment as a condition under which the major properties of the complex systems have to appear. The question is how general our approach is – is it general enough to include traditional physical systems as part of a more general scheme and if so what is this scheme?
We have found out that the response of a complex system comprises two general components: a specific one, called homeostatic, which is robust to small environmental variations, and a universal one, whose properties are insensitive to the statistics of environmental fluctuations. It should be highlighted that both components commence from the collective behavior of a system: the homeostatic part of the response is to be associated with the corresponding spatio-temporal pattern whereas the universal part is substantiated by the basic inter-level feedback which spontaneously emerges at every higher hierarchical level as we have demonstrated in the previous Chapter. Alongside, the central for our approach proof that consists of additive superimposing in the power spectrum of the corresponding characteristics of the homeostatic and the universal part of the response renders the constant accuracy of reoccurrence of the homeostasis in an ever-changing environment. Further, by means of delineating the corresponding basin-of-attraction, we are able to figure out the domain of “robustness” of each spatio-temporal pattern which characterizes a given homeostatic pattern. Thus, we substantiate a fundamental generalization of the notion of a law: a pre-determined environment is not necessary for an event to be sustained and to reoccur!
The knowledge about the structure and functioning of the complex systems in a non-predetermined environment poses the question what kind of measure to apply to cases when a system approaches the thresholds of stability by means of exerting fluctuations. At first sight it seems that this task is similar to the traditional task of finding out the probability for a catastrophic event to happen. The traditional probabilistic approach substantiates this idea by means of global predetermination of the environmental impact imposed by assessing the latter through certain probabilistic distribution. Then, assuming that the response is local, i.e. after each session of environmental impact the system returns to the same equilibrium, the probability for happening of a catastrophic event is explicitly determined by the distribution of the environmental impact. Our approach to complex systems is completely different because: (i) the hypothesis of boundedness is incompatible with the idea of thermodynamic equilibrium viewed as single state where a system resides for arbitrary long time because the concept of boundedness requires boundedness of the distance between successive states. Therefore, obviously the system must reside in a stable way in any of the states through which it passes; (ii) both parts of a complex system response, the specific homeostatic and the universal noise ones, commence from the self-organized behavior of the system. Therefore, it is to be expected that the catastrophic events actually happen gradually and thus there must be early “warning” signs for their occurrence. The question is whether these early warning signs are specific or universal. The answer to be given is that there are both specific early warning signs, considered in Chapter 2, but also there exists a universal measure for reaching certain point in the state space. Next in this Chapter we will demonstrate that this measure appears as Boltzmann-Gibbs distribution.