Towards a Theory of Complexity Matching
Today there is still much debate regarding the definition of a complex system and complexity. We limit ourselves to two-state systems which jump randomly from one state to another and thus give rise to a dichotomous stochastic process. Our definition of a complex system is based on two properties: power-law statistics and renewal. The former implies that the waiting time distribution for both states is an inverse-power law with a finite exponent. The latter implies is a condition in which the time of permanence in one state is completely independent from the time of permanence in the other. If the system obeys power-law statistics with an exponent smaller than 2, then it is always complex. If, on the other hand, the exponent is greater than 2, then the system is complex only in the non-stationary stage. Recently, it has been shown that the linear response of a complex system to a coherent perturbation vanishes in the long-time limit. This result, together with the hypothesis that complex systems can be excited only by other complex systems, is the key motivation for the work presented in this thesis. It is part of an ongoing search for a theory of complexity matching, a theory showing that complex systems respond only if they are excited by other complex systems and that otherwise the response is attenuated.
In the first part of the thesis we explore the possibility of coupling two Poissonian processes. Our approach is based on the experience obtained in the field of stochastic resonance. We try to perturb a system that obeys ordinary Poisson statistics using Poissonian signals with different rates. To this end, we adopt a simple model that reproduces aperiodic stochastic resonance and we show that such a phenomenon is present only under certain conditions. Furthermore, we adopt the concept of events and use it to study the interaction of Poisson systems. We discover that, when the system produces events at a lower rate with respect to the perturbation, the events of the perturbation become attractors of the system events and vice versa. We use the term rate matching to identify the condition when the two rates of event production are the same.
The second part of the thesis deals with signals produced by complex systems. In order to present the full theory of complexity matching, a new fluctuation-dissipation theorem must be in-troduced, but this goes beyond the scope of the work presented here. However, the understanding of the non-ergodic nature of complex systems is fundamental to the application of the new fluctu-ation dissipfluctu-ation theorem. Therefore, here we study the power-spectrum of complex signals and show that 1/f -noise is produced by systems that lie on the border that separates ergodic systems from non-ergodic ones. We do this by generalising the Wiener-Khinchin theorem and extending
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it to non-stationary non-ergodic processes. We distinguish between two different types of trunca-tion effects: the physical truncatrunca-tion, where we use a truncated waiting time distributrunca-tion, and an observation-induced effect, which is a consequence of finite acquisition times. It is the finite obser-vation time that allows us to apply the generalised Wiener-Khinchin theorem in the non-ergodic case. Our final results show that the power-spectrum is inversly proportional to the frequency and that, in the non-ergodic condition, the power-spectrum also depends on the observation time.
