Information-theoretic perspective on complex systems

Up to now, complex systems are far from being scientifically understood, and hence are basically without reach of any realistic control. This is a highly unsatisfactory situation, since many complex systems are of biological and societal relevance, such as the economy, financial systems, ecosystems, the brain, or the immune system. Many of these systems are evolutionary systems in the sense that they change the environment along with their own progression over time. It would be highly desirable to have more control over these systems, especially under what conditions they might collapse. In this project we try to develop a set of new mathematical tools that allow us to take a perspective on complex systems that has so-far been neglected. We will build these tools from basic building blocks that have been used in physics and computer science (information theory) to describe mostly non- complex systems. These tools can in principle not be used naively for complex systems but, must be tailored with extreme caution toward the particular complex system of interest. In this project we have selected several examples where we systematically want to apply and test our new methodology: Systems that are evolutionary and create and destroy diversity in a system, financial markets, and models of sandpiles, which are toy systems to understand earth quakes better. Finally, we might be also able to address a fundamental question in physics, which arises as a natural consequence of these novel techniques, when they are applied to classical mechanics: the origin of special relativity from a path-integral based statistics. The project will be carried out by two groups, one in Prague and one in Vienna, because their joint knowledge on the mathematical and statistical tools makes it possible to tackle the difficulty of the envisioned tasks. It would be very difficult for the individual groups to carry out their parts of the project alone.

Complex systems are systems consisting of many subsystems interacting with each other. They are ubiquitous in nature, ranging from physical and chemical systems over biological systems to socio-economic systems. This project aimed to develop, analyze, and implement robust and efficient techniques for the treatment of complex dynamical systems based on methods combining information theory and (non-equilibrium) statistical physics. So far, this topic has not been carried out systematically on a theoretical basis. The main results of the project can be divided into several groups. First, we have shown that statistical and thermodynamic properties of a broad class of complex systems can be described within a framework of the so-called generalized entropies, which can grasp the complex structure of many systems exhibiting long-range interactions. Second, with the help of information geometry, we were able to distinguish three classes of complex systems according to their scaling properties: strongly-correlated systems (as, e.g., gravitational systems or entangled particles), weakly-correlated systems (e.g., ordinary gases), and systems with emergent structures (as, e.g., molecules formed of atoms, self-assembly of colloid particles). The aforementioned results motivated us to study the thermodynamic properties of structure- forming systems and derived the corresponding thermodynamic quantities, including entropy. Finally, we focused on the properties of complex systems out of equilibrium, i.e., in the presence of external driving or the presence of more than one heat reservoir (as in the case of thermal engines). We showed a natural connection between non-linear systems and generalized entropies and were able to derive their equilibrium and non-equilibrium properties.