Non-equilibrium statistical mechanics of complex systems

Many complex systems are stochastic systems that can be described with methods from statistical physics once the methods take into account the complex structure of these systems. One difficulty with complex systems is that with increasing size (number of degrees of freedom), they often show a non-exponential growth of phase space volume. It is caused either by the presence of inherent correlations in the system caused, e.g., by the long-range interactions (as for quantum entanglement or gravitating objects), or by the emergence of higher-order structures that appear with increasing size of the system. It leads to a number of sometimes unexpected thermodynamical properties of complex systems. While sub- exponential, i.e., correlated, systems have been studied intensively in the past decades, so far, super-exponential ones have not received attention. These include important stochastic systems that can form structures. One can find examples of these systems in chemistry (atoms forming molecules), biology and biochemistry (micelles and polymers), soft matter (self- assembly), or even in social systems (emergence of social groups). To date, there is limited knowledge on how to understand their statistical and thermodynamical properties, particularly how to adequately describe their non-equilibrium states that play a role in small systems. The main aim of this project is to establish a systematic classification scheme of complex systems of this kind with a focus on their statistical properties out-of-equilibrium. We further want to contribute to a consistent thermodynamic framework for such systems. Finally, we aim to compare the theoretical results with real-world examples, such as small chemical reaction networks. To this end, we will complement our analytical approaches with computer simulations.