A complex system is typically characterised by the difficulty of describing it if one tries to understand it from its single components. In other words, a complex system is usually made up of many micro-systems, but its behaviour cannot be explained by studying these micro-systems individually. A simple but universally understandable example is given by groups of animals, such as a school of fish: it is composed of a multitude of fish, but if you study each individual fish, you completely lose sight of the behaviour of the school and you will never be able to predict its overall movement. In complex systems, therefore, there are what are called emergent properties, i.e. global properties of a system, which emerge from the interactions between the parts of the system but which are not specific to the individual parts. Another interesting example is given directly by Parisi, in a documentary entitled Giorgio Parisi e la fisica della complessità (Giorgio Parisi and the physics of complexity)[3]: inside the underground stations, when they are not particularly crowded, people move in disjointed trajectories, sometimes colliding; as the number of people increases, rows form and a more orderly behaviour emerges, in which people moving in a common direction tend to group together and move in unison, leaving space for those coming in the opposite direction. There is no dictated order from above, there is not even an explicit consciousness on behalf of the people, yet they find themselves behaving collectively. Something similar happens in economics, to name but one field around which the world is now oriented, where predicting the behaviour of individual investors is dictated by well-known rules of microeconomics. However, the transition from microeconomics to macroeconomics is far from trivial, and predictions can have disastrous results.
A second fundamental characteristic of complex systems is that their dynamics, i.e. their time evolution, is extremely responsive and easily modified. This is due to the fact that, for these systems, many different evolutions are possible and a small variation in the initial conditions can lead to enormous differences in the final state of the system. In addition to this we must also consider the so-called non-linearity, which means that it is impossible for a complex system to predict the response to a combined stimulus from the response to individual stimuli. In other words, if a complex system responds in manner A to one stimulus and in manner B to another, the system’s response to both stimuli will not be A+B. These two factors – sensitivity to the initial state of the system and non-linearity – make complex systems extremely difficult to handle within a scientific theory. Examples include atmospheric climate (of which we all recognise the difficulty of forecasting in the mid- and long-term), or the movements of the Earth’s crust (considering the unforeseeable nature of seismic events), not to mention all the phenomena involving mankind: from economics to politics and even social relationships.
Parisi has played the role of ‘pathfinder’ in several studies of complex systems, especially of disordered systems at an atomic level such as glass, which contains many diverse atoms mixed together in a complex and disordered way. One of his missions, therefore, was a theoretical/mathematical study on the properties of certain types of materials based on such disorder. However, as mentioned above, complex systems belong to a wide range of phenomena, so Parisi’s mathematical studies led him to take an interest in phenomena even extremely different from atoms in microscopic systems, such as the effect of fluctuations on our planet’s ice ages. At the time, Parisi was working on quantum dynamics, though one day he happened to attend a lecture on ice ages. The model proposed in the lecture was purely deterministic and dictated by the periodic movements of the planet; whereas Parisi realised that the effect of fluctuations were to be added to these movements due to random climatic events in the Earth’s atmosphere. These fluctuations, associated with what Parisi and his colleagues later called stochastic resonances, have also found applications in electronics, medicine and neuroscience.

Historically, physics has always sought to break down systems that are difficult to understand into their individual parts, in order to approach nature through an understanding of its fundamental units. The approach of Parisi and other physicists to complex systems is in a certain way a new approach to physics, dictated by the need to explain complex systems through a holistic study, which includes the phenomenon by embracing it in its entirety. This approach, so different from what scientists of the past have always imagined doing, could be the key to mathematical modelling and thus interpretation of currently seemingly indecipherable phenomena.