-  Roberto Benzi. La Teoria del Caos (Lezioni di Fisica #10). Le iniziative del Corriere Della Sera. RCS MediaGroup SpA 2018.
-  Angelo Vulpiani. Determinismo e Caos. Ed. Carocci. 2004
-  R Benzi, M Sbragaglia, P Perlekar, M Bernaschi, S Succi, F Toschi. Direct evidence of plastic events and dynamic heterogeneities in soft-glasses. Soft Matter. 2014
-  Pinaki Kumar, Evangelos Korkolis, Roberto Benzi, Dmitry Denisov, André Niemeijer, Peter Schall, Federico Toschi, Jeannot Trampert. On interevent time distributions of avalanche dynamics. Scientific Reports. Nature Pubb. 2020
-  Federico Toschi, Eberhard Bodenschatz. Lagrangian properties of particles in turbulence. Annual review of fluid mechanics. 2009
-  Abigail Plummer, Roberto Benzi, David R Nelson, Federico Toschi. Fixation probabilities in weakly compressible fluid flows. Proceedings of the National Academy of Sciences. 2019
-  M Buzzicotti, L Biferale, F Toschi. Statistical properties of turbulence in the presence of a smart small-scale control. Phys. Rev. Lett. 2020
02. COMPLESSITÀ NELLA “SOFT MATTER”
One of the most deeply rooted ideas underlying the scientific revolution of the last few centuries is that natural phenomena, as they appear to us, are understandable on the basis of well-defined rules. Natural rules or laws are therefore seen in antithesis to chaos or disorder. Discovering and validating natural laws is the aim of science as we understand it today. Yet the complexity of the world we observe is clearly at odds with the simplicity of natural laws that scientists were first able to understand.
In the second half of the last century, these two aspects, the complexity of natural phenomena and the simplicity of the underlying laws, were somehow reconciled thanks to the discovery that chaos is inherent in the laws themselves. Chaos theory explains why chaos is an integral part of natural law, and understanding it has produced a profound conceptual revolution in science [1,2]. Along with quantum mechanics and relativity theory, chaos theory is one of the three great scientific paradigms that have changed the way we see and understand our surroundings. Beyond physics, chaos theory finds application in almost all scientific disciplines, including economics and the social sciences. It is almost impossible to make a somewhat exhaustive summary of the problems that are being faced and partly solved in the various disciplines. However, it is possible to understand how, also thanks to chaos theory, the way of doing research has partly changed and what are, potentially, the routes and directions that today we think can yield significant results.
Chaos theory has made a fundamental contribution to the development of the concept of a complex system. In reality, there is no shared definition of what is meant by a complex system. There are various definitions, each depending on the point of view. For some, a complex system is a system whose behavior depends crucially on the details of the system itself. On the other hand, a computer simulation of a chaotic system is not exact but represents an approximation of the exact solution and of the rules that govern it. Under these conditions, we can understand what are the general characteristics of the system, i.e., the probability distribution that characterizes it, and which, to some extent, are independent of the details themselves.
In this context, also thanks to technological developments (e.g., supercomputers) and new experimental investigation techniques, scientific interest in the study of the physical characteristics of “soft” matter has developed and it is rapidly growing. The research activity in this area takes the name, perhaps an understatement, of “Soft Matter”, and it does not only concern physics, but it is highly interdisciplinary involving biology, chemistry, ecology, geophysics, medicine and the social sciences.
Within physics, the term “Soft Matter” applies to systems such as the behavior of complex fluids, amorphous materials (e.g., emulsions, glasses, granular systems, etc.)
The figure shows the density field of an emulsion, where the dark part is slightly denser than the light one, obtained using a numerical simulation based on Boltzmann’s equations on the lattice and implementing the molecular interaction forces at the mesoscopic level. This type of simulation enables study of the dynamic behavior of emulsions when external forces act .
Probability distribution of the time elapsed between one avalanche and the next as observed experimentally in granular systems (symbols with circles and diamonds) in numerical simulations of emulsions (symbols with triangles) similar to those reported in FIGURE 1 and as observed in the statistics of seismic events (Corral distribution solid black line). This figure is a typical example of an interdisciplinary study in the Soft Matter sector where numerical simulations and / or laboratory experiments can be used to understand some geophysical phenomena .
Trajectory of a particle transported by a turbulent velocity field obtained from the numerical simulation of the velocity field. The colors refer to the acceleration undergone by the particle (in the simulation the maximum observed is about three orders of magnitude more intense than the acceleration of gravity). This type of simulations is the basis of the study of the dynamic behavior of macromolecules within a moving fluid and has applications in various research fields such as, for example, atmospheric physics, industrial engineering, medicine and biology 
The figure shows the evolution of two populations of bacteria (green and purple colors). The horizontal axis represents the spatial position of the population and the vertical axis represents time. This figure was obtained from a numerical simulation for the behavior of individual bacteria (agent simulation) with characteristics similar to those that make up marine phytoplankton. In this case the population of bacteria is subjected to phenomena of upwelling (dashed line) and downwelling (solid line) which are very frequent along the sea coasts. The population represented in green is initially close to the upwelling zone and receives a selective advantage over the one represented in purple. This result explains, in part, the effect of sea currents on the dynamics of phytoplankton. It should be remembered that phytoplankton is at the base of the food chain at sea and is an important element of balance for the climate in the cycle of formation and absorption of greenhouse gases .
The studies related to the chemical-physical behavior of macromolecules, to the study of cell membranes, to the dynamics of populations and related genetic evolution, to the development of models of neural networks are inserted between biology and physics.
Concerning Soft Matter, CREF intends to pursue the development and application of numerical simulations suitable for reproducing some of the most relevant phenomena and possibly predicting new ones. It should be emphasized that some of the simulation techniques developed in the last twenty years have been conceived and developed by groups of excellence in scientific research operating in Italian universities and/or institutes and in particular in the Roman area. Collaborations with these research groups are therefore the first step to be implemented to give substance to this activity.
In addition to the computational component, another sector that is emerging significantly at an international level concerns the application, in the context of soft matter, of new artificial intelligence technologies and in particular of the vast class of techniques contained under the name of Machine Learning . The need for the development and application of these techniques is motivated by the fact that the traditional tools of data analysis (even enriched with new concepts developed by scientific research) are not always able to quantitatively identify some phenomena (for example the recognition of the spatial configurations of an amorphous system close to a crisis event). In other cases, such as the behavior of “active” particles within a fluid, Machine Learning techniques allow the development of learning protocols suitable for promoting particular functions and of fundamental importance in many applications (for example drug delivery). Finally, these techniques can assist the possibility of predicting “critical” phenomena within the system in question. The development and application of Machine Learning techniques is the second research activity of the Fermi Center in the context of Soft Matter.
Given the number and complexity of possible phenomena on which to focus research in the coming years, it is necessary, also through collaborations with other universities and research institutes, to circumscribe a more specific field on which the CREF can produce significant results in realistic times. However, it is important to note that the research activities to be carried out are a natural terrain of trade-union within the three-year plan with the other projects of the Center (for example the study of the brain and Economic Complexity) and that therefore they can be developed in a cooperative with the other research lines identified.