The Nobel Committee has decided to award this year's Nobel Prize for pioneering contributions to the general understanding of complex systems. Half of the prize went to prof. Giorgi Parisi of La Sapienza University in Rome for his discovery of the interaction of disorder and fluctuations in physical systems from atomic to planetary scales.

Disorder is ubiquitous in nature. Its influence on the resulting behaviour is all the greater the more diverse parts the system is composed of. Microscopic systems consist of almost identical atoms, which are usually in a well-defined state. In the microworld, disorder is expected to be minimal. However, Giorgio Parisi showed that the situation is significantly different when microscopic objects, atoms and molecules interact strongly with each other. The disorder of interacting microscopic objects leads to fluctuations in characteristic atomic quantities. These fluctuations are very fast and usually not directly observable in the global context. But what we can perceive is the influence of these microscopic fluctuations on the macroscopic properties of complex systems. Parisi found a way to demonstrate that the combined effects of microscopic disorder and interactions significantly influenced the resulting behaviour over large, macroscopically observable distances.

At low temperatures all large thermodynamic systems tend to order and create a homogeneous environment. A typical example is magnetism, where atomic magnetic moments are homogeneously ordered in one direction, which corresponds to the optimal state. More precisely, in the optimum of free energy. Microscopic disorder generally hinders the global order, which in turn leads to frustration, when the macroscopic system can no longer find one optimal equilibrium. Instead, a large number of different suboptimal, equally advantageous configurations are created. A typical representative of such a complex system is the so-called spin glass.

Spin glasses were experimentally discovered in the early 1970s. They are disordered magnetic materials that do not prefer either ferromagnetic or antiferromagnetic ordering. Nevertheless, at low temperatures, they change to a macroscopically ordered state. Understanding the origin and properties of the ordered state of spin glasses was at the birth of Parisi's fundamental contribution to the understanding of complex systems.

First, S. Edwards and P. W. Anderson proposed a model that was to simulate the behaviour of spin glasses. Shortly afterwards, D. Sherrington and S. Kirkpatrick found a solution to this model in approximating the statistical mean field. Although everything in this solution was formally flawless, the result was non-physical behaviour of the low-temperature phase. This paradox was removed by G. Parisi in his pioneering solution. He showed that the ordered phase of spin glass cannot be limited to the configurations of atomic magnetic moments that make up the system. The full description requires the introduction of replicas of the original system and new unmeasurable parameters that characterize the non-trivially formed hierarchical structure of the ordered state. Parisi's solution was so revolutionary that it took almost thirty years to be fully understood and generally accepted as physically correct. A mathematical proof of the exactness of Parisi's solution was also found, which dispelled the last doubts.

Parisi's construction is so universal that it finds use not only in physics, but in general in complex systems in which a large number of interacting elements are subject to a number of hidden bonds that need to be revealed. Such problems occur in mathematics (solvability of a large number of equations with many variables), computer science (machine learning, self-correcting codes), biophysics and artificial intelligence (neural networks) or in mathematical economics (theory of interacting economic agents). Last but not least, Parisi’s results can be applied to such a complex system as the ecosystem of our planet.

Complex systems have also been studied at the Institute of Physics of the Czech Academy of Sciences since the second half of the 1990s. These are mainly models of spin glasses and their variants in the approximation of the statistical mean field, which can be important for modelling real glasses and some dielectric materials [1]. We proposed, among other things, an original physical derivation and interpretation of the Parisi's hierarchical solution [2]. We personally collaborated with G. Parisi on some of the problems of complex systems [3].

Václav Janiš and František Slanina

[1] V. Janiš: Introduction to Mean-Field Theory of Spin Glass Models, Chap. 8 in Many-Body Physics: From Kondo to Hubbard Modeling and Simulation Vol. 5, E. Pavarini, E. Koch, and P. Coleman (eds.), Forschungszentrum Jülich, 2015.

[2] V. Janiš: Free-energy functional for the Sherrington-Kirkpatrick model: The Parisi formula completed, Phys. Rev. B B 77, 104417 (2008).

[3] G. Parisi and F. Slanina: Toy model for the mean-field theory of hard-sphere liquids, Phys. Rev. E 62, 6554 (2000).