Complex systems rarely yield to classical statistical mechanics. From the atoms in a spin glass, which were the subject of this year’s Nobel prize in physics, to the stars in a galaxy, complex systems are typically characterized by long-range interactions between their component parts — interactions that are not satisfactorily accounted for in the simplifying assumptions of Boltzmann-Gibbs statistical mechanics.

Physicist Constantino Tsallis* thought of a possible solution to the problem in 1988 when he proposed what came to be known as generalized or nonextensive statistical mechanics, where global interactions between components change how the information in a system scales with size. Tsallis’s statistics did for complex systems what the celebrated Boltzmann-Gibbs theory did for simpler systems, like air in a container, which Tsallis views as a special case in the wider world of complex systems. By generalizing the logarithm in Boltzmann-Gibbs statistical mechanics, Tsallis was able to describe entropy and the distributions of velocities and energies for complex systems — three major signatures of the classical, and powerful, Boltzmann-Gibbs statistics.

In February of 2020, Tsallis, mathematician Nihat Ay,* and physicist Ugur Tirnakli* convened a small SFI working group to focus on the mathematical counterpart of the third signature of nonextensive statistical mechanics, namely the distribution of energies. They have published their results in a recent paper in the journal Nonlinear Dynamics. 

In the paper, they demonstrate how non-extensive statistical mechanics mirrors the large deviation theory, related to the long-awaited third signature in these statistics for complex systems. They go on to computationally test their approach on iconic models of complex systems that describe earthquakes, avalanches, and extinction of biological species.

Since its initial publication, nonextensive statistical mechanics has been applied to a wide range of systems — solar neutrinos, stock markets, patterns in literature, and medicine, to name but a few. Tsallis hopes that complex systems scientists will find still more uses for the work now that the missing piece is being better understood.

“Nonextensive statistical mechanics can be used by physicists, astrophysicists, geophysicists, and economists,” he says. “Anyone who studies complex systems."

Read the paper, “Approaching a large deviation theory for complex systems,” in Nonlinear Dynamics

*Constantino Tsallis is an SFI External Faculty Fellow, founder of the National Institute of Science and Technology of Complex Systems, and former head of the Department of Theoretical Physics at the Centro Brasileiro de Pesquisas Fisicas in Brazil; Nihat Ay is an SFI Professor and Founding Director of the Institute for Data Science Foundations at Hamburg University of Technology in Germany; Ugur Tirnakli is a professor in the physics department at Ege University in Turkey.