Complex systems don’t always follow the rules – often, they don’t even follow the rules of ordinary statistics.

A quarter-century ago, that revelation paved the way for a new approach to complex systems – from biology to high- energy particle physics – that takes into account the often strong correlations between systems’ component parts. 

In May, SFI External Professor Stefan Thurner convened a workshop with the help of two seminal figures in the eld – SFI External Professor Constantino Tsallis and SFI Distinguished Fellow Murray Gell-Mann – to revisit those ideas and assess the state of a field that some of the participants helped create.

The issue, Tsallis says, is the set of assumptions physicists make in conventional statistical mechanics. When studying a gas,
for example, the usual thing to do is nearly ignore the possibility of collisions between particles. Often that assumption works out fine, but it fails, for example, when work- ing with elementary particles in high-energy colliders – or when studying language, for that matter, where like the interactions of subatomic particles, grammar and semantics make some combinations of words more sensible than others.

Tsallis first addressed that problem in 1988 with the first example of what came to be known as generalized or nonextensive statistical mechanics, where nuclear forces, grammatical rules, or other interactions change how the information in a system scales with its size. Gell-Mann was an early proponent of the approach, and by 2002 the pair organized a workshop to see “if Nature likes that idea too,” Tsallis says.

Recent progress on generalized statistical mechanics and related ideas in information theory and machine learning made this the right time for a status update, Thurner says. Since the 2002 workshop, concepts that Tsallis and others developed have been applied to biology, social science, and ecology, not to mention physics.

Still, many questions remain. “One recurrent topic of the workshop was the question ‘to what extent is generalized statistical mechanics the theory behind specific models” in complex systems, Thurner says. In some cases, it may only be a good approximation, while in others, such as with path-dependent but still random processes, the theory may turn out to be exact, he says. 

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