Many complex systems are effectively described by more general versions of statistical mechanics.

An example is a statistical mechanics based on q-entropies, where the canonical distributions are q-exponentials.

One can obtain these distributions in a non-equilibrium setting if chi-square distributed inverse temperature fluctuations are assumed. But there are other possibilities of complex behavior, for example lognormally distributed temperature fluctuations.

The two cases lead to chi-square and lognormal superstatistics, respectively. In this talk I give examples where a complex system exhibits a transition from one superstatistics to another. One such example is the statistics of Lagrangian turbulence, which is well described by chi-square superstatistics in the quantum case and by lognormal superstatistics in the classical case.

Another example is an observed transition from chi-square to lognormal superstatistics for some financial time series as a function of the time scale of returns.

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Photos by : Ivan