Galaxies, like the M81 galaxy pictured above, may be better described by q-statistics than by standard Boltzmann-Gibbs statistics (Image: JPL)

Statistical mechanics is an essential tool for studying complexity because it describes systems with large numbers of interacting elements. But there’s an outstanding question of which statistics are best applied to which systems — the subject of a new paper by SFI External Professor Constantino Tsallis and Debarshee Bagchi of the Centro Brasileiro de Pesquisas Físicas in Brazil.

In their paper, published this month in the journal Physica A, the authors draw a critical distinction between systems where elements interact across short-range distances, as in a system of air molecules, and those with long-range interactions, such as a galaxy and its stars. They show that standard Boltzmann-Gibbs statistics is adequately applicable to short-range interactions whereas q-statistics, proposed by Tsallis himself in 1988, give far better quantitative results for long-range interactions, like those where gravity comes into play.

“Boltzmann-Gibbs either doesn’t work or works badly in gravitational systems,” Tsallis explains, “but q-statistics handles these scenarios very well.” He and Bagchi argue in the paper that each class of systems should have its own statistics.

Boltzmann-Gibbs statistical mechanics is acknowledged as one of the five pillars of modern theoretical physics along with Newtonian mechanics, electromagnetism, Einstein’s relativistic mechanics, and quantum mechanics. Within Boltzmann-Gibbs statistical mechanics, the celebrated Boltzmann weight reveals how energy is balanced amongst molecules within a system at a constant temperature.

According to Tsallis, q-statistics is to Boltzmann-Gibbs as relativity is to Newtonian mechanics. Einstein’s relativity equations give way to Newton’s equations when the masses start getting smaller than those of stars, and the speeds drop below the speed of light. Similarly, q-statistics give way to Boltzmann-Gibbs typically when the correlations strongly decay in space and time, in contrast with say gravity, which exerts a strong pull between distant elements within the system.

Computationally, Tsallis and Bagchi show the transition from Boltzmann to q-statistics by adjusting a term within the Fermi-Pasta-Ulam-Tsingou problem, which is famous in physics.

Since 1988, Tsallis’ q-statistics have been applied to a breadth of diverse systems including stock markets, literature, hydraulics, medicine, and even cosmic rays. These examples are drawn from the ~4.5 thousand citations of his 1988 paper, in which he first published q-statistics based on the non-additive entropies that he introduced, before working on various deeply related aspects with SFI's co-founder Murray Gell-Mann.

Tsallis likes to believe that, "like beauty, complexity is hard to define although kind of easy to identify, and even to handle if we adapt the tool, for instance the q-index, to the specific system."

Read the paper, "Fermi–Pasta–Ulam–Tsingou problems: Passage from Boltzmann to q-statistics" in Physica A (February 1, 2018)

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