Tsallis Entropy

I'm speechless.

Strongly ergodic/mixing phenomena are ubiquitous in Nature; essentially, they are driven by microscopic interactions which are short-ranged in spacetime (short-range forces, short-range memory, nonfractal boundary conditions); their basic geometry tends to be continuous, Euclidean-like; their thermodynamics is extensive; their central laws (energy distribution at equilibrium, time-relaxation towards equilibrium) are exponentials; and their thermostatistical foundation is Boltzmann-Gibbs statistics (i.e., q = 1). But weakly ergodic/mixing phenomena also are ubiquitous in Nature (e.g., biological, socio-economical, human cognitive phenomena, etc); essentially, they are driven by microscopic interactions which are long-ranged in space and time (long-range forces, nonmarkovian memory, fractal boundary conditions); their basic geometry tends to be discrete, multifractal-like; their thermodynamics is nonextensive; their central laws (energy distribution at equilibrium, time-relaxation towards equilibrium) are power-laws; and the thermostatistical foundation of (at least some of) them (hopefully) is the q != 1 statistics.

Review here.

Comments

[Mike Speiser]

When you find your tongue, what is the applicability of this in your mind?

[Michael F. Martin]

Tsallis entropy permits us to estimate (and so to some extent forecast) the large-scale, long-term behavior of a system that is built up from a variety of spatially separated local-scale, short-term dynamics. In the limit where the these spatially discrete regions merge into one, you get the entropy we learned in college.

The linked paper has a lot of examples. It doesn't say much about the nature of the varieties of local dynamics that give rise to these statistics though. Somebody has probably already checked to see whether, for example, the dynamical clustering that results in scale invariance obeys q-statistics.

This is all very new. I need to digest, but in principle this could offer a connection between some microeconomic and macroeconomic variables.