Maximum entropy with fluctuating constraints: The example of K-distributions

Abstract

We indicate that in a maximum entropy setting, the thermodynamic β and the observation contraint are linked, so that fluctuations of the latter imposes fluctuations of the former. This gives an alternate viewpoint to 'superstatistics'. While a Gamma model for fluctuations of the β parameter gives the so-called Tsallis distributions, we work out the case of a Gamma model for fluctuations of the observable, and show that this leads to K-distributions. We draw attention to the fact that these heavy-tailed distributions have high interest in physical applications, and we discuss them in some details.

Hierarchical maximum entropy principle for generalized superstatistical systems and Bose-Einstein condensation of light

“…( 4) and ( 5), respectively. Thus, given the constraints (8), the intensive parameter distribution (10), partition function (11), and Massieu function ( 14) depend only on β and ξ:…”

“…This is in full analogy with the case of the dynamics in a cell, when we may first set the mean energy U (β) and then find the corresponding intensive parameter β, or set β and then find U (β), which is more common. Incidentally, this duality allows one to alternatively formulate superstatistics by introducing the fluctuations of U (β) instead of those of β [11]. Note that the control parameter ξ has a more general nature than β, since β is exactly a Lagrange multiplier, while ξ, though controlling the Lagrange multiplier µ, may not coincide with µ.…”