Tales of Tails

Typical human-scaled considerations of thermodynamic states depend primarily on the core of associated speed or other relevant distributions, because the wings of those distributions are so improbable that they cannot contribute significantly to averages. However, for long timescale regimes (slow time), previous papers have shown otherwise. Fluctuating local equilibrium systems have been proven to have distributions with non-Gaussian tails demanding more careful treatment. That has not been needed in traditional statistical mechanics. The resulting non-Gaussian distributions do not admit notions such as temperature; that is, a global temperature is not defined even if local regimes have meaningful temperatures. A fluctuating local thermodynamic equilibrium implies that any local detector is exposed to sequences of local states which collectively induce the non-Gaussian forms. This paper shows why tail behavior is observationally challenging, how the convolutions that produce non-Gaussian behavior are directly linked to time-coarse graining, how a fluctuating local equilibrium system does not need to have a collective temperature, and how truncating the tails in the convolution probability density function (PDF) produces even more non-Gaussian behaviors.