Non-Gaussian statistics from the generalized uncertainty principle

In many quantum gravity theories, there is the emergence of a generalized uncertainty principle (GUP), implying a minimal length of the order of the Planck length. From the statistical mechanics point of view, this prescription enters into the phase space structure by modifying the elementary cell volume, which becomes momentum-dependent. In this letter, it is pointed out that if one assumes that the total phase space volume is not affected by the minimum length prescription, the statistics that maximize the entropy are non-Gaussian but exhibit a quadratic correction over Gaussian statistics. The departure from Gaussian statistics is significant for high energies. To substantiate our point, we apply these statistics to the Unruh effect and the Jeans gravitational instability and show that—in these cases—non-Gaussian statistics produce the same effect as the GUP and capture the underlying physics behind it.