Non-Gaussian statistics from the generalized uncertainty principle
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Non-Gaussian statistics from the generalized uncertainty principle Homa Shababi1,a
, Kamel Ourabah2,b
1 Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University, Chengdu
610065, China
2 Theoretical Physics Laboratory, Faculty of Physics, University of Bab-Ezzouar, USTHB, B.P. 32, El Alia,
16111 Algiers, Algeria Received: 22 January 2020 / Accepted: 28 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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.
1 Introduction Statistical mechanics, even in its classical formulation, incorporates some concepts from quantum theory. One important concept is the particle indiscernibility that, if ignored, leads to the Gibbs paradox. Another concept, to which quantum mechanics provides a justification, is the discreetness of phase space. In fact, in classical physics, the number of states is infinitely large; this infinity is regularized in statistical physics by dividing up the phase space into cells. Such a regularization was done by Boltzmann as a mathematical trick before the advent of quantum theory, but quantum mechanics shed light on this procedure through the Heisenberg uncertainty principle (HUP), stating that for a d-dimensional system, one cannot specify a phase space volume smaller than h d , h being the Plank constant. Although in classical statistical mechanics, the volume of the elementary phase space cell remains arbitrary, since it does not enter into most of the thermodynamic quantities, it does, however, enter in a few occasions as in the Sackur–Tetrode entropy of a mono-atomic classical ideal gas or in the chemical potential, because, in its very definition, the latter implies a variation in the number of particles [1,2].
a e-mail: [email protected] (corresponding author) b e-mail: [email protected]
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Eur. Phys. J. Plus
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In dealing with quantum gravity, the usual uncertainty principle is amended. This is because one of the main outcomes of different quantum gravity proposals such as string theory [3,5–9], noncommutative space-time [1
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