  More recently in [J. Phys. A: Math. Theor. 53, 115303 (2020)], we have
introduced a set of noncommutative algebra that describes the space-time at the
Planck scale. The interesting significant result we found is that the
generalized uncertainty principle induced a maximal length of quantum gravity
which has different physical implications to the one of generalized uncertainty
principle with minimal length. The emergence of a maximal length in this theory
revealed strong quantum gravitational effects at this scale and predicted the
detection of gravity particles with low energies. To make evidence of these
predictions, we study the dynamics of a free particle confined in an infinite
square well potential in one dimension of this space. Since the effects of
quantum gravity are strong in this space, we show that the energy spectrum of
this system is weakly proportional to the ordinary one of quantum mechanics
free of the theory of gravity. The states of this particle exhibit proprieties
similar to the standard coherent states which are consequences of quantum
fluctuation at this scale. Then, with the spectrum of this system at hand, we
analyze the thermodynamic quantities within the canonical and microcanonical
ensembles of an ideal gas made up of $N$ indistinguishable particles at the
Planck scale. The results show a complete consistency between both statistical
descriptions. Furthermore, a comparison with the results obtained in the
context of minimal length scenarios and black hole theories indicates that the
maximal length in this theory induces logarithmic corrections of deformed
parameters which are consequences of a strong quantum gravitational effect.
