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IR-deformed thermodynamics of quantum bouncers and the issue of dimensional reduction

We probe the low-temperature behavior of a system of quantum bouncers as a theoretical model for ultracold neutrons within a low energy modified version of the standard quantum mechanics, due to the gravitational effects. Working in one dimension, the energy spectrum and bound states of a deformed quantum bouncer are obtained using the first-order WKB approximation, granted the very low energy regime of the particle. In this manner, we can study energy levels of a system of ultracold neutrons as an informative probe towards exploring the low energy manifestation of semi-classical quantum gravitational effects. Our calculated energy levels of ultracold neutrons are in accordance with the observed energy levels, as obtained in the famous Nesvizhevsky \emph{et al.} experiment, with a negative constant deformation, as dependent on the deformation parameter. In advance, we tackle modified thermodynamics of a system of quantum bouncers in the infrared regime via an ensemble theory both in one dimension and also three dimensions, to seek for any trace of an effective, thermodynamic dimensional reduction in this low energy regime of semi-classical quantum gravity. While the issue of dimensional reduction has been essentially assigned to the high energy regime, here we show that there is a trace of an effective, thermodynamic dimensional reduction in infrared regime with one important difference: in the high energy regime, the dimensional reduction effectively occurs from $D=3$ to $D=1$, but here, in this low energy regime, there is a trace of thermodynamic dimensional reduction from $D=3$ to $D=2$.