  The Hulth\'{e}n potential is a short-range potential that has been widely
used in various fields of physics. In this paper, we investigate the
distribution functions for the Hulth\'{e}n potential by using statistical and
superstatistical methods. We first review the ordinary statistics and
superstatistics methods. We then consider some distribution functions, such as
uniform, 2-level, gamma, and log-normal and F distributions. Finally, we
investigate the behavior of the Hulth\'{e}n potential for statistical and
superstatistical methods and compare the results with each other. We use the
Tsallis statistics of the superstatistical system. We conclude that the Tsallis
behavior of different distribution functions for the Hulth\'{e}n potential
exhibits better results than the statistical method. We examined the thermal
properties of the Hulth\'{e}n potential for five different distributions:
Uniform, 2-level, Gamma, Log-normal, and F. We plotted the Helmholtz free
energy and the entropy as functions of temperature for various values of q. It
shows that the two uniform and 2-level distributions have the same results due
to the universal relationship and that the F distribution does not become
ordinary statistics at q=1. It also reveals that the curves of the Helmholtz
free energy and the entropy change their order and behavior as q increases and
that some distributions disappear or coincide at certain values of q. One can
discuss the physical implications of our results and their applications in
nuclear and atomic physics in the future.
