  Polymer quantum systems are mechanical models quantized similarly as loop
quantum gravity. It is actually in quantizing gravity that the polymer term
holds proper as the quantum geometry excitations yield a reminiscent of a
polymer material. In such an approach both non-singular cosmological models and
a microscopic basis for the entropy of some black holes have arisen. Also
important physical questions for these systems involve thermodynamics. With
this motivation, in this work, we study the statistical thermodynamics of two
one dimensional {\em polymer} quantum systems: an ensemble of oscillators that
describe a solid and a bunch of non-interacting particles in a box, which thus
form an ideal gas. We first study the spectra of these polymer systems. It
turns out useful for the analysis to consider the length scale required by the
quantization and which we shall refer to as polymer length. The dynamics of the
polymer oscillator can be given the form of that for the standard quantum
pendulum. Depending on the dominance of the polymer length we can distinguish
two regimes: vibrational and rotational. The first occur for small polymer
length and here the standard oscillator in Schr\"odinger quantization is
recovered at leading order. The second one, for large polymer length, features
dominant polymer effects. In the case of the polymer particles in the box, a
bounded and oscillating spectrum that presents a band structure and a Brillouin
zone is found. The thermodynamical quantities calculated with these spectra
have corrections with respect to standard ones and they depend on the polymer
length. For generic polymer length, thermodynamics of both systems present an
anomalous peak in their heat capacity $C_V$.
