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 Schrodinger 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 $C_V$.
The relativistic equilibrium velocity distribution plays a key role in describing several high-energy and astrophysical effects. Recently, computer simulations favored Juttners as the relativistic generalization of Maxwells distribution for d=1,2,3 s
patial dimensions and pointed to an invariant temperature. In this work we argue an invariant temperature naturally follows from manifest covariance. We present a new derivation of the manifestly covariant Juttners distribution and Equipartition Theorem. The standard procedure to get the equilibrium distribution as a solution of the relativistic Boltzmanns equation is here adopted. However, contrary to previous analysis, we use cartesian coordinates in d+1 momentum space, with d spatial components. The use of the multiplication theorem of Bessel functions turns crucial to regain the known invariant form of Juttners distribution. Since equilibrium kinetic theory results should agree with thermodynamics in the comoving frame to the gas the covariant pseudo-norm of a vector entering the distribution can be identified with the reciprocal of temperature in such comoving frame. Then by combining the covariant statistical moments of Juttners distribution a novel form of the Equipartition Theorem is advanced which also accommodates the invariant comoving temperature and it contains, as a particular case, a previous not manifestly covariant form.
