Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 110, 23 pages      arXiv:1109.0803      http://dx.doi.org/10.3842/SIGMA.2011.110
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Statistical Thermodynamics of Polymer Quantum Systems

Guillermo Chacón-Acosta a, Elisa Manrique b, Leonardo Dagdug c and Hugo A. Morales-Técotl c
a) Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana-Cuajimalpa, Artificios 40, México D. F. 01120, México
b) Institut für Physik, Johannes-Gutenberg-Universität, D-55099 Mainz, Germany
c) Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Méico D. F. 09340, México

Received September 01, 2011, in final form November 16, 2011; Published online December 02, 2011

Abstract
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 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 Schrödinger 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. When the polymer length is small such corrections resemble those coming from the phenomenological generalized uncertainty relation approach based on the idea of the existence of a minimal length. For generic polymer length, thermodynamics of both systems present an anomalous peak in their heat capacity CV. In the case of the polymer oscillators this peak separates the vibrational and rotational regimes, while in the ideal polymer gas it reflects the band structure which allows the existence of negative temperatures.

Key words: statistical thermodynamics; canonical quantization; loop quantum gravity.

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