Statistical Thermodynamics of Polymer Quantum Systems

Guillermo Chacón-Acosta ; Elisa Manrique ; Leonardo Dagdug ; Hugo A. Morales-Técotl

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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 $C_V$. 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.

Détail

Keywords: statistical thermodynamics, loop quantum gravity.
Mots-clés : canonical quantization

Bibliographie

[1] Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[2] Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007 | DOI | MR | Zbl

[3] Bojowald M., “Loop quantum cosmology”, Living Rev. Relativity, 8 (2005), 11, 99 pp. ; arXiv: gr-qc/0601085 | MR | Zbl

[4] Bojowald M., “Absence of a singularity in loop quantum cosmology”, Phys. Rev. Lett., 86 (2001), 5227–5230 ; arXiv: gr-qc/0102069 | DOI | MR

[5] Bojowald M., “Quantum nature of cosmological bounce”, Gen. Relativity Gravitation, 40 (2008), 2659–2683 ; arXiv: 0801.4001 | DOI | MR | Zbl

[6] Ashtekar A., Pawlowski T., Singh P., “Quantum nature of the big bang: improved dynamics”, Phys. Rev. D, 74 (2006), 084003, 23 pp. ; arXiv: gr-qc/0607039 | DOI | MR | Zbl

[7] Rovelli C., “Black hole entropy from loop quantum gravity”, Phys. Rev. Lett., 77 (1996), 3288–3291 ; arXiv: gr-qc/9603063 | DOI | MR | Zbl

[8] Ashtekar A., Baez J., Corichi A., Krasnov K., “Quantum geometry and black hole entropy”, Phys. Rev. Lett., 80 (1998), 904–907 ; arXiv: gr-qc/9710007 | DOI | MR | Zbl

[9] Domagala M., Lewandowski J., “Black-hole entropy from quantum geometry”, Classical Quantum Gravity, 21 (2004), 5233–5243 ; arXiv: gr-qc/0407051 | DOI | MR | Zbl

[10] Kolb E.W., Turner M.S., The early universe, Paperback Ed. Westview Press, 1994 | Zbl

[11] Weinberg S., Cosmology, Oxford University Press, Oxford, 2008 | MR | Zbl

[12] Grishchuk L.P., Sidorov Y.V., “Relic graviton and the birth of the universe”, Classical Quantum Gravity, 6 (1989), L155–L160 | DOI

[13] Grishchuk L.P., Sidorov Y.V., “Squeezed quantum states of relic gravitons and primordial density fluctuations”, Phys. Rev. D, 42 (1990), 3413–3421 | DOI | MR

[14] Mielczarek J., Szydłowski M., “Relic gravitons as the observable for loop quantum cosmology”, Phys. Lett. B, 657 (2007), 20–26 ; arXiv: 0705.4449 | DOI

[15] Mielczarek J., Szydłowski M., Relic gravitons from super-inflation, arXiv: 0710.2742

[16] Bojowald M., Hossain G.M., “Loop quantum gravity corrections to gravitational wave dispersion”, Phys. Rev. D, 77 (2008), 023508, 14 pp. ; arXiv: 0709.2365 | DOI | MR

[17] Mielczarek J., “Gravitational waves from the big bounce”, J. Cosmol. Astropart. Phys., 11 (2008), 011, 17 pp. ; arXiv: 0807.0712 | DOI | MR

[18] Ashtekar A., Taveras V., Varadarajan M., “Information is not lost in the evaporation of 2D black holes”, Phys. Rev. Lett., 100 (2008), 211302, 4 pp., arXiv: ; Ashtekar A., Pretorius F., Ramazanoğlu F.M., “Evaporation of two-dimensional black holes”, Phys. Rev. D, 83 (2011), 044040, 18 pp., arXiv: 0801.18111012.0077 | DOI | MR | DOI

[19] Ashtekar A., Fairhurst S., Willis J.L., “Quantum gravity, shadow states and quantum mechanics”, Classical Quantum Gravity, 20 (2003), 1031–1061 ; arXiv: gr-qc/0207106 | DOI | MR | Zbl

[20] Corichi A., Vukašinac T., Zapata J.A., “Polymer quantum mechanics and its continuum limit”, Phys. Rev. D, 76 (2007), 044016, 16 pp. ; arXiv: 0704.0007 | DOI | MR

[21] Corichi A., Vukašinac T., Zapata J.A., “Hamiltonian and physical Hilbert space in polymer quantum mechanics”, Classical Quantum Gravity, 24 (2007), 1495–1511 ; arXiv: gr-qc/0610072 | DOI | MR | Zbl

[22] Beaume R., Manuceau J., Pellet A., Sirugue M., “Translation invariant states in quantum mechanics”, Comm. Math. Phys., 38 (1974), 29–45 ; Acerbi F., Morchio G., Strocchi F., “Infrared singular fields and nonregular representations of canonical commutation relation algebras”, J. Math. Phys., 34 (1993), 899–914 ; Cavallaro F., Morchio G., Strocchi F., “A generalization of the Stone–von Neumann theorem to nonregular representations of the CCR-algebra”, Lett. Math. Phys., 47 (1999), 307–320 ; Halvorson H., “Complementarity of representations in quantum mechanics,”, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys., 35 (2004), 45–56, arXiv: quant-ph/0110102 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[23] Dobrev V., Doebner H.-D., Twarock R., “Quantum mechanics with difference operators”, Rep. Math. Phys., 50 (2002), 409–431 ; arXiv: quant-ph/0207077 | DOI | MR | Zbl

[24] Fredenhagen K., Reszewski F., “Polymer state approximation of Schrödinger wave functions”, Classical Quantum Gravity, 23 (2006), 6577–6584 ; arXiv: gr-qc/0606090 | DOI | MR | Zbl

[25] Velhinho J.M., “The quantum configuration space of loop quantum cosmology”, Classical Quantum Gravity, 24 (2007), 3745–3758 ; arXiv: 0704.2397 | DOI | MR | Zbl

[26] Chiou D.-W., “Galileo symmetries in polymer particle representation”, Classical Quantum Gravity, 24 (2007), 2603–2620 ; arXiv: gr-qc/0612155 | DOI | MR | Zbl

[27] Husain V., Louko J., Winkler O., “Quantum gravity and the Coulomb potential”, Phys. Rev. D, 76 (2007), 084002, 8 pp. ; arXiv: 0707.0273 | DOI | MR

[28] Kunstatter G., Louko J., Ziprick J., “Polymer quantization, singularity resolution, and the $1/r^2$ potential”, Phys. Rev. A, 79 (2009), 032104, 9 pp. ; arXiv: 0809.5098 | DOI | MR

[29] Amelino-Camelia G., Ellis J., Mavromatos N.E., Nanopoulos D.V., Sarkar S., “Tests of quantum gravity from observations of $\gamma$-ray bursts”, Nature, 393 (1998), 763–765 | DOI

[30] Amelino-Camelia G., “Introduction to quantum gravity phenomenology,”, Planck Scale Effects in Astrophysics and Cosmology, Lecture Notes in Physics, 669, Springer-Verlag, Berlin, 2005, 59–100 | DOI

[31] Kalayana Rama S., “Some consequences of the generalised uncertainty principle: statistical mechanical, cosmological, and varying speed of light”, Phys. Lett. B, 519 (2001), 103–110 ; arXiv: hep-th/0107255 | DOI | MR | Zbl

[32] Nozari K., Mehdipour H., “Implications of minimal length scale on the statistical mechanics of ideal gas”, Chaos Solitons Fractals, 32 (2007), 1637–1644 ; arXiv: hep-th/0601096 | DOI | MR

[33] Nozari K., Fazlpour B., “Generalized uncertainty principle, modified dispersion relations and the early universe thermodynamics”, Gen. Relativity Gravitation, 38 (2006), 1661–1679 ; arXiv: gr-qc/0601092 | DOI | MR | Zbl

[34] Nozari K., Sefidgar A.S., “The effect of modified dispersion relations on the thermodynamics of black-body radiation”, Chaos Solitons Fractals, 38 (2008), 339–347 | DOI

[35] Camacho A., Macías A., “Thermodynamics of a photon gas and deformed dispersion relations”, Gen. Relativity Gravitation, 39 (2007), 1175–1183 ; arXiv: gr-qc/0702150 | DOI | MR | Zbl

[36] Colladay D., McDonald P., “Statistical mechanics and Lorentz violation”, Phys. Rev. D, 70 (2004), 125007, 8 pp. ; arXiv: hep-ph/0407354 | DOI

[37] Yépez H., Romero J.M., Zamora A., Corrections to the Planck's radiation law from loop quantum gravity, arXiv: hep-th/0407072

[38] Alfaro J., Morales-Técotl H.A., Urrutia L.F., “Loop quantum gravity and light propagation”, Phys. Rev. D, 65 (2002), 103509, 18 pp., arXiv: hep-th/0108061 | DOI | MR

[39] Hossain G.M., Husain V., Seahra S.S., “Background-independent quantization and the uncertainty principle”, Classical Quantum Gravity, 27 (2010), 165013, 8 pp. ; arXiv: 1003.2207 | DOI | MR | Zbl

[40] Reed M., Simon B., Methods of modern mathematical physics, v. I, Functional analysis, Academic Press Inc., New York, 1980 | MR

[41] Peltola A., Kunstatter G., “Effective polymer dynamics of $D$-dimensional black hole interiors”, Phys. Rev. D, 80 (2009), 044031, 13 pp. ; arXiv: 0902.1746 | DOI | MR

[42] Hossain G.M., Husain V., Seahra S.S., “Nonsingular inflationary universe from polymer matter”, Phys. Rev. D, 81 (2010), 024005, 5 pp. ; arXiv: 0906.2798 | DOI

[43] Kunstatter G., Louko J., Peltola A., “Polymer quantization of the Einstein–Rosen wormhole throat”, Phys. Rev. D, 81 (2010), 024034, 10 pp. ; arXiv: 0910.3625 | DOI

[44] Battisti M.V., Lecian O.M., Montani G., “Polymer quantum dynamics of the Taub universe”, Phys. Rev. D, 78 (2008), 103514, 9 pp. ; arXiv: 0806.0768 | DOI | MR

[45] Battisti M.V., Lecian O.M., Montani G., GUP vs polymer quantum cosmology: the Taub model, arXiv: 0903.3836

[46] Creutz M., Quarks, gluons, and lattices, Cambridge University Press, Cambridge, 1983

[47] Hossain G.M., Husain V., Seahra S., “Propagator in polymer quantum field theory”, Phys. Rev. D, 82 (2010), 124032, 5 pp. | DOI

[48] Chalbaud E., Gallinar J.-P., Mata G., “The quantum harmonic oscillator on a lattice”, J. Phys. A: Math. Gen., 19 (1986), L385–L390 | DOI | MR

[49] Abramowitz M., Stegun I., Handbook of mathematical functions, Dover, 1968

[50] Baker G.L., Blackburn J.A., Smith H.J.T., “The quantum pendulum: small and large”, Amer. J. Phys., 70 (2002), 525–531 | DOI

[51] Martin J., Brandenberger R.H., “Trans-Planckian problem of inflationary cosmology”, Phys. Rev. D, 63 (2001), 123501, 16 pp. ; arXiv: hep-th/0005209 | DOI

[52] Doncheski M.A., Robinett R.W., “Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum”, Ann. Physics, 308 (2003), 578–598 ; arXiv: quant-ph/0307079 | DOI | Zbl

[53] Elaydi S., An introduction to difference equations, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1996 | MR | Zbl

[54] Gallinar J.P., Mattis D.C., “Motion of ‘hopping’ particles in a constant force field”, J. Phys. A: Math. Gen., 18 (1985), 2583–2589 | DOI

[55] Sakurai J.J., Modern quantum mechanics, Addison-Wesley, 1994

[56] Marder M.P., Condenseed matter physics, John Wiley Sons, Inc., 2010

[57] Kittel C., Introduction to solid state physics, John Wiley Sons, Inc., 1996

[58] Nozari K., Azizi T., “Some aspects of gravitational quantum mechanics”, Gen. Relativity Gravitation, 38 (2006), 735–742 ; arXiv: quant-ph/0507018 | DOI | MR | Zbl

[59] Pathria R.K., Statistical mechanics, Butterworth-Heinemann, 2001

[60] Swain J., “Exotic statistics for ordinary particles in quantum gravity”, Internat. J. Modern Phys. D, 17 (2008), 2475–2484 | DOI | MR | Zbl

[61] Chacón-Acosta G., Dagdug L., Morales-Técotl H., “On microstates counting in many body polymer quantum systems”, AIP Conf. Proc., 1396 (2011), 99–103 | DOI

[62] Naudts J., “Boltzmann entropy and the microcanonical ensemble”, Europhys. Lett., 69 (2005), 719–724 ; arXiv: cond-mat/0412683 | DOI

[63] Ramsey N.F., “Thermodynamics and statistical mechanics at negative absolute temperatures”, Phys. Rev., 103 (1956), 20–28 | DOI | Zbl

[64] Konopka T., Markopoulou F., Severini S., “Quantum graphity: a model of emergent locality”, Phys. Rev. D, 77 (2008), 104029, 15 pp., arXiv: 0801.0861 | DOI | MR

[65] Smoot G.F., Gorenstein M.V., Muller R.A., “Detection of anisotropy in the cosmic blackbody radiation”, Phys. Rev. Lett., 39 (1977), 898–901 ; Fixsen D.J. et al., “Cosmic microwave background dipole spectrum measured by the COBE FIRAS instrument”, Astrophys. J., 420 (1994), 445–449 ; Fixsen D.J. et al., “The cosmic microwave background spectrum from the full COBE FIRAS data set”, Astrophys. J., 473 (1996), 576–587, arXiv: astro-ph/9605054 | DOI | DOI | DOI

[66] Husain V., Mann R.B., “Thermodynamics and phases in quantum gravity”, Classical Quantum Gravity, 26 (2009), 075010, 6 pp. ; arXiv: 0812.0399 | DOI | MR | Zbl

[67] Agullo I., Barbero G. J.F., Borja E.F., Diaz-Polo J., Villaseñor E.J.S., “Detailed black hole state counting in loop quantum gravity”, Phys. Rev. D, 82 (2010), 084029, 31 pp. ; arXiv: 1101.3660 | DOI

[68] Li L.-F., Zhu J.-Y., “Thermodynamics in loop quantum cosmology”, Adv. High Energy Phys., 2009 (2009), 905705, 9 pp. ; arXiv: 0812.3544 | DOI | Zbl

[69] Bianchi E., “Black hole entropy, loop gravity, and polymer physics”, Classical Quantum Gravity, 28 (2011), 114006, 12 pp. ; arXiv: 1011.5628 | DOI | MR | Zbl

[70] Hamma A., Markopoulou F., “Background independent condensed matter models for quantum gravity”, New J. Phys., 13 (2011), 095006, 26 pp. ; arXiv: 1011.5754 | DOI

[71] Hamma A., Markopoulou F., Lloyd S., Caravelli F., Severini S., Markström K., “Quantum Bose–Hubbard model with an evolving graph as a toy model for emergent spacetime”, Phys. Rev. D, 81 (2010), 104032, 22 pp. ; arXiv: 0911.5075 | DOI

[72] Husain V., Seahra S., Webster E. (to appear)