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A model of Josephson junctions on boson systems – currents and entropy production rate. (English) Zbl 1402.82012
Summary: Non-equilibrium steady states, in the sense of D. Ruelle [Commun. Math. Phys. 224, No. 1, 3–16 (2001; Zbl 1051.82003)], of Boson systems with Bose-Einstein condensation are investigated with the aid of the \(C^{\ast}\)-algebraic method. The model consists of a quantum particle and several bosonic reservoirs. We show that the mean entropy production rate is strictly positive and independent of phase differences provided that the temperatures or the chemical potentials of reservoirs are different. Moreover, Josephson currents occur without entropy production, if the temperatures and the chemical potentials of reservoirs are identical.
©2018 American Institute of Physics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
81R15 Operator algebra methods applied to problems in quantum theory
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
94A17 Measures of information, entropy
46L06 Tensor products of \(C^*\)-algebras
Full Text: DOI
[1] Aschbacher, W.; Jakšić, V.; Pautrat, Y.; Pillet, C.-A., Topics in non-equilibrium quantum statistical mechanics, Open Quantum Systems III, 1-66, (2006), Springer: Springer, Berlin · Zbl 1126.82032
[2] Aschbacher, W.; Jakšić, V.; Pautrat, Y.; Pillet, C.-A., Transport properties of quasi-free fermions, J. Math. Phys., 48, 3, 032101, (2007) · Zbl 1137.82331
[3] Bratteli, O.; Robinson, D., Operator Algebras and Quantum Statistical Mechanics II, (1997), Springer
[4] Bruneau, L.; Jakšić, V.; Pillet, C.-A., Landauer-Büttiker formula and Schrödinger conjecture, Commun. Math. Phys., 319, 2, 501-513, (2013) · Zbl 1277.82055
[5] Bruneau, L.; Jakšić, V.; Last, Y.; Pillet, C.-A., “Landauer-Büttiker and Thouless conductance, Commun. Math. Phys., 338, 1, 347-366, (2015) · Zbl 1323.82026
[6] Burioni, R.; Cassi, D.; Rasetti, M.; Sodano, P.; Vezzani, A., Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At., Mol. Opt. Phys., 34, 4697-4710, (2001)
[7] Fidaleo, F.; Guido, D.; Isola, T., Bose-Einstein condensation on inhomogeneous amenable graphs, Infin. Dimens. Anal., Quantum Probab. Relat. Top., 14, 2, 149-197, (2011) · Zbl 1223.82012
[8] Fidaleo, F., Harmonic analysis on inhomogeneous amenable networks and the Bose-Einstein condensation, J. Stat. Phys., 160, 3, 715-759, (2015) · Zbl 1362.82053
[9] Jakšić, V.; Pillet, C.-A., On entropy production in quantum statistical mechanics, Commun. Math. Phys., 217, 2, 285-293, (2001) · Zbl 1042.82031
[10] Jakšić, V.; Pillet, C.-A., Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs, Commun. Math. Phys., 226, 1, 131-162, (2002) · Zbl 0990.82017
[11] Jakšić, V., Topics in spectral theory, Open Quantum Systems I, 235-312, (2006), Springer: Springer, Berlin · Zbl 1119.47006
[12] Jakšić, V.; Ogata, Y.; Pillet, C.-A., The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics, Commun. Math. Phys., 265, 3, 721-738, (2006) · Zbl 1104.82039
[13] Kanda, T., Remarks on BEC on graphs, Rev. Math. Phys., 29, 8, 1750024, (2017) · Zbl 1376.82007
[14] Lewis, J. T.; Pulè, J. V., The equilibrium states of the free Boson gas, Commun. Math. Phys., 36, 1-18, (1974)
[15] Măntoiu, M.; Richard, S.; Tiedra de Aldecoa, R., Spectral analysis for adjacency operators on graphs, Ann. Henri Poincaré., 8, 7, 1401-1423, (2007) · Zbl 1137.81013
[16] Matsui, T., BEC of free Bosons on networks, Infin. Dimens. Anal., Quantum Probab. Relat. Top., 9, 1, 1-26, (2006) · Zbl 1093.82003
[17] Pruitt, W. E., Eigenvalues of non-negative matrices, Ann. Math. Stat., 35, 1797-1800, (1964) · Zbl 0211.48502
[18] Rudin, W., Real and Complex Analysis, (1987), McGraw-Hill Book Co.: McGraw-Hill Book Co., New York · Zbl 0925.00005
[19] Ruelle, D., Entropy production in quantum spin systems, Commun. Math. Phys., 224, 1, 3-16, (2001) · Zbl 1051.82003
[20] Tasaki, S.; Accardi, L.; Accardi, L.; Ohya, M.; Watanabe, N., Nonequilibrium steady states for a harmonic oscillator interacting with two Bose fields—Stochastic approach and C*-algebraic approach, (2006), World Scientific · Zbl 1151.82372
[21] Tasaki, S.; Matsui, T., Nonequilibrium steady states with Bose-Einstein condensates, Stochastic Analysis: Classical and Quantum, 211-227, (2005), World Scientific Publication: World Scientific Publication, Hackensack, NJ · Zbl 1122.82026
[22] Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, (1986), Chelsea Publishing Co.: Chelsea Publishing Co., New York · Zbl 0017.40404
[23] Verbeure, A. F., Many-body Boson systems, Half a Century Later, x+189, (2011), Springer-Verlag London, Ltd.: Springer-Verlag London, Ltd., London · Zbl 1210.82002
[24] Yafaev, D. R.; Schulenberger, J. R., Mathematical Scattering Theory: General Theory, x+341, (1992), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 0761.47001
[25] Yafaev, D. R., Mathematical Scattering Theory: Analytic Theory, (2010), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 1197.35006
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