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Undistinguishing statistics of objectively distinguishable objects: Thermodynamics and superfluidity of classical gas

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Abstract

In the present paper, we describe an approach to thermodynamics that does not involve Bogolyubov chains or Gibbs ensembles. We present isotherms, isochores, and isobars of various pure gases, as well as binodals, i.e., lines along which gas becomes liquid, and spinodals (endpoints of isotherms). We study supercritical phenomena for values of temperature and pressure above the critical ones. A lot of attention is paid to the region of negative pressures. The superfluid component for supercritical phenomena is described, as well as the thermodynamics of nanostructures and superfluidity in nanotubes.

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References

  1. V. V. Kozlov, Thermal Equilibrium According to Gibbs and Poincaré (Institute for Computer Studies,, Moscow, 2002) [in Russian].

    Google Scholar 

  2. The European Physical Journal 37(4) (2012).

    Google Scholar 

  3. V. A. Malyshev and R. A. Minlos, Gibbs Random Fields: Cluster Expansions Mathematics and Its Applications (Kluwer Academic Publishers, Dordrecht-Boston, 1991).

    MATH  Google Scholar 

  4. A. A. Lykov and V. A. Malyshev, “Convergence to Gibbs equilibrium-unveiling the mystery,” Markov Processes and Related Fields 19(4) (2013).

    Google Scholar 

  5. A. M. Vershik, “Statistical mechanics of combinatorial partitions, and their limit shapes,” Funktsional. Anal. Prilozhen. 30(2), 19–39 (1996) [Functional Anal. Appl. 30 (2), 90–105 (1996)].

    MathSciNet  Google Scholar 

  6. V. P. Maslov, “Quasithermodynamics and a correction to the Stefan-Boltzmann law,” Mat. Zametki 83(1), 77–85 (2008) [Math. Notes 83 (1) 72–79 (2008)].

    MathSciNet  Google Scholar 

  7. V. P. Maslov, “Quasithermodynamic correction to the Stefan-Boltzmann law,” Teoret. Mat. Fiz. 154(1), 207–208 (2008) [Theoret. and Math. Phys. 154 (1), 175–176 (2008)].

    MathSciNet  Google Scholar 

  8. H.-O. Georgii, GibbsMeasures and Phase Transitions (Walter de Gruyter, Berlin, 2011).

    Google Scholar 

  9. D. Ruelle, Statistical Mechanics (W. A. Benjamin, Inc., New York-Amsterdam, 1969; Mir, Moscow, 1971).

    MATH  Google Scholar 

  10. Ya. G. Sinai, Theory of Phase Transitions (Nauka, Moscow, 1980).

    Google Scholar 

  11. B. Ya. Frenkel, Yakov Il’ich Frenkel (Nauka Publ., Moscow-Leningrad, 1966) [in Russian].

    Google Scholar 

  12. V. P. Maslov, “On a Serious Mathematical Error in the “Mathematical Encyclopedia” Related to the Solution of the Gibbs Paradox,” Math. Notes 93(5), 732–739 (2013).

    MATH  Google Scholar 

  13. V. P. Maslov, “Old Mathematical Errors in Statistical Physics,” Russian J. Math. Phys. 20(2), 214–229 (2013).

    MATH  Google Scholar 

  14. V. P. Maslov, “The Law of Preference of Cluster Formation over Passage to Liquid State,” Math. Notes 94(1) 115–126 (2013).

    MATH  Google Scholar 

  15. V. P. Maslov, “The Law of Preference of Cluster Formation over Passage to Liquid State II,” Math. Notes 94(3) 364–368 (2013).

    Google Scholar 

  16. L. D. Landau and E.M. Lifshits, Statistical Physics (Nauka, Moscow, 1964) [in Russian].

    Google Scholar 

  17. Mathematical Encyclopedia (Soviet Encyclopedia, Moscow, 1984), Vol. 4 [in Russian].

  18. W.-S. Dai and M. **e, “Gentile statistics with a large maximum occupation number,” Annals of Physics 309, 295–305 (2004).

    MATH  MathSciNet  Google Scholar 

  19. V. P. Maslov, Thermodynamics as a Multistep Relaxation Process and the Role of Observables in Different Scales of Quantities, ar**v:1303.5307v2 [physics.gen-ph], 25 Mar 2013.

    Google Scholar 

  20. V. P. Maslov, “Unbounded Probability Theory and Multistep Relaxation Processes,” Math. Notes 93(3) 451–459 (2013).

    MATH  Google Scholar 

  21. V. P. Maslov, “On I.M. Gelfand’s 100th Anniversary,” Math. Notes 94(6), 841–842 (2013).

    Google Scholar 

  22. V. P. Maslov, Threshold Levels in Economics, ar**v:0903.4783v2 [q-fin.ST] 3 Apr 2009.

    Google Scholar 

  23. V.P. Maslov, Daring to Touch Radha (Academic Express, Lviv, 1993); also on http://viktor.maslovs.co.uk/en/home/articles/hist/my-hist.

    Google Scholar 

  24. A. I. Anselm, Foundations of Statistical Physics and Thermodynamics (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  25. H. Eyring, Basic Chemical Kinetics (Mir, Moscow, 1985) [in Russian].

    Google Scholar 

  26. L. I. Mandelshtam and M. A. Leontovich, “On the theory of absorption of sound in liquid,” Zh. Éksp. Teoret. Fiz. 7(3), 438–449 (1937)

    Google Scholar 

  27. V. P. Maslov, Quantization of Thermodynamics and Ultrasecond Quantization (Institute for Computer Studies, Moscow, 2001) [in Russian].

    Google Scholar 

  28. S. T. Kuroda, “Scattering theory for differential operators. III,” Lecture Notes in Mathematics 448, 227–241 (1975).

    MathSciNet  Google Scholar 

  29. G. A. Martynov, Fundamental theory of liquids: method of distribution functions, (A. Hilger, Bristol-Philadelphia-New York, 1992).

    Google Scholar 

  30. G. L. Litvinov, “Maslov dequantization, idempotent and tropicalmathematics: a brief introduction,” Journal of Math. Sciences 140(3), 426–444 (2007).

    MathSciNet  Google Scholar 

  31. E. M. Apfelbaum, V. S. Vorob’ev, “Correspondence between the critical and the Zeno-line parameters for classical and quantum liquids,” J.Phys.Chem. B 113, 3521–3526 (2009).

    Google Scholar 

  32. V. P. Maslov, “Thermodynamics of fluids: The law of redestribution of energy, two-dimensional condensate, and T-map**,” Teoret.Mat. Fiz. 161(3), 422–456 (2009) [Theoret. and Math. Phys. 161 (3), 1681–1713 (2009)].

    Google Scholar 

  33. E. M. Apfelbaum, V. S. Vorobtev, and G. A. Martynov,, “Triangle of Liquid-Gas States,” J. Phys. Chem. B 110, 8474–8480 (2006).

    Google Scholar 

  34. V. P. Maslov, “Thermodynamics of fluids for imperfect gases with Lennard-Jones interaction potential: I,” Math. Notes 86(3–4), 522–529 (2009).

    MATH  Google Scholar 

  35. V. P. Maslov, “Thermodynamics of fluids for imperfect gases with Lennard-Jones interaction potential: II (the law of redistribution of energies),” Math. Notes 86(5–6), 605–611 (2009).

    MATH  Google Scholar 

  36. V. P. Maslov, “Thermodynamics of fluids for imperfect gases with Lennard-Jones interaction potential: III,” Math. Notes 87(1–2), 79–87 (2010).

    MATH  Google Scholar 

  37. V. P. Maslov, “The mathematical theory of classical thermodynamics,” Math. Notes 93(1) 102–136 (2013).

    MATH  Google Scholar 

  38. V.P. Maslov, “Mixture of new ideal gases and the solution of problems in Gibbs and Einstein paradoxes,” Russian J. Math. Phys. 18(1), 83–101 (2011).

    MATH  Google Scholar 

  39. G. Birkhoff, Hydrodynamics (Princeton, 1950; Inostr. Lit., Moscow, 1954).

    MATH  Google Scholar 

  40. W.-G. Dong and J. Lienhard, “Corresponding states correlation of saturated and metastable properties,” Canadian J. Chem. Eng. 64, 158–161 (1986).

    Google Scholar 

  41. V. P. Maslov and P. P. Mosolov, Nonlinear Wave Equations Perturbed by Viscous Term, De Gruyter Expositions in Mathematics 31 (Walter de Gruyter, Berlin-New York, 2000).

    Google Scholar 

  42. S. M. Stishov and A. V. Smirnov, “Study of expanded substance at low temperature by the Monte-Carlo method,” Pis’ma Zh. Éksp. Teoret. Fiz. 57(11), 715–720 (1993).

    Google Scholar 

  43. R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals (New York, 1965; Mir, Moscow, 1968).

    Google Scholar 

  44. V. P. Maslov, Perturbation Theory and Asymptotical Methods (Izd. Moskov. Univ., Moscow, 1965; Dunod, Paris, 1972) [in Russian and French].

    Google Scholar 

  45. N. Hurt, Geometric Quantization in Action (Reidel, Dordrecht, 1983; Mir,Moscow, 1985).

    MATH  Google Scholar 

  46. V. P. Maslov, Asymptotic Methods and Perturbation Theory (Nauka, Moscow,1988) [in Russian].

    MATH  Google Scholar 

  47. V. P. Maslov, “Nonstandard characteristics in asymptotic problems,” Uspekhi Mat. Nauk 38(6), 3–36 (1983).

    MathSciNet  Google Scholar 

  48. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 3: Quantu Mechanics: Non-Relativistic Theory, 2nd ed. (Nauka, Moscow, 1964; translation of the 1st ed., Pergamon Press, London-Paris and Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958).

    Google Scholar 

  49. V. V. Brazhkin, A. G. Lyapin, V. N. Ryzhov, K. Trachenko, Yu. D. Fomin, and E. N. Tsiok, “Where is the supercritical fluid on the phase diagram?” Uspekhi Fiz. Nauk 182(11), 1137–1156 (2012) [Physics-Uspekhi 55 (11), 1061–1079 (2012)].

    Google Scholar 

  50. E. M. Apfelbaumand V. S. Vorob’ev, “Regarding the universality of some consequences of the van der Waals equation in the supercritical domain,” Journ. Phys. Chem. B 117(25), 7750–7755 (2013).

    Google Scholar 

  51. E. M. Apfelbaum and V. S. Vorob’ev, “The saturation pressure for different objects in reduced variables and the justification of some empirical relations set from the van der Waals equation,” Chem. Phys. Lett. (2013) (in press).

    Google Scholar 

  52. V. P. Maslov, “Critical indices as a consequence of Wiener quantization of thermodynamics,” Teoret. Mat. Fiz. 170(3), 458–470 (2012) [Theoret. and Math. Phys. 170 (3), 384–393 (2012)].

    Google Scholar 

  53. D. Yu. Ivanov, Critical Behavior of Non-Ideal Systems (Fizmatlit, Moscow, 2003; Wiley-VCH, 2008).

    Google Scholar 

  54. V. P. Maslov, “A mathematical theory of the supercritical state serving as an effective means of destruction of chemical warfare agents,” Math. Notes 94(4) 532–546 (2013).

    Google Scholar 

  55. V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, “Universal crossover of liquid dynamics in supercritical region,” Pis’ma Zh. Eksp.Teor.Fiz. 95(3), 179–184 (2012)

    Google Scholar 

  56. V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, “Two liquid states of matter: A dynamic line on a phase diagram,” Phys. Rev. E 85(3), 031203–215 (2012).

    Google Scholar 

  57. V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and K. Trachenko, “Liquid-Gas” Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics,” Phys. Rev. Lett. 111(14), 145901–905 (2013).

    Google Scholar 

  58. A.M. Stishov, “Phase transition in expanded matter,” Pis’ma Zh. Eksp.Teor.Fiz. 57(3) 189–194 (1993)

    Google Scholar 

  59. V. P. Maslov and T. V. Maslova, “Economics as an analog of thermodynamics: conjugate variables,” Math. Notes 91(3) 442–444 (2012).

    MathSciNet  Google Scholar 

  60. V. P. Maslov, “Quasi-particles associated with Lagrangian manifolds corresponding to semiclassical self consistent fields. III,” Russian J. Math. Phys., 3(2) 271–276 (1995).

    MATH  MathSciNet  Google Scholar 

  61. V. P. Maslov and O. Yu. Shvedov, The Complex Germ Method in Many-Particle Problems and in Quantum Field Theory (Editorial URSS, Moscow, 2000) [in Russian]

    Google Scholar 

  62. V. P. Maslov, The Complex WKB Method in Nonlinear Equations (Nauka, Moscow, 1977) [in Russian].

    MATH  Google Scholar 

  63. A. A. Vlasov, “On the vibrational properties of an electronic gas,” Zh. Éxper. Teoret. Fiz. 8, 291–238 (1938).

    MATH  Google Scholar 

  64. Mathematical Encyclopedic Dictionary (Soviet Encyclopedia, Moscow, 1988) [in Russian].

  65. Ph. Blanchard and M. Sirugue, “Large deviations from classical paths,” Comm. Math. Phys. 101, 173–185 (1985).

    MATH  MathSciNet  Google Scholar 

  66. V. P. Maslov, Complex Markov Chains and the Feynman Path Integral for Nonlinear Equations (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  67. V. P. Maslov and O. Yu. Shvedov, “An asymptotic formula for the N-particle density function as N →∞ and a violation of the chaos hypothesis,” Russ. J. Math. Phys. 2(2), 217–234 (1994).

    MATH  MathSciNet  Google Scholar 

  68. V. P. Maslov and P. P. Mosolov, “Asymptotic behavior as N →∞ of trajectories of N point masses interacting according to Newton’s gravitation law,” Izv. Akad. Nauk SSSR Ser. Mat. 42(5), 1063–1100 (1978).

    MATH  MathSciNet  Google Scholar 

  69. V. P. Maslov, “Quasiparticles associated with Lagrangian manifolds and corresponding to classical selfconsistent fields. II,” Russ. J. Math. Phys. 3(1), 123–132 (1995).

    MATH  Google Scholar 

  70. V. P. Maslov and O. Yu. Shvedov, “Quantization in the neighborhood of classical solution in the N-particle problem and superfluidity,” Theoret. and Math. Phys. 98(2), 181–196 (1994).

    MathSciNet  Google Scholar 

  71. V. P. Maslov, O. Yu. Shvedov, “Complex WKB-method in the Fock space,” Doklady Akad. Nauk 340(1), 42–47 (1995).

    MathSciNet  Google Scholar 

  72. V. P. Maslov and O. Yu. Shvedov, “Large deviations in the many-body problem,” Mat. Zametki 57(1), 133–137 (1995).

    MathSciNet  Google Scholar 

  73. V. P. Maslov, “Mathematical aspects of weakly nonideal Bose and Fermi gases on a crystal base,” Funktsional. Anal. i Prilozhen. 37(2), 16–27 (2003) [Functional Anal. Appl. 37 (2), (2003)].

    MathSciNet  Google Scholar 

  74. V. P. Maslov, “A note on computer-oriented language,” Problemy Peredachi Informatsii 39(3),72–76 (2003).

    Google Scholar 

  75. V. P. Maslov, “Spectral series and quantization of thermodynamics,” Russ. J. Math. Phys. 9(1), 112–122 (2002).

    MATH  MathSciNet  Google Scholar 

  76. V. P. Maslov, “A model of weakly nonideal Bose gas. Phase transition in superfluid state spouting effect,” Vestnik Moskov. Univ. Ser. Fiz. Astron. 1, 3–11 (2003).

    Google Scholar 

  77. V. P. Maslov, “Quasi-particles associated with Lagrangian manifolds corresponding to semiclassical self-consistent fields. IV,” Russian J. Math. Phys. 3(3), 401–406 (1995).

    MATH  MathSciNet  Google Scholar 

  78. V. P. Maslov, “Quasi-particles associated with Lagrangian manifolds corresponding to semiclassical self-consistent fields. V-XI,” Russian J. Math. Phys. 3, 529–534 (1995); 4, 117–122, 265–270, 539–546 (1996); 5, 123–130, 273–278, 405–412 (1997).

    MATH  MathSciNet  Google Scholar 

  79. V. P. Maslov, “Analytic continuation of asymptotic formulas and axiomatic foundations of thermodynamics and quasithermodynamics,” Functional Anal. Appl. 28(4), 28–41 (1994).

    Google Scholar 

  80. N. N. Bogolyubov, On the Theory of Superfluidity, in Selected Works (Naukova Dumka, Kiev, 1970), Vol. 2 [in Russian].

    Google Scholar 

  81. V. P. Maslov, “Spectral series, superfluidity, and high-temperature superconductivity,” Mat. Zametki 58(6), 933–936 (1995) [Math. Notes 58 (6), 1349–1352 (1995)].

    Google Scholar 

  82. V. P. Maslov, “Spectral series and quantization of thermodynamics,” Russian J. Math. Phys. 9(1), 112–122 (2002).

    MATH  Google Scholar 

  83. Physical Encyclopedic Dictionary (Sovetskaya Entsiklopediya, Moscow, 1966), Vol. 5 [in Russian].

  84. V. P. Maslov, “On the dispersion law of the form \(\varepsilon (p) = \hbar ^2 p^2 /2m + \tilde V(p) - \tilde V(0)\) for elementary excitations of a nonideal Fermi gas in the pair interaction approximation (i ↔ j), V (|x i − x j|),” Mat. Zametki 82(5), 690–708 (2007) [Math. Notes 82 (5), 619–634 (2007)].

    MathSciNet  Google Scholar 

  85. F. A. Berezin, The Method of Second Quantization (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

  86. L. D. Landau, “On the theory of superfluidity,” in Collected Papers (Nauka, Moscow, 1969), Vol. 2, pp. 42–46 [in Russian].

    Google Scholar 

  87. V. P. Maslov, “On the appearance of the λ-point in a weakly nonideal Bose gas and the two-liquid Thiess-Landau model,” Russian J. Math. Phys. 16(2), 146–165 (2009).

    MATH  Google Scholar 

  88. V. P. Maslov, “On the Bose condensate in the two-dimensional case, λ-point and the Thiess-Landau twoliquid model,” Teoret. Mat. Fiz. 159(1), 20–23 (2009).

    Google Scholar 

  89. V. P. Maslov, “Generalization of the second quantization method to the case of special tensor products of Fock space and quantization of free energy,” Funktsional. Anal. Prilozhen. 34(4), 35–48 (2000).

    Google Scholar 

  90. V. P. Maslov, “Supersecond quantization and quantization of entropy with preserved charge,” UspekhiMat. Nauk 55(6), 145–146 (2000).

    Google Scholar 

  91. V. P. Maslov, “Some identities for ultrasecond-quantized operators,” Russ. J. Math. Phys. 8(3), 309–321 (2001).

    MATH  MathSciNet  Google Scholar 

  92. V. P. Maslov, “Quantization of thermodynamics, ultrasecondary quantization and a new variational principle,” Russ. J. Math. Phys. 8(1), 55–82 (2001)

    MATH  MathSciNet  Google Scholar 

  93. V. P. Maslov, “Ultratertiary quantization of thermodynamics,” Teoret. Mat. Fiz. 132(3), 388–398 (2002) [Theoret. and Math. Phys. 132 (3), 1222–1232 (2002)].

    MathSciNet  Google Scholar 

  94. E. M. Lifshits and L. P. Pitaevskii, Statistical Physics, Part 2: Theory of Condensed State (Nauka, Moscow, 1978; Pergamon, Oxford, 1980).

    Google Scholar 

  95. V. P. Maslov, “On the dependence of the criterion for superfluidity from the radius of the capillary,” Teoret. Mat. Fiz. 143(3), 307–327 (2005) [Theoret. and Math. Phys. 143 (3), 741–759 (2005)].

    MathSciNet  Google Scholar 

  96. V. P. Maslov, “Resonance between one-particle (Bogolyubov) and two-particle series in a superfluid liquid in a capillary,” Russ. J. Math. Phys. 12(3), 369–379 (2005).

    MATH  MathSciNet  Google Scholar 

  97. V. P. Maslov, “The scattering problem in the quasiclassical approximation,” Dokl. Akad. Nauk SSSR 151, 306–309 (1963) [Soviet Physics Dokl. 8, 666–668 (1964)].

    Google Scholar 

  98. Yu. S. Barash, Van der Waals Forces (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

  99. V. P. Maslov, “Quantization of Boltzmann entropy, pairs, and the correlation function,” Teoret. Mat. Fiz. 131(2), 261–277 (2002) [Theoret. and Math. Phys. 131 (2), 666–680 (2002)].

    MathSciNet  Google Scholar 

  100. V. P. Maslov, “On the averaging method for a large number of clusters. Phase transitions,” Teoret. Mat. Fiz. 125(2), 297–314 (2000) [Theoret. and Math. Phys. 125 (2), 1552—1567 (2000)].

    MathSciNet  Google Scholar 

  101. V. P. Maslov, “Ultrasecond quantization and ‘ghosts’ in quantized entropy,” Teoret. Mat. Fiz. 129(3), 464–490 (2001) [Theoret. and Math. Phys. 129 (3), 1694–1716 (2001)].

    MathSciNet  Google Scholar 

  102. V. P. Maslov, Operator Methods (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  103. D. S. Golikov and V. P. Maslov, “On the exact solution of the four-row matrix corresponding to the variational equations for ultrasecond quantization problems,” Mat. Zametki 83(2), 305–309 (2008) [Math. Notes 83 (2), 274–278 (2008)].

    MathSciNet  Google Scholar 

  104. V. P. Maslov, “Revision of probability theory from the point of view of quantum statistics,” Russian J. Math. Phys., 14(1), 66–95 (2007).

    MATH  Google Scholar 

  105. H. Liu, J. He, J. Tang, et al., “Translocation of single-stranded DNA through single-walled carbon nanotubes,” Science 327(64), 64–67 (2010).

    Google Scholar 

  106. Ch. Y. Lee, W. Choi, J.-H. Han, and M. Strano, “Coherence resonance in a single-walled carbon nanotube ion channel,” Science 329(10) 1320–1324 (2010).

    Google Scholar 

  107. W. Choi, Z.W. Ulissi, S. Shimizu, et al., “Diameter-dependent ion transport through the interior of isolated single-walled carbon nanotube,” Nature Communications (2013) (accepted 5 Aug 2013) (in press).

    Google Scholar 

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  1. Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow, Russia

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Maslov, V.P. Undistinguishing statistics of objectively distinguishable objects: Thermodynamics and superfluidity of classical gas. Math Notes 94, 722–813 (2013). https://doi.org/10.1134/S0001434613110138

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