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Chaotic Quantum Vortices in He II: Thermodynamic Equilibrium and Turbulence

Abstract

The use of superfluid helium as a refrigerant in cryogenic systems is governed by the presence of a chaotic tangle of quantum filaments in the superfluid component of helium. Therefore, to describe any hydrodynamic phenomena (in particular, heat transfer) in quantumliquids containing vortex tangles, it is necessary to have information on their structure and statistics. The paper discusses two possible statistical configurations of chaotic vortices: the thermodynamic equilibrium and the highly nonequilibrium turbulent state, as well as the transition between them. Basing on the Langevin approach, we discuss the mechanism of establishment of thermodynamic equilibrium for a chaotic set of quantum vortex filaments. The corresponding Fokker–Planck equation for the probability density functional has a solution in the form of the Gibbs distribution. Basing on the above analysis, we discuss the possible causes and mechanisms of violation of thermodynamic equilibrium and transition to the turbulent regime.

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References

  1. 1.

    Nemirovskii, S.K. and Tsoi, A.N., Transient Thermal and Hydrodynamic Processes in Superfluid Helium, Cryogenics, 1989, vol. 29, no. 10, pp. 985–994.

    ADS  Article  Google Scholar 

  2. 2.

    Lebrun, Ph., Serio, L., Tavian, L., and van Weelderen, R., Cooling Strings of SuperconductingDevices Below 2 K: The Helium II Bayonet Heat Exchanger, Boston,MA: Springer, 1998, pp. 419–426.

    Google Scholar 

  3. 3.

    Gorter, C.J. and Mellink, J.H., On the Irreversible Processes in Liquid Helium II, Physica, 1949, vol. 15, nos. 3/4, pp. 285–304.

    ADS  Article  Google Scholar 

  4. 4.

    Feynman, R.P., Progress in Low Temperature Physics, vol. 1, North-Holland, Amsterdam, 1955.

    Google Scholar 

  5. 5.

    Donnelly, R.J., Quantized Vortices in Helium II, Cambridge, UK: Cambridge University Press, 1991.

    Google Scholar 

  6. 6.

    Khalatnikov, I.M., An Introduction to the Theory of Superfluidity, New York: Benjamin, 1965.

    Google Scholar 

  7. 7.

    Nemirovskii, S.K. and Fiszdon, W., Chaotic Quantized Vortices and Hydrodynamic Processes in Superfluid Helium, Rev. Mod. Phys., 1995, vol. 67, no. 1, pp. 37–84.

    ADS  Article  Google Scholar 

  8. 8.

    Schwarz, K.W., Three-Dimensional Vortex Dynamics in Superfluid He4: Homogeneous Superfluid Turbulence, Phys. Rev. B, 1988, vol. 38, no. 4, pp. 2398–2417.

    ADS  Article  Google Scholar 

  9. 9.

    Nore, C., Abid, M., and Brachet, M.E., Kolmogorov Turbulence in Low-Temperature Superflows, Phys. Rev. Lett., 1997, vol. 78, no. 20, pp. 3896–3899.

    ADS  Article  Google Scholar 

  10. 10.

    Svistunov, B.V., Superfluid Turbulence in the Low-Temperature Limit, Phys. Rev. B, 1995, vol. 52, no. 5, pp. 3647–3653.

    ADS  Article  Google Scholar 

  11. 11.

    Vinen, W.F., Mutual Friction in a Heat Current in Liquid Helium II. III. Theory of the Mutual Friction, Proc. R. Soc. London A, 1957, vol. 242, pp. 493–515.

    ADS  Article  Google Scholar 

  12. 12.

    Nemirovskii, S.K., Quantum Turbulence: Theoretical and Numerical Problems, Phys. Rep., 2013, vol. 524, no. 3, pp. 85–202.

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Nemirovskii, S.K., Modeling of Classical Turbulence by Quantized Vortices, J. Eng. Therm., 2017, vol. 26, no. 4, pp. 476–484.

    Article  Google Scholar 

  14. 14.

    Araki, T. and Tsubota, M., Cascade Process of Vortex Tangle Dynamics in Superfluid 4He without Mutual Friction, J. Low Temp. Phys., 2000, vol. 121, pp. 405–410.

    ADS  Article  Google Scholar 

  15. 15.

    Antunes, N.D., Bettencourt, L.M.A., and Hindmarsh, M., Thermodynamics of Cosmic String Densities in U (1) Scalar Field Theory, Phys. Rev. Lett., 1988, vol. 80, no. 5, p.908.

    ADS  Article  Google Scholar 

  16. 16.

    Copeland, E.J., Kibble, T.W.B., and Steer, D.A., Evolution of a Network of Cosmic String Loops, Phys. Rev. D, 1988, vol. 58, no. 4, p. 043508.

    ADS  Article  Google Scholar 

  17. 17.

    Zurek, W.H., Cosmological Experiments in Condensed Matter Systems, Phys. Rep., 1996, vol. 276, no. 4, pp. 177–221.

    ADS  Article  Google Scholar 

  18. 18.

    Nemirovskii, S.K., Kinetics of a Network of Vortex Loops in HeII and a Theory of Superfluid Turbulence, Phys. Rev. B, 2008, vol. 77, no. 21, p. 214509.

    ADS  Article  Google Scholar 

  19. 19.

    Migdal, A.A., Fokker–Planck Vortex Equation, Vopr. Kib., 1986, p.122.

    Google Scholar 

  20. 20.

    Onsager, L., Statistical Hydrodynamics. Il Nuovo Cimento (1943–1954), 1949, vol. 6, pp. 279–287.

    MathSciNet  Article  Google Scholar 

  21. 21.

    Alekseenko, S.V., Kuibin, P.A., and Okulov, V.L., Theory of Concentrated Vortices, Springer, 2007.

    MATH  Google Scholar 

  22. 22.

    Saffman, P.G., Vortex Dynamics, Cambridge, UK: Cambridge University Press, 1992.

    MATH  Google Scholar 

  23. 23.

    Batchelor, G.K., An Introduction to FluidMechanics, Cambridge, UK: Cambridge University Press, 1967.

    Google Scholar 

  24. 24.

    Ambegaokar, V., Halperin, B.I., Nelson, D.R., and Siggia, E.D., Dynamics of Superfluid Films, Phys. Rev. B, 1980, vol. 21, no. 5, pp. 1806–1826.

    ADS  Article  Google Scholar 

  25. 25.

    Zinn-Justin, J., Quantum Field Theory and Critical Phenomena, Oxford: Clarendon Press, 2002.

    Book  MATH  Google Scholar 

  26. 26.

    Nemirovskii, S.K., Thermodynamic Equilibrium in the System of Chaotic Quantized Vortices in a Weakly Imperfect Bose Gas, Theor. Math. Phys., 2004, vol. 141, pp. 1452–1460.

    Article  MATH  Google Scholar 

  27. 27.

    Ma, S. and Mazenko, G.F., Critical Dynamics of Ferromagnets in 6 − ϵ Dimensions:General Discussion and Detailed Calculation, Phys. Rev. B, 1975, vol. 11, pp. 4077–4100.

    ADS  Article  Google Scholar 

  28. 28.

    Landau, L.D. and Lifshitz, E.M., Course of Theoretical Physics, vol. 5, Statistical Physics, part 1, Oxford: Pergamon Press, 3rd ed., 1980.

    Google Scholar 

  29. 29.

    Nemirovskii, S.K., Pakleza, J., and Poppe, W., Notes et Documents LIMSI, Laboratoire d’Informatique pour la Mecanique et les Sciences de l’Ingenieur, 1991, no. 91–14.

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Nemirovskii, S.K. Chaotic Quantum Vortices in He II: Thermodynamic Equilibrium and Turbulence. J. Engin. Thermophys. 27, 415–421 (2018). https://doi.org/10.1134/S1810232818040057

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