Journal of Low Temperature Physics

, Volume 156, Issue 3–6, pp 182–192 | Cite as

Langevin Dynamics of Vortex Lines and Thermodynamic Equilibrium of Vortex Tangle



Langevin dynamics—stochastic motion of vortex filaments under action of random force is studied analytically and numerically. We introduce a Langevin-type equation of motion of the line with a stirring force satisfying the fluctuation-dissipation theorem. The respective Fokker-Planck equation for probability functional ℘({s(ξ)}) in vortex loop configuration space is shown to have a solution of the form \(\mathcal{P}(\{\mathbf{s}(\xi )\})=\mathcal{N}\exp (-H\{\mathbf{s}\}/T),\) where \(\mathcal{N}\) is a normalizing factor and H{s} is energy of vortex line configurations. Numerical calculations are performed on base of the full Biot-Savart law for different intensities of the Langevin force. A new algorithm, which is based on consideration of crossing lines, is used for vortex reconnection procedure. After some transient period the vortex tangle develops into the stationary state characterizing by the developed fluctuations of various physical quantities, such as total length, energy etc. We tested this state to learn whether or not it the thermodynamic equilibrium is reached. With the use of a special treatment, so called method of weighted histograms, we process the distribution energy of the vortex system. The results obtained demonstrate that the thermodynamical equilibrium state is reached.


Quantum vortices Superfluid turbulence Vortex tangle Langevin dynamics Thermal equilibrium 


47.37.+q 98.80.Cq 


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  1. 1.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Claberson Press, Oxford, 1992) Google Scholar
  2. 2.
    S.-K. Ma, Modern Theory of Critical Phenomena (Westview Press, New York, 2000) Google Scholar
  3. 3.
    M.P. Solf, T.A. Vilgis, Phys. Rev. E 55, 3037–3043 (1997) CrossRefADSGoogle Scholar
  4. 4.
    W.D. McComb, The Physics of Fluid Turbulence (Oxford University Press, London, 1990) Google Scholar
  5. 5.
    R.J. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991) Google Scholar
  6. 6.
    V. Ambegaokar, B.I. Halperin, D.R. Nelson, E.D. Siggia, Phys. Rev. B 21, 1806 (1980) CrossRefADSGoogle Scholar
  7. 7.
    G.A. Williams, Vortex-loop phase transitions in liquid helium, cosmic strings, and high-T c superconductors. Phys. Rev. Lett. 82(6), 1201 (1999) CrossRefADSGoogle Scholar
  8. 8.
    S.K. Nemirovskii, J. Pakleza, W. Poppe, Russ. J. Eng. Thermophys. 3, 369 (1993) Google Scholar
  9. 9.
    S.K. Nemirovskii, W. Fiszdon, Chaotic quantized vortices and hydrodynamic processes superfluid helium. Rev. Mod. Phys. 67(1), 37 (1995) CrossRefADSGoogle Scholar
  10. 10.
    S.K. Nemirovskii, Thermodynamic equilibrium in the system of chaotic quantized vortices in a weakly imperfect Bose gas. Teor. Mat. Phys. 141(1), 141 (2004) Google Scholar
  11. 11.
    S.K. Nemirovskii, Phys. Rev. B 77, 214509 (2008) CrossRefADSGoogle Scholar
  12. 12.
    A.P. Finne, T. Araki, R. Blaauwgeers, V.B. Eltsov, N.B. Kopnin, M. Krusius, L. Skrbek, M. Tsubota, G.E. Volovik, Nature 424, 1022 (2003) CrossRefADSGoogle Scholar
  13. 13.
    R.G.M. Aarts, A Numerical Study of Quantized Vortices in HeII (Tech. Univer., Eindhoven, 1993) Google Scholar
  14. 14.
    L.P. Kondaurova, S.K. Nemirovskii, Full Biot-Savart numerical simulation of vortices in He II. J. Low Temp. Phys. 138(3/4), 555 (2005) CrossRefADSGoogle Scholar
  15. 15.
    A.M. Ferrenberg, R.H. Swendsen, New Monte Carlo technique for studying phase transitions. Phys. Rev. Lett. 61, 2635 (1988) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Sergey K. Nemirovskii
    • 1
    • 2
  • Luiza P. Kondaurova
    • 1
    • 2
  1. 1.Institute of ThermophysicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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