Abstract
Turbulent excitations have been observed in superfluid liquid helium, a Bose Einstein system, that obeys the Kolmogoroc \(\frac{5}{3}\) scaling law for its energy spectrum. In recent joint work with Kouskik Ray of IACS we show how by regarding superfluid helium as a Bose Einstein quantum field theory with local 3 dimensional scale invariance leads to the Kolmogorov scaling law observed. In order to get local 3 dimensional scale invariance geometrical methods are needed, while in order to derive the observed turbulence scaling law the Bose Einstein condensation description of superfluid helium and Zakharov’s weak wave turbulence method are used.
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References
K. Ray, S. Sen, Scale invariance and superfluid turbulence. Nucl. Phys. B [FS], 637 (2013)
H. Weyl, Gravitation and electricity. Sitzungsber. Preuss. Akad. Berlin 465 (1918) (Collected in [3])
L. O’Raifeartaigh, The Dawning of Gauge Theory, Princeton Series in Physics (1997)
F. London, Quantum-mechanical interpretation of Weyl’s theory. Zeit. F. Phys. 42, 375 (1927)
M. Tsubota, Quantum turbulence. J. Phys. Soc. Jpn. 77, 111006 (2008) arXiv:0806.2737(cond-mat)
M. Paoletti et al., Velocity statistics distinguish quantum turbulence from classical turbulence. Phys. Rev. Lett. 101, 154501 (2008)
E. Gross, Structure of a quantized vortex in boson systems, Il. Nuovo Cimento 20, 454 (1961)
L. Pitaevskii, Vortex Lines in an Imperfect Bose Gas, Soviet Physics JETP-USSR, vol. 13, pp. 451 (American Institute of Physics, New York, 1961)
K. Schwarz, Three-dimensional vortex dynamics in superfluid \({}^4\text{ He }\): line-line and line-boundary interactions. Phys. Rev. B 31, 5782 (1985)
D. Kivotides et al., Velocity spectra of superfluid turbulence. Europhys. Lett. 57, 845 (2002)
M. Kobayashi, M. Tsubota, Kolmogorov spectrum of superfluid turbulence: numerical analysis of the Gross-Pitaevskii equation with a small-scale dissipation. Phys. Rev. Lett. 94, 065302 (2005)
C. Connaughton, S. Sen, Local scale invariance and weak wave Turbulenc (Unpublished)
T. Padmanabhan, Conformal invariance, gravity and massive gauge theories. Class. Quantum Grav. 2, L105 (1985)
A. Iorio, L. O’Raifeartaigh, I. Sachs et al., Weyl gauging and conformal invariance. Nucl. Phys. B 495, 433 (1997) [hep-th/9607110]
S. Musher, A. Rubenchik, V. Zakharov, Weak Langmuir turbulence. Phys. Rep. 252, 177 (1995)
M. Rakowski, S. Sen, Quantum kinetic equation in weak turbulence. Phys. Rev. E 53, 586590 (1996). arXiv:cond-mat/9510107
V. Zakharov, V. L’vov, G. Falkovich, Kolmogorov Spectra of Turbulence (Springer, New York, 1992)
A.M. Balk, On the Kolmogorov Zakharov spectra of weak turbulence. Phys. D 139, 137157 (2000)
D. Sanyal, S. Sen, Quantum weak turbulence. Ann. Phys. 321 1327 (2006). arXiv:cond-mat/0402395
Acknowledgments
SS acknowledges the hospitality of the Department of Theoretical Physics, IACS, during the period this work and Koushik Ray for an enjoyable collaboration.
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Sen, S. (2015). Bose Einstein Condensation, Geometry of Local Scale Invariance, and Turbulence. In: Sarkar, S., Basu, U., De, S. (eds) Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 146. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2547-8_5
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DOI: https://doi.org/10.1007/978-81-322-2547-8_5
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