zbMATH — the first resource for mathematics

The time dependent Ginzburg-Landau equation in fractal space-time. (English) Zbl 1248.35202
Summary: We use the hydrodynamic formulation of scale relativity theory to analyze the TDGL equation. As a result, London equations come naturally from the system, when equating to zero the real velocity, the imaginary one turns real, the superconducting fluid act as a subquantum medium energy accumulator, the vector potential, the real and the imaginary velocity are all written in terms of the elliptic function. When solving the resulted system by means of WKBJ method, we get tunneling and quantization. In other words, scale transformation laws produce, on the motion equation of particles governed by the TDGL equation, under some peculiar assumptions, effects which are analogous to those of a “macroscopic quantum mechanics”.

35Q56 Ginzburg-Landau equations
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
82D55 Statistical mechanical studies of superconductors
Full Text: DOI
[1] El Naschie, M.S., Chaos solitons fractals, 41, 2635, (2009)
[2] El Naschie, M.S.; Rossler, O.; Prigogine, I., Quantum mechanics, diffusion and chaotic fractals, (1995), Pergamon Oxford · Zbl 0830.58001
[3] Crnjac, M.I., Chaos solitons fractals, 41, 5, 2697, (2009)
[4] Crnjac, M.I., Chaos solitons fractals, 20, 4, 669, (2004)
[5] Iovane, G., Chaos solitons fractals, 20, 4, 657, (2004)
[6] Iovane, G.; Laserra, E.; Tortoriello, F.S, Chaos solitons fractals, 20, 3, 415, (2004)
[7] Nottale, L.; Nottale, L.; Da Rocha, D.; Nottale, L.; Nottale, L., Fractal space – time and microphysics: towards a theory of scale relativity, Int. J. mod. phys. A, Chaos solitons fractals, Chaos solitons fractals, 25, 797, (2005), World Scientific Singapore · Zbl 1069.81025
[8] Mandelbrot, B., The fractal geometry of nature, (1982), Freeman San Francisco · Zbl 0504.28001
[9] Ord, G.N., J. phys A: math. gen., 16, 1869, (1983)
[10] Abbot, L.F.; Wise, M.B., Am. J. phys., 49, 37, (1981)
[11] Feynman, R.P.; Hibbs, A.R., Quantum mechanics and path integrals, (1965), McGraw-Hill New York · Zbl 0176.54902
[12] Buzea, C.Gh.; Bejinariu, C.; Boris, C.; Vizureanu, P.; Agop, M., Int. J. nonlinear sci. numer. simul., 10, 11-12, 1399, (2009)
[13] Ginzburg, V.L.; Landau, L.D., (), 20, 138-167, (1950), English translation
[14] Gorkov, L.P.; Eliashberg, G.M., Zh. eksp. teor. fiz., Soviet phys., JETP, 27, 328, (1968)
[15] Galbreath, N., Parallel solution of the three-dimensional time-dependent ginzburg – landau equation, ()
[16] Calidonna, C.R.; Naddeo, A.; Alicki, R.; Horodecki, M.; Horodecki, P.; Horodecki, R., Phys. lett. A, Open syst. inform. dyn., 11, 205, (2004)
[17] Agop, M.; Ioannou, P.D.; Buzea, C.Gh.; Nica, P., Physica C, 390, 37, (2003)
[18] Poole, C.P.; Farach, H.A.; Geswich, R.J., Superconductivity, (1995), Academic Press San Diego, New York, Boston, London, Sydney, Tokyo, Toronto
[19] Buzea, C.Gh.; Agop, M.; Galusca, G.; Vizureanu, P.; Ionita, I., Chaos solitons fractals, 34, 4, 1060, (2006)
[20] Smirnov, V.I., Analiza matematica, vol. III, (1963), Ed. Tehnica Bucuresti
[21] Bowman, F., Introduction to elliptic functions with applications, (1953), English Univ. Press Ltd. London · Zbl 0052.07102
[22] McCormack, P.D.; Crane, L., Physical, Fluid mechanics, (1973), Academic Press London, New York · Zbl 0289.76001
[23] Nolting, W., Quantenmechanik – methoden und anwendungen, Grundkurs theoret. physik., (2004), Springer
[24] Park, C.-S.; Park, C.S.; Jeong, M.G.; Yoo, S.-K.; Park, D.K., J. Korean phys. soc., 42, 6, 830, (1998)
[25] Landauer, R., IBM J. res. dev., 5, 183, (1961)
[26] Lloyd, S., Phys. rev. A, 39, 5378, (1981)
[27] Popov, V.S.; Karnakov, B.M.; Mur, V.D.; Eltschka, C.; Friedrich, H.; Moritz, M.J.; Trost, J.; Moritz, M.J., Phys. lett. A, Phys. rev. A, Phys. rev. A, 60, 832, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.