Abstract
Considering that the motions of the particles take place on continuous but non-differentiable curves, i.e. on fractals with constant fractal dimension, an extended scale relativity model in its hydrodynamic version is built. In this approach, static (particle in a box and harmonic oscillator) and time-dependent (free particle etc.) systems are analyzed. The static systems can be associated with a coherent fractal fluid (of superconductor or of super-fluid types behavior), whose particles are moving on stationary trajectories. The complex speed field of the fractal fluid proves to be essential: the zero value of the real (differentiable) part specifies the coherence of the fractal fluid, while the non-zero value of the imaginary (non-differentiable or fractal) part selects, through some “quantization” relations, the “stationary” trajectories (that may correspond to the observables from quantum mechanics) of the fractal fluid particles. Moreover, the momentum transfer in the fractal fluid is achieved only through the fractal component of the complex speed field. The free time-dependent systems can be associated with an incoherent fractal fluid, and both the differentiable and fractal components of complex speed field are inhomogeneous in fractal coordinates due to the action of a fractal potential. It exist momentum transfer on both speed components and the “observable” in the form of an uniform motion is generated through a specific mechanism of “vacuum” polarization induced by the same fractal potential. The analysis on the fractal fluid specifies conductive properties in the case of movements synchronization both on differentiable and fractal scales, and convective properties in the absence of synchronization.
Similar content being viewed by others
References
B. Madelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982)
E. Nelson, Quantum Fluctuations (Princeton University Press, Princeton, NY, 1985)
J. Feder, A. Aharony, Fractals in Physics (North Holland, Amsterdam, 1990)
J.F. Gouyet, Physique et structures fractales (Masson, Paris, 1992)
L. Nottale, Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity (World Scientific, Singapore, 1993)
Quantum Mechanics, Diffusion and Chaotic Fractals, edited by M.S. El Naschie, O.E. Rösler, I. Prigogine (Elsevier, Oxford, 1995)
J. Argyris, C. Ciubotariu, G. Mattatis, Chaos Solit. Fract. 12, 1 (2001)
Space-time Physics and Fractality, edited by P. Weibel, G. Ord, G. Rössler (Vienna, New York, Springer, 2005)
C.P. Cristescu, Nonlinear Dynamics and Chaos in Science and Engineering (Bucharest, Academy Publishing House, 2008)
J. Cresson, F. Ben Adda, Chaos Solit. Fract. 19, 1323 (2004)
J. Cresson, J. Math. Anal. Appl. 307, 48 (2005)
G.N. Ord, J. Phys. A 16, 1869 (1983)
M.S. El Naschie, Chaos Solit. Fract. 27, 39 (2006)
M.S. El Naschie, Chaos Solit. Fract. 25, 969 (2005)
L. Nottale, Astron. Astrophys. 327, 867 (1997)
L. Nottale, Chaos Solit. Fract. 9, 1051 (1980)
L. Nottale, Chaos Solit. Fract. 10, 459 (1999)
L. Nottale, Chaos Solit. Fract. 16, 539 (2003)
D. Da Rocha, L. Nottale, Chaos Solit. Fract. 16, 565 (2003)
L. Nottale, Chaos Solit. Fract. 25, 797 (2005)
L. Nottale, M.N. Célérier, T. Lehner, J. Math. Phys. 47, 032303 (2006)
M.N. Célérier, L. Nottale, J. Phys. A:Math. Gen. 37, 931 (2004)
I. Gottlieb, M. Agop, G. Ciobanu, A. Stroe, Chaos Solit. Fract. 30, 380 (2006)
M. Agop, P.D. Ioannou, P. Nica, J. Math. Phys. 46, 062110 (2005)
M. Agop, P.E. Nica, P.D. Ioannou, A. Antici, V.P. Paun, Eur. Phys. J. D 49, 239 (2008)
M. Agop, P. Nica, M. Girtu, Gen. Rel. Grav. 40, 35 (2008)
M. Agop, P. Nica, P.D. Ioannou, O. Malandraki, I. Gavanas-Pahomi, Chaos Solit. Fract. 34, 1704 (2007)
A.G. Agnese, R. Festa, Phys. Lett. A 227, 165 (1997)
E. Schrödinger, Collected Papers on Wave Mechanics (WM Deans, London, 1928)
R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals (Mc Graw Hill, New York, 1965)
F. Halbwachs, Théorie relativiste des fluides à spin (Gauthier-Villars, Paris, 1960)
J. Argyris, C. Marin, C. Ciubotariu, Physics of Gravitation and the Universe (Tehnica-Info and Spiru Haret Publishing Houses, Iasi, 2006)
E.A. Jackson, Perspectives in Nonlinear Dynamics (Cambridge University Press, Cambridge, 1991), Vols. I, II
A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal Processing (Prentince-Hall, 1989)
M. Agop, C. Murgulet, Nonlinear Dynamics, Ball Lightning and Cosmic Structures (Ars Longa Publishing House, Iasi, 2006)
V. Chiroiu, P. Stiuca, L. Munteanu, S. Danescu, Introduction in Nanomechanics (Romanian Academy Publishing House, Bucharest, 2005)
D.K. Ferry, S.M. Goodnick, Transport in Nanostructures (Cambridge University Press, Cambridge, 1997)
L.E. Ballentine, Quantum mechanics. A Modern Development (World Scientific, Singapore, 1998)
A.C. Phillips, Introduction to Quantum Mechanics (John Wiley and Sons, New York, 2003)
H.E. Wilhem, Phys. Rev. D 1, 2278 (1970)
L. Conde, L. Leon, Phys. Plasmas 1, 2441 (1994)
C. Ionita, D.G. Dimitriu, R. Schrittwieser, Int. J. Mass Spectrom. 233, 343 (2004)
A. Nikitorov, V. Ouvarov, Éléments de la théorie des fonctions spéciales (Mir, Moskow, 1974)
I. Alcaide, P.C. Balam, L. Conde, C. Ionita, R. Schrittwieser, Contrib. Plasma Phys. 43, 373 (2003)
P. Nica, P. Vizureanu, M. Agop, S. Gurlui, C. Focsa, N. Forna, P.D. Ioannou, Z. Borsos, Jpn J. Appl. Phys. 48, 066001 (2009)
P. Mora, Phys. Rev. Lett. 90, 185002 (2003)
M. Murakami, Y.G. Kang, K. Nishihara, H. Nishimura, Phys. Plasmas 12, 062706 (2005)
L. Landau, E. Lifshitz, Fluid Mechanics (Butterworth-Heinemann, Oxford, 1987)
L. Spitzer, Physics of Fully Ionized Gases (Wiley, New York, 1962)
I.C.E. Turcu, J.B. Dance, X-Rays from Laser Plasmas (Wiley, Chichester, UK, 1998)
O.C. Zienkievicz, R.L. Taylor, The Finite Element Method (McGraw-Hill, New York, 1991)
S. Gurlui, M. Agop, P. Nica, M. Ziskind, C. Focsa, Phys. Rev. E 78, 062706 (2008)
S.S. Harilal, C.V. Bindhu, M.S. Tillack, F. Najmabadi, A.C. Gaeris, J. Appl. Phys. 93, 2380 (2003)
A.V. Bulgakov, N.M. Bulgakova, J. Phys. D 31, 693 (1998)
P. Cristescu, Nonlinear Dynamics and Chaos. Theoretical Fundaments and Applications (Academy Publishing House, Bucharest, 2008)
A.R. El-Nabulsi, Chaos Solit. Fract. 42, 2384 (2009)
A.R. El-Nabulsi, Chaos Solit. Fract. 42, 2924 (2009)
B.M. Hambly, M.L. Lapidus, Trans. Amer. Math. Soc. 358, 285 (2006)
A. Arneodo, F. Argoul, E. Bacry, J.F. Muzy, Phys. Rev. Lett. 68, 3456 (1992)
A. Arneodo, F. Argoul, J.F. Muzy, M. Tabard, E. Bacry, Fractals 1, 629 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Agop, M., Nica, P., Gurlui, S. et al. Implications of an extended fractal hydrodynamic model. Eur. Phys. J. D 56, 405–419 (2010). https://doi.org/10.1140/epjd/e2009-00304-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjd/e2009-00304-5