Spectral dimension
The spectral dimension is a real-valued quantity that characterizes a spacetime geometry and topology. It characterizes a spread into space over time, e.g. a ink drop diffusing in a water glass or the evolution of a pandemic in a population. Its definition is as follow: if a phenomenon spreads as [math]\displaystyle{ t^n }[/math], with [math]\displaystyle{ t }[/math] the time, then the spectral dimension is [math]\displaystyle{ 2n }[/math]. The spectral dimension depends on the topology of the space, e.g., the distribution of neighbors in a population, and the diffusion rate.
In physics, the concept of spectral dimension is used, among other things, in quantum gravity,[1][2][3][4][5] percolation theory, superstring theory,[6] or quantum field theory.[7]
Examples
The diffusion of ink in an isotropic homogeneous medium like still water evolves as [math]\displaystyle{ t^{3/2} }[/math], giving a spectral dimension of 3.
Ink in a 2D Sierpiński triangle diffuses following a more complicated path and thus more slowly, as [math]\displaystyle{ t^{0.6826} }[/math], giving a spectral dimension of 1.3652.[8]
See also
References
- ↑ Ambjørn, J.; Jurkiewicz, J.; Loll, R. (2005-10-20). "The Spectral Dimension of the Universe is Scale Dependent". Physical Review Letters 95 (17): 171301. doi:10.1103/physrevlett.95.171301. ISSN 0031-9007. PMID 16383815. Bibcode: 2005PhRvL..95q1301A.
- ↑ Modesto, Leonardo (2009-11-24). "Fractal spacetime from the area spectrum". Classical and Quantum Gravity 26 (24): 242002. doi:10.1088/0264-9381/26/24/242002. ISSN 0264-9381.
- ↑ Hořava, Petr (2009-04-20). "Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point". Physical Review Letters 102 (16): 161301. doi:10.1103/physrevlett.102.161301. ISSN 0031-9007. PMID 19518693. Bibcode: 2009PhRvL.102p1301H.
- ↑ Lauscher, Oliver; Reuter, Martin (2001). "Ultraviolet fixed point and generalized flow equation of quantum gravity". Physical Review D 65 (2): 025013. doi:10.1103/PhysRevD.65.025013. Bibcode: 2001PhRvD..65b5013L.
- ↑ Lauscher, Oliver; Reuter, Martin (2005). "Fractal spacetime structure in asymptotically safe gravity". Journal of High Energy Physics 2005 (10): 050. doi:10.1088/1126-6708/2005/10/050. Bibcode: 2005JHEP...10..050L.
- ↑ Atick, Joseph J.; Witten, Edward (1988). "The Hagedorn transition and the number of degrees of freedom of string theory". Nuclear Physics B (Elsevier BV) 310 (2): 291–334. doi:10.1016/0550-3213(88)90151-4. ISSN 0550-3213. Bibcode: 1988NuPhB.310..291A.
- ↑ Lauscher, Oliver; Reuter, Martin (2005-10-18). "Fractal spacetime structure in asymptotically safe gravity". Journal of High Energy Physics 2005 (10): 050. doi:10.1088/1126-6708/2005/10/050. ISSN 1029-8479. Bibcode: 2005JHEP...10..050L.
- ↑ R. Hilfer and A. Blumen (1984) “Renormalisation on Sierpinski-type fractals” J. Phys. A: Math. Gen. 17
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