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]

## Contents

* 1 Examples
* 2 Other usage of the term
* 3 See also
* 4 References

## 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]

## Other usage of the term

The term spectral dimension is also used to denote the dimension of the variable in a spectral analysis, therefore it is in that case typically synonymous with the frequency dimension, as in, e.g., the sentence "small instrumental shifts in the spectral dimension" from the wikipedia page on Data binning), or in "where x and y represent two spatial dimensions of the scene, and λ represents the spectral dimension (comprising a range of wavelengths)" from the wikipedia page on Hyperspectral imaging.

## See also

* Dimension
* Fractal dimension
* Hausdorff dimension

## References

1. ↑ 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.
2. ↑ 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.
3. ↑ 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.
4. ↑ 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: 2002PhRvD..65b5013L.
5. ↑ 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.
6. ↑ 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.
7. ↑ 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.
8. ↑ R. Hilfer and A. Blumen (1984) “Renormalisation on Sierpinski-type fractals” J. Phys. A: Math. Gen. 17

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Spectral dimension. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Spectral_dimension&oldid=954394"