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
A pandemic () is an epidemic of an infectious disease that has spread across a large region, for instance multiple continents or worldwide, affecting a substantial number of individuals. A widespread endemic (epidemiology), endemic disease wi ...
in a population. Its definition is as follow: if a phenomenon spreads as
, with
the time, then the spectral dimension is
. 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
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
,
percolation theory,
superstring theory,
or
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
.
Examples
The diffusion of ink in an
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
homogeneous medium like still water evolves as
, giving a spectral dimension of 3.
Ink in a 2D
Sierpiński triangle diffuses following a more complicated path and thus more slowly, as
, giving a spectral dimension of 1.3652.
[R. Hilfer and A. Blumen (1984) “Renormalisation on Sierpinski-type fractals”](_blank)
J. Phys. A: Math. Gen. 17
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 th
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 th
wikipedia page on Hyperspectral imaging
See also
*
Dimension
*
Fractal dimension
*
Hausdorff dimension
References
{{reflist
Geometry
Diffusion
Quantum gravity
Power laws