The spectral dimension is a

real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...

quantity that characterizes a

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

and

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

. It characterizes a spread into space over time, e.g. a ink drop

diffusing Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of ...

in a water glass or the evolution of a pandemic in a population. Its definition is as follow: if a phenomenon spreads as

t^n

, with

t

the time, then the spectral dimension is

2n

. 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 Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, the concept of spectral dimension is used, among other things, in quantum gravity,

percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnecte ...

superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string t ...

, or quantum field theory.

Examples

The diffusion of ink in an isotropic homogeneous medium like still water evolves as

t^

, giving a spectral dimension of 3. Ink in a 2D

Sierpiński triangle The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equi ...

diffuses following a more complicated path and thus more slowly, as

t^

, giving a spectral dimension of 1.3652.R. Hilfer and A. Blumen (1984) “Renormalisation on Sierpinski-type fractals”
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 Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

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

References

{{reflist Geometry Diffusion Quantum gravity Power laws

Examples

Other usage of the term

See also

References