Spectral Dimension
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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 ...
quantity that characterizes a
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
. It characterizes a spread into space over time, e.g. an ink drop
diffusing Molecular diffusion is the motion of atoms, molecules, or other particles of a gas or liquid at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid, size and density (or their product, ...
in a water glass or the evolution of a
pandemic A pandemic ( ) is an epidemic of an infectious disease that has a sudden increase in cases and spreads across a large region, for instance multiple continents or worldwide, affecting a substantial number of individuals. Widespread endemic (epi ...
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 scientific study of matter, its Elementary particle, 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 whi ...
, 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 v ...
,
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, disconnected ...
,
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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
.


Examples

The diffusion of ink in an
isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
homogeneous medium like still water evolves as t^, giving a spectral dimension of 3. Ink in a 2D
Sierpiński triangle The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal with the overall shape of an equilateral triangle, subdivided recursion, recursively into smaller equilateral triangles. Originally constructed as a ...
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


See also

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Dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
*
Fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...
*
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...


References

{{reflist Geometry Diffusion Quantum gravity Power laws