Spectral dimension

This article is about the spacetime concept and is not to be confused with the independent variable in spectral analysis.

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. an 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 ${\displaystyle t^{n}}$, with ${\displaystyle t}$ the time, then the spectral dimension is ${\displaystyle 2n}$. 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, 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 ${\displaystyle t^{3/2}}$, giving a spectral dimension of 3.

Ink in a 2D Sierpiński triangle diffuses following a more complicated path and thus more slowly, as ${\displaystyle t^{0.6826}}$, giving a spectral dimension of 1.3652.^[8]

See also

• Dimension
• Fractal dimension
• Hausdorff dimension