Spectral dimension

This article is about the spacetime concept; it 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 $t^n$, 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, the concept of spectral dimension is used, among other things, in quantum gravity, percolation theory, superstring theory, or quantum field theory.

Examples

The diffusion of ink in an isotropic homogeneous medium like still water evolves as $t^{3/2}$, giving a spectral dimension of 3.

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

See also

• Dimension
• Fractal dimension
• Hausdorff dimension