Spectral dimension and diffusion in multiscale spacetimes

Starting from a classical-mechanics stochastic model encoded in a Langevin equation, we derive the natural diffusion equation associated with three classes of multiscale spacetimes (with weighted, ordinary, and “$q$-Poincaré” symmetries). As a consistency check, the same result is obtained by inspecting the propagation of a quantum-mechanical particle in a disordered environment. The solution of the diffusion equation displays a time-dependent diffusion coefficient and represents a probabilistic process, classified according to the statistics of the noise in the Langevin equation. We thus illustrate, also with pictorial aids, how spacetime geometries can be more completely catalogued not only through their Hausdorff and spectral dimension, but also by a stochastic process. The spectral dimension of multifractional spacetimes is then computed and compared with what was found in previous studies, where a diffusion equation with some open issues was assumed rather than derived. These issues are here discussed and solved, and they point towards the model with $q$-Poincaré symmetries.

Figure 1

Dark (black) curve: Example of trajectory of the process (71) in a fractional spacetime with weighted Laplacian, diffusion measure weight (58), $\nu =1$, and $\beta =1/2$ (top panel), $\beta =1$ (middle panel, ordinary Brownian motion), and $\beta =3/2$ (bottom panel). Light (blue) curve: Example of trajectory of ordinary Brownian motion in ordinary spacetime, plotted for reference (notice the different scaling of the vertical axes).

Figure 2

Thick (black) curve: Example of trajectory of scaled Brownian motion (76) in ordinary spacetime with $\nu =1/2$ (top panel), $\nu =1$ (middle panel, ordinary Brownian motion), and $\nu =3/2$ (bottom panel). The same trajectories represent the process (86) in a fractional spacetime with ordinary Laplacian with $\nu =1$ and, respectively, $\beta =3/2$, 1, $1/2$. Thin (blue) curve: Example of trajectory of ordinary Brownian motion in ordinary spacetime, plotted for reference (notice the different scaling of the vertical axes). The code implements the algorithm presented in Eq. (A2) of 38.

Figure 3

Thick (black) curve: Example of trajectory of the process (101) for the diffusion measure weight (58), with $\beta =1=\alpha$ (top panel, ordinary Brownian motion in ordinary spacetime) and $\beta =1/2=\alpha$ (bottom panel, fractional spacetime with $q$-Laplacian). Thin (blue) curve: Example of trajectory of ordinary Brownian motion in ordinary spacetime, plotted for reference (notice the different scaling of the vertical axes).