Diffusion in multiscale spacetimes

We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples, and the most general spectral-dimension profile of multifractional spaces is constructed.

Figure 1

Typical single-scale profile of the spectral dimension as a function of the probed length scale. ($A$) Asymptotic regimes where $d_{\mathrm{S}}\sim \mathrm{const}$ (plateaux) and the values of $d_{\mathrm{S}}$ therein are universal or almost universal. ($B$) Also intermediate plateaux, possibly reduced to local extrema, are robust within a given physical system, but different mathematical realizations of the same system cannot produce extra plateaux or transient features such as bumps, glitches, and so on. ($C$) Details of dimensional flow such as the monotonic slope of the profile between different regimes can change with the mathematical realization but they are physically unimportant.

Figure 2

The probability densities (58) (dashed curve) and $\tilde{P}=u_{\beta }/v_{\alpha }\left(x\right)$ (solid curve) with $\beta =1/2=\alpha$, $x^{\prime }=1$ and $\sigma =1$. The plot of $P=u_{\beta }/\sqrt{v_{\alpha }\left(x\right)v_{\alpha }\left(x^{\prime }\right)}$ is very similar to the one for $\tilde{P}$.

Figure 3

The solution (74) of the quartic diffusion equation (84) ($\beta =1/2$, dashed curve) and $\tilde{P}=u_{1/2}/v_{\alpha }\left(x\right)$ (solid curve) with $D=4,\alpha =1/2,x^{\prime }=1$ and $\sigma =1$. $\tilde{P}$ and $u_{1/2}$ are non-negative definite and can be interpreted as probability distributions.

Figure 4

The spectral dimension $d_{\mathrm{S}}\left(\ell \right)$ of a $D=1$ multifractal space whose stochastic properties are given by a fractional telegraph process.

Figure 5

The spectral dimension $d_{\mathrm{S}}\left(\ell \right)$ in $D=4$ for a multifractional model and normal diffusion ($\beta =1=\gamma$, $d_{\mathrm{S}}=d_{\mathrm{H}}$) in four dimensions, with profile (133) (a) and (135) (b).