Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem

Strong anomalous diffusion, where $⟨|x(t){|}^q⟩\sim t^{q\nu (q)}$ with a nonlinear spectrum $\nu (q)\ne \text{const}$, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomenon is related to infinite covariant densities; i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multifractal anomalous diffusion, as it is complementary to the central limit theorem.

Figure 1

Rescaled PDFs $t^{\alpha }P(x,t)$ (open symbols) versus $x/t$ for the jump model with $\alpha =3/2$, $A=v_0=1$ [42] for three different times. The bold black line depicts the ICD of the process Eq. (11). Error bars are smaller then the size of the symbols. Big filled symbols indicate the locations of the crossover velocities ${\overline{v}}_c$ and horizontal lines are Lévy’s central limit theory for $t^{\alpha }P(x=0,t)$ (see the main text for more details).

Figure 2

ICDs for different Lévy walk models, with $\alpha =3/2$. All densities exhibit characteristic non-integrability on $x/t\to 0$ and collapse for small $x/t$ on a master curve Eq. (17). Open blue squares (velocity model) and black triangles (jump model) are simulation results for $t=10^6$.