Fig. 3: Binding memory in fractal systems with anomalous diffusion.

Source data are provided as a Source Data file. a A liquid system of dynamic binding particles walled by a Hilbert curve. The particles are shown in red. Zooming in on the system does not change the fractal structure. b A different fractal system constructed by two-dimensional diffusion-limited aggregation. Mean squared displacements (MSDs) of particles in both systems are shown in the rightmost subpanels. c Data points represent averages over 60 independent simulations for BAF characterization in the 2D DLA system and 80 for the 2D Hilbert system. Error bars indicate the standard deviation across these simulations. Power-law fitting of the averaged BAFs was conducted over the time window from \(7\times {10}^{5}\) to \(1\times {10}^{7}\). The resulting scaling exponents, together with the corresponding fitting errors, are displayed in the figure, showing that the two fractal binding environments give rise to BAFs with different scaling exponents. Inset: Agreement between simulation and theory on the scaling of BAFs in fractal systems. d Probability density functions (PDFs) of molecular displacements in the Hilbert geometry at two different time points collapse onto a Gaussian form, \(\frac{1}{\sqrt{4\pi {D}_{\alpha }{t}^{\alpha }}}\exp \left(-\frac{{x}^{2}}{4{D}_{\alpha }{t}^{\alpha }}\right)\), with consistent parameters \({D}_{\alpha }=0.048\), \(\alpha=0.69\), confirming \({t}^{\alpha }\)-scaling.