Collective Hard Core Interactions Leave Multiscale Signatures in Number Fluctuation Spectra

Abstract

A full understanding of transport in dense, interacting suspensions requires analysis frameworks sensitive to self and collective dynamics across all relevant spatial and temporal scales. Here we introduce a trajectory-free approach to address this problem based on the power spectral density of particle number fluctuations (N-PSD). By combining colloidal experiments and theory we show that the N-PSD naturally probes behaviour across multiple important dynamic regimes and we fully uncover the mechanistic origins of characteristic spectral scalings and timescales. In particular, we demonstrate that while high-frequency scalings link to self-diffusion, low-frequency scalings sensitively capture long-lived correlations and collective dynamics. In this regime, interactions lead to non-trivial spectral signatures, governed by pairwise particle exchange at small length scales and collective rearrangements over large scales. Our findings thus provide important insight into the effect of interactions on microscopic dynamics and fluctuation phenomena and establish a powerful new tool with which to probe dynamics in complex systems.

Figures (4)

• Figure 1: Interactions modify low-frequency signatures in the power spectral density of particle number fluctuations (a) Sections of microscopy images at low ($\phi=0.02$) and intermediate ($\phi=0.39$) packing fractions, with representative particle counts $N(t)$ for a square box of size $L=25~\mu$m. Power spectral densities of particle number $N(t)$ at (b) $\phi=0.02$ and (c) $\phi=0.39$ with rescaled amplitude and frequency. Points show eight experimental spectra for logarithmically spaced box sizes from $L = 0.3$$\mu$m(dark) to $46.1~\mu$m (light). (Insets) Same data on linear-logarithmic axes, with unscaled frequency axis in (c). Red solid lines show theoretical results from sDFT (without interactions) using $D$ as input. In (c), blue lines are from sDFT with interactions SuppMat and the dashed red line shows the large box size limit using the collective diffusion coefficient $D_{\rm coll}=D(1+\phi)/(1-\phi)^3$Carter2025.
• Figure 2: Persistent particle returns in dilute systems induce low-frequency noise. (a) N-PSD for a single box with $L=0.7\sigma$ from experiment (blue) and Brownian Dynamics simulations (yellow) with (open) and without (closed) artificial particle repositioning. (b) N-PSDs for particle counts in rectangular boxes of fixed aspect ratio ($L_2/L_1=100)$ and variable size from $L_1=0.002\,\sigma$ (dark blue) to $L_1=0.8\,\sigma$ (light blue). Points show experimental data; red line shows sDFT theory (without interactions) SuppMat.
• Figure 3: Single particle return probabilities do not qualitatively change with density. (a) $P_{\mathrm{in}}(t)$ at different $\phi$ for a small box at $L=\sigma$ with the predicted limiting slope $P_{\rm in}(t)=L^2/4\pi D_l t$ (red dot-dashed line). (b) PSD of $N_{\rm tag}(t)$ for particles randomly tagged at density $\phi_{\rm tag}$ in the system at total $\phi = 0.39$. Points show experimental data with increasing percentage of tagged particles from $2~\%$ (pale grey) to $100~\%$ (blue stars); Red lines show sDFT theory (without interactions) using $D$ (solid line) or $D_l$ (dot-dashed). Inset shows the corresponding experimental data for fixed $\phi_{\rm tag}=0.02$ and varying $L$, rescaled as in Fig. \ref{['fig:fig1']}(b). (Red line) sDFT theory without interactions using $D_l$SuppMat.
• Figure 4: Interparticle correlations introduce new scalings at small length scales. (a) Single particle ($P_{\rm in}(t)$, grey points), interparticle ($N_{\rm new}^{(\phi )}(t)$, purple points) and total ($P_{\rm in}(t) + N_{\rm new}^{(\phi )}(t)$, blue stars) correlations in a dense ($\phi=0.39$) experimental system for $L=\,\sigma$. (b) For all packing fractions, $N_{\rm new}^{(\phi )}(t)$ tends to $N_{L}$ at long time as $1/t$.