Family-Vicsek universality of the binary intrinsic dimension of nonequilibrium data

TL;DR

The paper shows that the binary intrinsic dimension (BID) of nonequilibrium growth data remains informative after aggressive data reduction. By binarizing height profiles relative to the mean and fitting the Hamming-distance distribution with $P(r)= \frac{\mathcal{C}}{2^{d(r)}} {d(r) \choose r}$ and $d(r) \approx d_0 + d_1 r$, BID extracted via $D_{\mathrm{KL}}(P_{emp}||P)$ with flexible cutoffs captures Family-Vicsek scaling: $d_{\mathrm{BID}}/L^D$ decays as $t^{-\beta'}$ in the growth regime and saturates as $L^{-\\alpha'}$, with exponent relations depending on dimension. In 1D RSOS/RDSD, $\alpha' \approx \alpha$ and $\beta' \approx \beta$, while in 2D the relations become $\alpha' \approx \tfrac{1}{2}\alpha$ and $\beta' \approx \tfrac{1}{2}\beta$, supported by data collapse using FV scaling. The results establish BID as a universal, compression-friendly observable that preserves key correlations in nonequilibrium dynamics and suggests broad applicability to other high-dimensional or quantum systems.

Abstract

The intrinsic dimension (ID) is a powerful tool to detect and quantify correlations from data. Recently, it has been successfully applied to study statistical and many-body systems in equilibrium, yet its application to systems away from equilibrium remains largely unexplored. Here we study the ID of nonequilibrium growth dynamics data, and show that even after reducing these data to binary form, their binary intrinsic dimension (BID) retains essential physical information. Specifically, we find that, akin to the surface width, it exhibits Family-Vicsek dynamical scaling -- a fundamental feature to describe universality in surface roughness phenomena. These findings highlight the ability of the BID to correctly discern key properties and correlations in nonequilibrium data, and open an avenue for alternative characterizations of out-of-equilibrium dynamics.

Figures (13)

• Figure 1: Typical surface profiles (rough curves) of a nonequilibrium discrete growth process at different times, juxtaposed with their binary representation (rows of light and dark boxes, representing $+1$ and $-1$ values, respectively). Such a binary representation is obtained by checking if the value of each height variable $h_i$ is below or above (or equal to) the average height $\overline{h}$ of the given surface profile [see Eq. \ref{['eq:binary']}].
• Figure 2: Numerical results for the RSOS model ($n=1$) in $D=1$. The upper panels show the scaling behavior of the width $W$ for different system sizes $L$, in both the intermediate dynamical regime (left) and the stationary regime (right). The lower panels show the corresponding plots for $d_\mathrm{BID}/L$, which also follows an intermediate decaying regime as correlations build up in the system (left), and a saturation regime (right). The data shown are computed on datasets with $2000$ samples, at $84$ observation times, which are logarithmically spaced from 1 to $10^4$ (in units of $\overline{h}$). The saturation values are estimated as the average over an ad hoc time window for each system size. The dashed and dotted curves are included for visual reference and not obtained from fitting the data.
• Figure 3: Same as in Fig. \ref{['fig:RSOS_N1']}, but for the RDSD model in $D=1$. The data shown are computed on datasets with $2000$ samples, at $100$ observation times, which are logarithmically spaced from 1 to $2 \times 10^4$ (in units of $\overline{h}$).
• Figure 4: Same as in Fig. \ref{['fig:RSOS_N1']}, but for the RSOS model ($n=1$) in $D=2$. Note that here $d_\mathrm{BID}$ is normalized by $L^2$.
• Figure 5: Collapse of the data displayed in Figs. \ref{['fig:RSOS_N1']} and \ref{['fig:RDSD']}, for the RSOS and RDSD models in $D=1$, respectively. The axes have been rescaled according to the Family-Vicsek scaling relation in Eq. \ref{['eq:FV_BID']}.