Stationary $1/f^α$ noise in discrete models of the Kardar-Parisi-Zhang class

Abstract

In discrete models describing growing rough interfaces of the Kardar-Parisi-Zhang universality class, we examine height fluctuations at a fixed site as a function of time in the monolayer unit. For small systems, we show that it is possible to reach the stationary state. We compute the two-time autocorrelation and power spectra independently. The correlation function remains non-exponential and vanishes after a correlation time that diverges with system size. As a result, the power spectra display a lower cutoff that maintains constant power. In the nontrivial frequency regime, we observe $1/f^α$-type scaling with the spectral exponent 5/3. Finite-size scaling reveals that the temporal correlation function follows a dynamic scaling. Our findings, supported by scaling-theoretical arguments, establish that the fluctuations are wide-sense stationary, implying applicability of the Wiener-Khinchin theorem.

Figures (5)

• Figure 1: A typical profile of the height noise signal $\delta h_L(t)$ in the KK model with the system size $L=2^8$. The signal $\delta h_L(t)$ represents the difference of the height at a site and the mean height of all the sites at time $t$. The time unit is considered monolayer, i.e., one Monte Carlo time step is equivalent to $L$ deposition attempts.
• Figure 2: (a) The two-time autocorrelation function $C(\tau, L)$ of the signal $\delta h_L(t)$ [cf. Eqs. (\ref{['corr_def']}) and (\ref{['corr_funct_1']})]. The lag time $\tau$ ranges from 1 to $10^5$, and the signal length $T$ is $2^{22}$. We used 120 independent realizations of the process for the ensemble average. (b) As the system size increases, the zero-lag correlation $C(0, L)$ grows linearly. (c) The data collapse is obtained by plotting $C(\tau, L)/C(0, L)$ with $\tau/L^z$, where $z$ is the dynamic exponent.
• Figure 3: Inset: The power spectrum $S(f, L)$ of the height noise $\delta h_L(t)$ in the KK model. The system size varies from $2^5$ to $2^{10}$. The initial $10^3$ measurements are discarded as transient, and the signal length is $T = 2^{20}$. To reduce noise in the PSD for each curve, we perform an ensemble average over $10^4$ independent realizations. Main panel: The data collapse of the power spectra as shown in the inset [cf. Eqs. (\ref{['eq_psd_1']})-(\ref{['eq_psd_3']})]. The dashed (black) line guides the slope of 5/3.
• Figure 4: The system size scaling of the power in the low frequency regime below the cutoff frequency $\sim L^{-z}$. The lower curve shows the scaling of total power with system size. We numerically find the dynamic exponent $z = 1.51(2)$, which agrees well with the theoretical value 3/2.
• Figure 5: The data collapse of the power spectrum $S(f, L)$ for the signal $\delta h_L(t)$ in different models. The BD and EM models belong to the KPZ universality class, and the results are consistent with the presented framework.