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

This paper demonstrates that height fluctuations in discrete Kardar-Parisi-Zark models exhibit wide-sense stationary $1/f^{5/3}$ noise in the nontrivial frequency regime, supported by finite-size scaling analysis and the validity of the Wiener-Khinchin theorem.

The Gist

Imagine you are watching a crowd of people trying to build a sandcastle together on a beach. Every few seconds, someone throws a bucket of sand onto a specific spot. Sometimes the sand lands perfectly flat; other times, it piles up high next to a hole, creating a bumpy, uneven surface. This is a simplified version of what physicists call a growing interface.

In the world of physics, there's a famous rulebook (called the Kardar-Parisi-Zhang or KPZ class) that describes how these bumpy surfaces grow, whether it's sand, a thin film of metal, or even the edge of a bacterial colony.

For a long time, scientists have been listening to the "noise" of these growing surfaces. If you record the height of the sand at one specific spot over time, it doesn't just go up steadily; it jiggles and fluctuates. When scientists analyzed this jiggling, they found a strange pattern: the noise followed a 1/f rule.

The Mystery of the "1/f" Noise

Think of 1/f noise like the sound of a busy city street.

• High frequencies are the sharp, quick sounds (like a car horn or a bird chirping).
• Low frequencies are the deep, rumbling sounds (like distant traffic or a heavy truck).

In a "1/f" world, the loudness of the sound is perfectly balanced so that the deep rumbles and the sharp chirps have a specific mathematical relationship. It's the kind of noise found in everything from heartbeats to stock markets. It's so common that it's often called "flicker noise."

The Big Debate: Is the System "Resting" or "Running"?

Here is where the story gets interesting. For decades, scientists studying these growing surfaces (especially in very large systems) thought the noise was non-stationary.

The Analogy:
Imagine a runner on a treadmill.

• Stationary (Resting): The runner is jogging at a steady pace. If you look at their heart rate over the last minute, it's stable. You can predict what it will do next based on what it just did.
• Non-Stationary (Running): The runner is sprinting and getting tired. Their heart rate is constantly changing, rising, and never settling down. If you look at the last minute, it's very different from the minute before.

Previous studies suggested that these growing surfaces were like the sprinting runner. Because the system is so huge (infinite in theory), it never really "settles down." It keeps changing its nature over time, meaning the old rules of physics (specifically the Wiener-Khinchin theorem, which links how things wiggle over time to the sounds they make) didn't seem to apply.

The New Discovery: The "Small System" Breakthrough

The authors of this paper, Rahul Chhimpa and Avinash Chand Yadav, decided to look at the problem from a different angle. Instead of looking at an infinite, impossible-to-measure beach, they looked at small, manageable sandcastles (small systems) and watched them for a very long time.

The Analogy:
Imagine you take a small, contained garden patch and watch the plants grow for a year. Eventually, the garden reaches a "steady state." The plants stop growing wildly and just sway gently in the wind. The garden has settled.

The researchers found that:

1. Small systems CAN settle: If the system is small enough and you wait long enough, the "runner" stops sprinting and starts jogging at a steady pace. The system becomes stationary.
2. The Rules Apply: Once the system settles, the old rules (Wiener-Khinchin) work perfectly again. The "noise" they heard was indeed a stable 1/f noise.
3. The Magic Number: They calculated exactly how the noise behaves and found a specific "spectral exponent" of 5/3. Think of this as the unique "fingerprint" of KPZ noise. It's like finding out that all these different growing surfaces (sand, bacteria, liquid crystals) all hum the exact same musical note, just at different volumes.

Why Does This Matter?

You might ask, "Why do we care if a small sandcastle settles?"

The answer is about perspective.

• The Problem: In real-world experiments (like watching a liquid crystal film), the "correlation time" (the time it takes for the system to settle) is so long that it looks like the system never settles. It looks like the "sprinting runner."
• The Insight: The authors show that this "never settling" behavior is actually an illusion caused by the system being too big or the observation time being too short. If you zoom in on a small piece or wait long enough, the system is stationary.

The Takeaway

This paper is like finding a map that explains why a city looks chaotic from a helicopter (non-stationary) but follows strict traffic laws if you stand on a street corner and watch for an hour (stationary).

They proved that for small systems, the fluctuations of growing surfaces are stable and predictable. They confirmed that the "noise" follows a beautiful, universal mathematical pattern (1/f with a 5/3 exponent) and that the standard laws of physics can be used to understand it.

In short: They showed that even in the chaotic, growing world of rough surfaces, if you look at the right scale for the right amount of time, everything settles down into a rhythmic, predictable dance.

Technical Summary

1. Problem Statement

The paper addresses a fundamental debate regarding the nature of temporal fluctuations in the Kardar-Parisi-Zhang (KPZ) universality class, specifically concerning the stationarity of height fluctuations in growing interfaces.

• The Conflict: Previous studies (e.g., by Takeuchi and Henkel et al.) on KPZ interfaces, particularly in the limit of infinite system size ($L \to \infty$), suggested that the dynamics are non-stationary. These studies observed "aging" behavior, where the two-time autocorrelation function depends on both absolute times rather than just the time difference, and the Power Spectral Density (PSD) lacked a lower cutoff frequency. Consequently, the Wiener-Khinchin theorem (which links autocorrelation to PSD) was considered inapplicable.
• The Gap: It remained unclear whether this non-stationarity is an intrinsic property of the KPZ class or an artifact of finite observation times relative to the diverging correlation time in large systems. The authors hypothesize that for finite (small) systems observed over sufficiently long times, the system can reach a stationary state, allowing for the application of standard spectral analysis.

2. Methodology

The authors employ numerical simulations of discrete lattice models belonging to the KPZ universality class in one spatial dimension ($1+1$ dimensions).

• Models Simulated:
  1. Kim-Kosterlitz (KK) model: A restricted solid-on-solid model with slope constraints.
  2. Ballistic Deposition (BD) model: Particles fall vertically and stick to the first contact.
  3. Etching Model (EM): A deposition version where a column grows and neighbors adjust.
  4. Single-Step (SS) model: A model with groove initial conditions and specific growth probabilities.
• Simulation Setup:
  • Systems of size $L$ with periodic boundary conditions.
  • Time is measured in "monolayer units" (one Monte Carlo step = $L$ deposition attempts).
  • Signal Definition: The authors analyze the height noise at a fixed site $x$: $\delta h_L(t) = h(x, t) - \bar{h}(t)$, where $\bar{h}(t)$ is the spatially averaged height.
• Key Condition: The observation time $T$ is set to be significantly larger than the intrinsic correlation time of the system ($T \gg L^z$, where $z$ is the dynamic exponent). This ensures the system reaches the stationary saturation regime.
• Analysis Techniques:
  • Two-time Autocorrelation: Computed via ensemble and time averaging: $C(\tau, L) = \langle \delta h_L(t) \delta h_L(t+\tau) \rangle$.
  • Power Spectral Density (PSD): Computed via Fast Fourier Transform (FFT) of the time series.
  • Finite-Size Scaling (FSS): Used to collapse data for different system sizes ($L$) to extract critical exponents.

3. Key Contributions

1. Establishment of Stationarity in Finite Systems: The paper demonstrates that while infinite systems exhibit aging (non-stationarity), finite KPZ systems do reach a stationary state if observed long enough ($T \gg L^z$).
2. Validation of Wiener-Khinchin Theorem: By proving stationarity, the authors validate the use of the Wiener-Khinchin theorem for these systems, establishing a direct Fourier relationship between the autocorrelation function and the PSD.
3. Unified Scaling Framework: The authors derive a scaling ansatz that links the temporal correlation function and the power spectrum, showing they are consistent descriptions of the same underlying physics.
4. Extraction of Spectral Exponent: They confirm the spectral exponent $\alpha = 5/3$ for the KPZ class, derived theoretically as $\alpha = 1 + 1/z$ (using $z=3/2$).

4. Results

• Autocorrelation Behavior:
  • The correlation function $C(\tau, L)$ is non-exponential.
  • It exhibits a power-law decay $C(\tau) \sim \tau^{1/z}$ for $\tau \ll L^z$ and vanishes for $\tau \gg L^z$.
  • The zero-lag correlation (variance) scales linearly with system size: $C(0, L) \sim L$.
  • Data Collapse: When plotted as $C(\tau, L)/C(0, L)$ vs. $\tau/L^z$, data for different $L$ collapse onto a single master curve, confirming dynamic scaling.
• Power Spectral Density (PSD):
  • Lower Cutoff: The PSD exhibits a distinct lower cutoff frequency $f_c \sim L^{-z}$. Below this frequency, the power remains constant (white noise behavior).
  • Scaling Regime: In the non-trivial frequency regime ($L^{-z} \ll f \ll 1/2$), the PSD follows a power law: $S(f) \sim f^{-\alpha}$.
  • Exponent Value: The numerical analysis yields a spectral exponent $\alpha \approx 5/3$ (specifically $\alpha z \approx 2.53$ and $z \approx 1.51$, consistent with theoretical $z=3/2$).
  • System Size Dependence: The total power and low-frequency power scale with system size as $P(L) \sim L^{(\alpha-1)z}$.
• Universality: The results hold consistently across all four discrete models (KK, BD, EM, SS), confirming that the stationary 1/f$^\alpha$ noise is a universal feature of the KPZ class, not specific to one model.

5. Significance

• Resolution of Contradictions: The paper reconciles previous conflicting findings. It clarifies that the "non-stationary" behavior observed in earlier works was likely due to insufficient observation times relative to the system size (i.e., the system hadn't reached the stationary limit).
• Theoretical Implication: It confirms that the Wiener-Khinchin theorem is applicable to KPZ interfaces in the stationary regime, allowing for robust spectral analysis of growth fluctuations.
• Experimental Relevance: The authors note that experimental limitations often prevent observing the "lower cutoff" frequency because the correlation time diverges with system size. This work provides a theoretical baseline for interpreting experimental data (e.g., in liquid crystal turbulence or thin film deposition) where finite observation windows might mask the stationary nature of the noise.
• Generalizability: The findings suggest that "aging" is a transient feature of large systems, while "stationarity" is the true asymptotic state for finite systems, offering a clearer path for modeling non-equilibrium growth phenomena.