🌀 nonlinear sciences

Dynamical phase transition for a logarithm observable in chaotic systems

This paper investigates the full distribution of a logarithmic dynamical observable in chaotic maps, revealing that while typical fluctuations are Gaussian, the left tail exhibits an anomalous $N^{1/3}$ scaling with a universal large-deviation function that undergoes a condensation-like phase transition at a critical value, separating a typical phase from one where a few trajectory points dominate the fluctuation.

Original authors: Lucianno Defaveri, Naftali R. Smith

Published 2026-10-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Lucianno Defaveri, Naftali R. Smith

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of complex systems, scientists often look for patterns in chaos. They examine how systems that appear random and unpredictable, like the swirling of a storm or the fluctuation of a stock market, actually follow strict rules. A central idea in this field is that while individual events are hard to predict, the overall behavior of a system over a long time tends to settle into a predictable average. However, nature is full of rare, extreme events that defy these averages. These are the outliers: the sudden heatwaves, the massive earthquakes, or the catastrophic market crashes. Understanding how these rare events happen, and how likely they are to occur, is crucial for both predicting disasters and understanding the fundamental laws of physics. For decades, researchers have studied these "large deviations" in systems driven by random chance, where the unpredictability comes from the process itself. But many real-world systems, including weather patterns, are not truly random; they are deterministic, meaning their future is entirely fixed by their starting point, yet they still appear chaotic. A lingering question has been whether these deterministic chaotic systems behave the same way as random ones when it comes to extreme fluctuations.

A team of physicists at the Hebrew University of Jerusalem has now explored this question by watching how a simple, chaotic system behaves over time. They focused on a specific type of mathematical model known as a chaotic map, which generates a sequence of numbers based on a fixed rule. They chose three classic examples: the doubling map, the tent map, and the logistic map. These are simple equations that take a number, apply a rule, and produce a new number, which then becomes the input for the next step. The researchers were interested in a specific measurement derived from these sequences: the sum of the negative logarithms of the numbers generated. In plain terms, this measurement grows larger whenever the sequence produces very small numbers. By running these maps for thousands of steps, the team could observe the full range of outcomes, from the most common results to the most extreme, rare ones.

What they found was a clear distinction between the "normal" behavior of the system and its extreme behavior. When the system produced typical results, the fluctuations around the average followed a standard bell curve, just as one would expect from many natural phenomena. The researchers confirmed that the left side of the distribution, representing outcomes smaller than the average, behaved exactly as predicted by established theories for chaotic systems. However, the right side of the distribution, representing outcomes much larger than the average, told a completely different story. Instead of fading away smoothly like a standard bell curve, the probability of these extreme events followed a strange, slower decay. This meant that extreme fluctuations were far more likely to happen than standard theories would suggest.

The key to this anomaly lies in how the system generates these large numbers. In a typical fluctuation, the entire sequence of numbers contributes a little bit to the final result, with each step adding a small amount of variation. But for the extreme events on the right tail, the system behaves differently. The researchers discovered that these massive fluctuations are usually caused by a single, localized event: a brief period where the sequence of numbers drops to an incredibly small value and stays there for a while. Because the measurement is based on the logarithm of these numbers, a tiny value creates a huge spike in the total sum. This is a phenomenon known as the "big jump" principle, where one single event dominates the entire outcome, rather than the whole system contributing equally.

Perhaps the most striking discovery was a sharp transition between these two modes of behavior. As the researchers looked at fluctuations of increasing size, they found a critical point where the system suddenly switches from the "normal" mode, where the whole sequence contributes, to the "condensed" mode, where a single small burst of activity takes over. This transition is not a gradual shift but a sudden change, similar to how water instantly freezes into ice at a specific temperature. The researchers identified this as a dynamical phase transition. Below this critical point, the system behaves like a collection of many small, independent parts. Above it, the system effectively "condenses" its energy into a single, massive fluctuation driven by a tiny fraction of the sequence.

To verify these findings, the team ran extensive computer simulations, generating millions of data points to map out the probability of these rare events. They used advanced statistical techniques to ensure they could see the rarest outcomes clearly. The simulations confirmed that the transition exists and that the mathematical description of the extreme tail is universal, meaning it looks the same for all three types of maps they tested, differing only in a few scaling factors. Interestingly, despite this strange, non-standard behavior in the extreme tails, the researchers found that the overall statistical properties of the system, such as its average and variance, still grew in a simple, linear way as the sequence got longer. This suggests that while the rare events are wild and unpredictable, the system's fundamental structure remains orderly.

This work bridges a gap between the study of random systems and deterministic chaos. It shows that even in a system with no inherent randomness, where every step is determined by the previous one, the rules governing rare events can be just as complex and surprising as those in truly random systems. The discovery of this condensation transition in chaotic maps suggests that such behavior might be a universal feature of many complex systems, from fluid dynamics to financial markets. By understanding how these systems switch from collective behavior to single-event dominance, scientists may gain new insights into predicting and managing the most extreme and dangerous events in the natural world. The study does not claim to solve the problem of predicting specific disasters, but it provides a clearer picture of the underlying mechanics that make such events possible.

Technical Summary: Dynamical Phase Transition for a Logarithm Observable in Chaotic Systems

Problem Statement
This study investigates the full probability distribution $P_N(A)$ of a dynamical observable $A = -\sum_{i=1}^N \ln x_i$ generated by discrete-time, one-dimensional chaotic maps ( $x_{n+1} = f(x_n)$ ). The authors focus on the limit of large trajectory lengths ( $N \gg 1$ ) for three specific chaotic systems: the doubling map, the tent map, and the logistic map. While typical fluctuations of such observables are known to be Gaussian with mean and variance scaling linearly with $N$ , the behavior of rare events (large deviations) in deterministic chaotic systems remains less understood compared to stochastic processes. Specifically, the paper addresses the breakdown of standard Large Deviation Principles (LDP) for unbounded observables and the potential existence of "condensation transitions," where a single localized event dominates the fluctuation.

Methodology
The authors employ a combination of theoretical analysis and numerical simulation:

Theoretical Framework:
- Left Tail ( $A < \langle A \rangle$ ): The study utilizes the Donsker-Varadhan (DV) theory, adapted for discrete chaotic dynamics. The rate function $I(a)$ is derived via the Legendre-Fenchel transform of the Scaled Cumulant Generating Function (SCGF), $\lambda(k)$ .
- SCGF Calculation: $\lambda(k)$ is calculated as the logarithm of the largest eigenvalue of a "tilted" Frobenius-Perron operator, $L_k$ . For $k \le 0$ , the authors solve the eigenvalue problem numerically using the power method. For the logistic map, analytical solutions are found for negative integer $k$ where the eigenvalue corresponds to the largest real root of a polynomial; however, for $k \le -3$ , these roots are determined numerically.
- Right Tail ( $A > \langle A \rangle$ ): Recognizing that the observable $g(x) = -\ln x$ is unbounded from above, the authors analyze the "big-jump" mechanism. They derive the distribution of a single localized excursion (a sequence of small $x_i$ values) and convolve it with the homogeneous Gaussian distribution of the remaining trajectory.
- Scaling Analysis: The authors apply saddle-point approximations to the convolution integral to identify the scaling regime where the transition between homogeneous and condensed phases occurs ( $A - \langle A \rangle \sim N^{2/3}$ ).
Numerical Simulations:
- Importance Sampling: To sample the rare events in the right tail efficiently, the authors use importance-sampling techniques.
- Cumulant Verification: Direct trajectory simulations are used to verify the linear growth of cumulants with $N$ .
- Convergence Checks: Simulations are performed for $N$ ranging from 50 to 1600, with theoretical extrapolations for much larger $N$ to assess convergence rates.

Key Contributions and Results

Left Tail and Typical Fluctuations:
- The left tail of the distribution ( $A < \langle A \rangle$ ) obeys the standard LDP scaling $P_N(A) \sim e^{-NI(A/N)}$ .
- The rate function $I(a)$ is convex, with a minimum at the ensemble average $a^*$ . Near this minimum, $I(a)$ is parabolic, consistent with the Central Limit Theorem.
- The SCGF $\lambda(k)$ is well-defined for $k \le 0$ but diverges for $k > 0$ , reflecting the sub-exponential decay of the right tail.
- Cumulants: Despite the breakdown of LDP for the right tail, the authors demonstrate that all cumulants of $P_N(A)$ grow linearly with $N$ ( $\langle A^n \rangle_c \sim \lambda_n N$ ) for large $N$ . The coefficients $\lambda_n$ are derived from the derivatives of $\lambda(k)$ at $k=0$ .
Right Tail and Anomalous Scaling:
- The right tail ( $A > \langle A \rangle$ ) exhibits anomalous scaling, deviating from the standard LDP form. Instead of $e^{-N}$ , the probability decays as a stretched exponential: $P_N(A) \sim e^{-c\sqrt{A}}$ .
- This behavior is driven by the "big-jump" principle: large fluctuations are dominated by trajectories where the system spends a prolonged time near the fixed point $x=0$ (where $-\ln x$ is large), rather than by collective contributions from the entire sequence.
Dynamical Phase Transition (Condensation):
- The paper identifies a critical value $y_c$ (where $y = (A-\langle A \rangle)/N^{2/3}$ ) marking a dynamical phase transition.
- Subcritical Phase ( $y < y_c$ ): Fluctuations are "homogeneous," arising from contributions across the entire trajectory. The large-deviation function $F(y)$ is parabolic ( $F(y) \propto y^2$ ).
- Supercritical Phase ( $y > y_c$ ): A "condensation" occurs. A small fraction of the trajectory (a localized burst of small $x_i$ ) provides a macroscopic contribution to the fluctuation.
- Singularity: The large-deviation function $F(y)$ exhibits a corner singularity (a jump in the first derivative) at $y_c$ , characteristic of a first-order phase transition. The function takes a universal form across the three maps, differing only by scaling constants.

Significance and Claims
The authors claim to demonstrate that condensation transitions, previously observed in stochastic processes (e.g., Ornstein-Uhlenbeck processes, stochastic resetting) and sums of i.i.d. variables with heavy tails, also occur in deterministic chaotic systems.

Universality: The study suggests that the anomalous scaling and the specific form of the large-deviation function $F(y)$ are universal features for a broad class of systems with unbounded observables and stretched-exponential tails, regardless of whether the underlying dynamics are stochastic or deterministic.
Correlation Effects: The work highlights that temporal correlations in chaotic dynamics significantly affect fluctuations compared to the i.i.d. case, yet the qualitative nature of the condensation transition remains robust.
Cumulant Behavior: A notable finding is that the linear growth of cumulants with $N$ persists even when the standard large-deviation theory (which typically governs the tails) breaks down. This contrasts with the intuition that anomalous tail behavior might disrupt the regular scaling of moments.

The paper concludes that while the analytical methods used are not mathematically rigorous in the strictest sense (relying on numerical eigenvalue solutions and asymptotic approximations), the agreement between the subleading-order theoretical predictions and numerical simulations provides strong evidence for the existence of this dynamical phase transition in chaotic maps.

Technical Summary: Dynamical Phase Transition for a Logarithm Observable in Chaotic Systems

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