Speed-Fisher Information: Chaos and Irreversibility in Classical and Quantum Dynamics

In the study of how things move and change, scientists have long been fascinated by the difference between order and chaos. In a regular system, like a clock or a planet orbiting a star, the future is predictable; if you know the starting position and the forces at play, you can calculate exactly where the object will be a moment later. In a chaotic system, however, tiny differences in the starting conditions grow rapidly, making long-term prediction impossible. This unpredictability is a hallmark of chaos in the classical world of everyday objects. But when physicists try to apply this same definition to the quantum world of atoms and subatomic particles, the rules seem to break down. In quantum mechanics, the mathematical description of a system evolves in a way that preserves its internal structure, meaning two quantum states that start close together never truly drift apart in the way classical trajectories do. This has made it difficult to find a single, unified way to describe chaos that works for both the large, visible world and the tiny, invisible one.

A team of researchers at Boston University has proposed a new way to look at this problem by focusing not on how states drift apart over time, but on how a system reacts when it is gently pushed. Instead of asking how a system behaves after a long time, they asked what happens when a system is subjected to a slow, repeating change, like a pendulum whose length is slowly lengthened and shortened in a cycle. They found that the way a system absorbs energy during this slow push reveals whether it is chaotic or regular. They introduced a concept they call "speed-Fisher information," which essentially measures how sensitive a system's final state is to the speed at which the push is applied. If the system is regular, the final state remains very close to the starting state regardless of how fast the push happens, as long as the push is slow enough. But if the system is chaotic, the final state becomes increasingly different from the start as the speed of the push increases, and this difference grows in a specific, predictable way.

The researchers showed that this sensitivity is directly linked to the amount of energy the system dissipates, or loses as heat, during the cycle. In a chaotic system, even a very slow push generates a measurable amount of friction, causing the system to heat up. In a regular system, this friction vanishes as the push becomes slower. By measuring this dissipated energy, scientists can determine if a system is chaotic without needing to track every single particle or solve complex equations that describe the system's entire history. This approach works for both classical systems, like a double pendulum, and quantum systems, like collections of atoms or spins. The team demonstrated that for chaotic systems that eventually reach a thermal equilibrium, this energy loss grows in a specific pattern as the duration of the push increases. For regular systems, the energy loss drops away to nothing.

One of the most striking findings of the study is the discovery of a purely quantum behavior that appears when the push happens very quickly, but still within a specific short timeframe. In this regime, the way the system absorbs energy follows a different rule than anything seen in classical physics. This quantum scaling only occurs when the duration of the push is shorter than a fundamental time limit set by the temperature of the system, a scale known as the Planckian time. Below this time limit, the system behaves in a way that has no classical equivalent, showing a much sharper increase in energy absorption. As the push slows down and the duration extends beyond this quantum limit, the system's behavior gradually shifts until it matches the classical pattern of energy loss. This crossover provides a clear window into how quantum chaos transitions into the familiar chaos of the macroscopic world.

To test their ideas, the researchers simulated a variety of systems, ranging from simple models with just a few parts to complex many-body systems with dozens of interacting particles. They looked at classical systems like a double pendulum, which can swing in both regular and chaotic patterns depending on its energy, and quantum systems like chains of interacting spins or atoms in a lattice. In every case, the results matched their predictions. For the regular, predictable systems, the speed-Fisher information remained small and finite. For the chaotic systems, it grew large and followed the specific mathematical patterns they had derived. They also found that the behavior depended on the type of disturbance applied; some disturbances revealed the chaos clearly, while others, if they were too closely related to the system's natural conservation laws, failed to trigger the chaotic response. This suggests that chaos is not just a property of the system itself, but also of how we choose to observe or disturb it.

The work offers a practical tool for experimentalists who study these systems. Because the speed-Fisher information is tied to the dissipated work, it can be measured directly in the lab by observing how much energy a system absorbs during a slow, cyclic change. This is particularly relevant for experiments with ultracold atoms trapped in optical lattices, where researchers can precisely control the interactions and external fields. By measuring the energy absorbed during these controlled cycles, they can determine whether the atoms are behaving in a chaotic or regular manner. The study also clarifies the relationship between chaos and the famous "eigenstate thermalization hypothesis," which suggests that chaotic quantum systems naturally evolve to look like they are in thermal equilibrium. The researchers found that systems satisfying this hypothesis show a robust, predictable increase in energy dissipation, confirming the link between thermalization and chaotic response.

Ultimately, this research provides a unified language for describing chaos across the divide between the classical and quantum worlds. It moves away from the idea that chaos is defined solely by the unpredictability of trajectories over infinite time, and instead focuses on the immediate, measurable response of a system to a gentle nudge. By connecting this response to the fundamental concepts of information and thermodynamics, the study shows that chaos is a tangible, measurable property that manifests as a form of friction. Whether in a swinging pendulum or a cloud of quantum atoms, the signature of chaos is the same: a system that cannot help but dissipate energy when pushed, revealing its inner disorder through the heat it generates.