# Describing chaotic systems

LMU researchers have found indications that chaotic many-body systems of the quantum realm can be described using fluctuating hydrodynamics.

Systems consisting of many interacting small particles can be highly complex and chaotic. Nevertheless, some can be described by simple theories. However, does this also extend to the quantum world? A research team led by Professor Monika Aidelsburger and Professor Immanuel Bloch from the LMU Faculty of Physics and the Max Planck Institute of Quantum Optics investigated this question in relation to quantum many-body systems. They found indications that such systems can be described macroscopically through simple diffusion equations with random noise. The study was recently published in Nature Physics.

“If you want to describe the flow behaviour of water, you don’t need to start with the physics of the individual water molecules. Instead, you can formulate flow equations and analyse the problem on a purely macroscopic basis,” explains Julian Wienand, a doctoral candidate in Monika Aidelsburger’s research team and lead author of the new study. This approach is known as hydrodynamics. However, when we observe the motion of small particles in water, we notice that they are not only carried by the flow, but also undergo small erratic movements known as Brownian motion.

These fluctuations arise as a direct result of random collisions between the particles and individual water molecules. “Because these erratic movements are random, we can describe them as white noise, and hydrodynamics becomes fluctuating hydrodynamics (FHD),” says Wienand. “Remarkably, this FHD theory suggests that, under certain cconditions, the entire behaviour of a system may be determined by a single quantity: the diffusion constant – even though the underlying physics are very complex and chaotic at the microscopic level.” This greatly simplifies the macroscopic description of such systems and eliminates the need for detailed description of microscopic particle interactions.

## Does this also apply to quantum systems?

It is suspected that chaotic systems could generally be described by FHD. But whether and to what extent this also holds for chaotic quantum systems remains largely an open question. The laws of physics that determine how quantum particles interact are fundamentally different from those governing classical particles and are characterised by phenomena like “uncertainty” and “entanglement,” which defy everyday intuition. At the same time, quantum systems are even more difficult to calculate and could therefore particularly benefit from an FHD description.

The research team explored this question by studying the behaviour of chaotic many-body quantum systems under a microscope. To observe the dynamics, the team prepared a quantum system of ultracold caesium atoms in optical lattices in a non-equilibrium initial state and allowed it to evolve freely. “The high resolution of our imaging system allows us to measure not only the average density of the particles in the lattice sites, but also their fluctuations,” says Wienand. “Thus, we were able to measure how the fluctuations and density correlations grew over time and concluded that FHD describes our system both qualitatively and quantitatively.” The researchers consider this to be an important indication that chaotic quantum systems, despite their microscopic complexity, can be described simply as a macroscopic diffusion process – much like Brownian motion.