Normally the word “chaos” evokes a lack of order: a hectic day, a teenager’s bedroom, the tax season. And the physical meaning of chaos is not far off. Something is something that is extremely difficult to predict, like the weather. Chaos allows a small shock (crash of a butterfly wing) to escalate into a major consequence (a typhoon halfway around the world), which explains why weather forecasts for more than a few days in the future may be unreliable. Individual air molecules, which are constantly jumping, are also chaotic – it is almost impossible to determine where a single molecule may be at any given moment.

Now, you may be wondering why anyone would care about the exact location of a single air molecule. But you may be interested in a property separated by a whole bunch of molecules, such as their temperature. Perhaps peacefully, it is the chaotic nature of molecules that allows them to fill a room and reach a single temperature. Individual chaos ultimately creates the collective order.

Being able to use a single number (temperature) to describe a bunch of particles jumping around in a crazy, unpredictable way is extremely appropriate, but it does not always happen. So a team of theoretical physicists at JQI decided to figure out when this description applies.

“The ambitious goal here is to understand how the chaos and universal tendency of most physical systems to achieve thermal equilibrium arises from the fundamental laws of physics,” says JQI Member Victor Galitski, who is also a professor of physics at the University of Maryland. (UMD))

As a first step towards this ambitious goal, Galitski and two colleagues set out to understand what happens when many particles, each of which is chaotic in its own right, come together. For example, the movement of a single ball in an air hockey game, jumping incessantly from the walls, is chaotic. But what happens when many of these pucs are left loose on the table? And moreover, what would happen if the pins obeyed the rules of quantum physics?

In a recently published journal work *Physical review letters*, the team studied this problem of air hockey in the quantum field. They found that the quantum version of the problem (where pucs are really quantum particles like atoms or electrons) was neither regulated nor chaotic, but a bit of both, according to a common way of measuring chaos. Their theory was general enough to describe a range of physical settings, including molecules in a container, a game of quantum air, and electrons jumping around in a disordered metal, such as copper wire on your laptop.

“We always thought it was a problem that was solved a long time ago in some textbooks,” says Yunxiang Liao, a JQI postdoctoral fellow and first author on the paper. “It turns out to be a more difficult problem than we imagined, but the results are also more interesting than we imagined.”

One reason this problem has remained unresolved for so long is that once quantum mechanics enters photography, the usual definitions of chaos do not apply. Classically, the butterfly effect – small changes in initial conditions that cause drastic changes below the line – is often used as a definition. But in quantum mechanics, the notion of initial or final position makes no sense. The uncertainty principle states that the position and velocity of a quantum particle cannot be known exactly at the same time. So the particle trajectory is not very well defined, making it impossible to trace how different initial conditions lead to different results.

One tactic to study quantum chaos is to take something classic chaotic, like a ball bouncing around an air hockey table, and treat it mechanically quantum. Surely, classic chaos needs to be translated. And indeed, yes. But when you place more than one quantum ball, things become less clear.

Classically, if pucks can jump on each other, exchanging energy, they will eventually all reach a single temperature, exposing the collective order of underlying chaos. But if the pucs do not collide with each other, and instead pass each other like ghosts, their energies will never change: hot will stay hot, cold will stay cold, and they never will not reach the same temperature. Since pucks do not interact, the collective order cannot emerge from chaos.

The team threw this ghost air hockey game into the quantum mechanical field expecting the same behavior – chaos for a quantum particle, but not the collective order when there are many. To control this scratch, they chose one of the oldest and most widely used (though not the most intuitive) proofs of quantum chaos.

Quantum particles cannot simply have any energy, the available levels are ‘quantized’, which basically means that they are limited to specific values. In the 1970s, physicists discovered that if quantum particles behaved in predictable ways, their energy levels were completely independent of each other – potential values did not tend to accumulate or propagate, on average. But if the quantum particles were chaotic, the energy levels seemed to dodge each other, propagating in distinctive ways. This aversion to energy level is now often used as one of the definitions of quantum chaos.

Since their hockey balls did not interact, Liao and her associates did not expect them to agree on a temperature, which means they would not see any indication of underlying chaos. Energy levels, they thought, they would not care about each other at all.

Not only did they find theoretical evidence of several levels of disgust, a hallmark of quantum chaos, but they also found that some of the levels tended to clump together rather than bounce back, a new phenomenon they could not explain at all. This simple deceptive problem turned out to be neither commissioned nor chaotic, but a curious combination of the two that had not been seen before.

The team was able to discover this hybrid using an innovative mathematical approach. “In previous numerical studies, researchers were only able to include 20 or 30 particles,” says Liao. “But using our mathematical approach from the theory of random matrices, we can include 500 or more. And this approach also allows us to calculate analytical behavior for a very large system. “

Armed with this mathematical framework and smart interest, researchers are now extending their calculations to gradually allow hockey pucks to interact little by little. “Our preliminary results show that thermalization can occur through spontaneous breakdown of reversibility – the past becomes mathematically distinct from the future,” says Galitski. “We see small concerns growing exponentially and destroying all the remaining signatures of the order. But that’s another story. ”

Reference: “Statistics of the level of many bodies of one-particle quantum chaos” by Yunxiang Liao, Amit Vikram and Victor Galitski, December 18, 2020, *Physical review letters*.

DOI: 10.1103 / PhysRevLett.125.250601

In addition to Liao and Galitski, Amit Vikram, a JQI graduate student in physics at UMD, was an author on paper.