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Physicists in Germany have experimentally confirmed a major scientific theory that has remained unresolved for nearly four decades. Researchers at the University of Würzburg say they have successfully demonstrated that the Kardar-Parisi-Zhang, or KPZ, equation works in two-dimensional systems, a breakthrough that helps explain how many different things grow and evolve over time.
First introduced in 1986, the KPZ equation was designed to describe growth in systems that change in irregular and unpredictable ways. Scientists have applied it to a surprisingly wide range of phenomena, including crystal formation, bacterial growth, flame propagation, and even aspects of machine learning. While the theory has been influential for decades, proving it experimentally in realistic two-dimensional environments remained a major challenge, as detailed in findings published through University of Würzburg.
The main difficulty comes from the nature of growth itself. Many growing systems are “out of equilibrium,” meaning they constantly change in complex and random ways. Tracking those changes in both space and time requires extremely precise measurements, often on ultrashort timescales.
To solve the problem, the researchers built a carefully controlled quantum experiment using a semiconductor made from gallium arsenide. The material was cooled to around minus 269 degrees Celsius and continuously stimulated with a laser. Under those conditions, unusual particles known as polaritons formed inside the material.
Polaritons are temporary hybrid particles that combine light and matter. Because they appear and disappear within picoseconds, they allowed scientists to observe rapid growth behavior in real time. By measuring how the polaritons evolved across the material, the team found that the system followed the predictions of the KPZ model.
The experiment builds on earlier theoretical work proposed in 2015 by physicist Sebastian Diehl and his research group. Scientists in Paris had previously confirmed KPZ behavior in one-dimensional systems in 2022, but extending the proof to two dimensions proved significantly harder.
Researchers say the breakthrough was only possible because of advances in material engineering and quantum control. The team constructed highly reflective mirror layers capable of trapping photons inside a microscopic quantum structure. They also used molecular beam epitaxy, a technique that allows materials to be built atom by atom with extreme precision.
The result strengthens the idea that very different systems may obey the same fundamental mathematical rules when they grow. That universality is one reason the KPZ equation has become so important in modern physics.
Beyond theoretical interest, the findings could improve understanding of non-equilibrium systems more broadly, an area relevant to quantum technologies, advanced materials, and complex computational models.
After nearly 40 years, one of physics’ most influential growth theories now has the experimental evidence scientists had been searching for.