Although Navier–Stokes equations are the foundation of modern hydrodynamics, adapting them to quantum systems has so far been a major challenge. Researchers from the Faculty of Physics at the University of Warsaw, Maciej Łebek, M.Sc. and Miłosz Panfil, Ph.D., Prof., have shown that these equations can be generalized to quantum systems, specifically quantum liquids, in which the motion of particles is restricted to one dimension.

This discovery opens up new avenues for research into transport in one-dimensional quantum systems. The resulting paper, published in Physical Review Letters, was awarded an Editors’ Suggestion.

Liquids are among the basic states of matter and play a key role in nature and technology. The equations of hydrodynamics, known as the Navier–Stokes equations, describe their motion and interactions with the environment. Solutions to these equations allow us to predict the behavior of fluids under various conditions, from the ocean currents and the blood flow in blood vessels, to the dynamics of quark-gluon plasma on subatomic scales.

Navier–Stokes equations, formulated in the 19th century based on the principles of conservation of mass, momentum and energy, belong to classical physics. However, the motion of particles is governed by the laws of quantum mechanics, which raises the question of whether these equations can be derived from the principles of quantum physics.

The relationship between hydrodynamics and the microscopic description of the movement of the particles forming a liquid is not only theoretical, but also of practical importance. Navier–Stokes equations contain certain quantities known as transport coefficients, which determine how quickly disturbances in the fluid dissipate, meaning how quickly the system returns to equilibrium.

Their values cannot be deduced without knowledge of the microscopic interactions between the molecules. Deriving these equations from the microscopic laws makes it possible to determine the relationship of the transport coefficients to the interactions between the molecules.

Navier–Stokes equations in quantum systems

The application of the Navier–Stokes equations to quantum systems has so far been a major challenge. The University of Warsaw researchers, from the Faculty of Physics, have addressed this issue in the context of quantum liquids, in which the motion of particles is restricted to one dimension.

Under specific conditions, such systems exhibit quantum integrability, i.e. the presence of multiple conservation laws. This feature has important consequences—it makes it possible to accurately describe the state of the fluid (using the wave function) and in the case of strong interactions between particles.

“In combination with numerous conservation laws, this has allowed the formulation of equations describing the hydrodynamics of these systems, called generalized hydrodynamics. The generalized hydrodynamics equations are much more complex than the Navier–Stokes equations. Despite their complexity, they have been confirmed in experiments with ultracold quantum gases and were the starting point of our work,” explains Łebek, the first author of the paper.

Another difference between Navier–Stokes equations and the generalized hydrodynamics equations is the range of applicability. Navier–Stokes equations hold for most liquids, whereas the generalized hydrodynamics equations apply only to integrable systems.

“In our study, we have taken into account the influence of additional interactions between particles that break integrability. If they are sufficiently weak, the dynamics of the system can still be described by the generalized hydrodynamics equations, supplemented with an additional term describing non-integrable interactions. As a result, the equations take a structure reminiscent of the Boltzmann kinetic equation,” explains Dr. Panfil.

In their paper, the researchers showed that Navier–Stokes equations are derived from generalized hydrodynamics with an additional Boltzmann term, and derived formulae for transport coefficients such as viscosity and thermal conductivity.

“Interestingly, the derived values of these coefficients have two contributions—one from integrable interactions and the other from interactions that break integrability. Classical kinetic theory for weakly interacting liquids predicts zero viscosity, which contradicts experimental results. Our method, on the other hand, provides a viscosity value different from zero, which is due to the subtle interplay between the two types of interactions,” the researcher explains.

Transport in quantum systems

The researchers’ results show that the ideas of hydrodynamics are also applicable in quantum conditions. They are an example of the microscopic derivation of transport coefficients in strongly interacting systems. They also have practical relevance for contemporary experiments on ultracold atoms conducted in laboratories around the world.

The discovery opens up new possibilities for research on transport in one-dimensional quantum systems. In the future, the researchers plan to extend the theory to more complex systems and to experimentally test the model’s predictions.