Physicists from Heinrich Heine University Düsseldorf (HHU), the Technical University of Darmstadt, Sapienza University in Rome and the University of Camerino (both Italy) have calculated the fundamental laws of thermodynamics for a gas made up of spinning particles. In the scientific journal Proceedings of the National Academy of Sciences (PNAS), they demonstrate that the pressure of this gas is similar to that of a normal gas, but at an elevated temperature. In addition, localised surface currents arise, which can be utilised for targeted particle transport.
The so-called ideal gas law is one of the fundamentals of thermodynamics and states that the pressure of a gas is proportional to its density and temperature. From a microscopic perspective, pressure is the average force per unit of area exerted by incoming particles on impact and reflection. At a higher temperature, the particles move more quickly and therefore exert greater pressure on impact.
A research team headed by Professor Dr Hartmut Löwen from the Institute for Theoretical Physics II at HHU has now asked the question whether the ideal gas law also applies for self-propelled, so-called “active” particles or whether the non-equilibrium state results in significant variances. In addition to the physicists in Düsseldorf, colleagues from Rome, Camerino and Darmstadt were also involved.
Professor Löwen comments on the findings published in PNAS: “The answer to this question is both yes and no: Although the ideal gas law no longer applies, we can make the interpretation that an effectively higher temperature exists in the non-equilibrium state.”
The focus lay in particular on “chiral” active particles: self-propelling objects, which do not move in a straight line, but rather drift systematically in an orbit, i.e. they spin. Such particles can often be found in nature – for example algae and rotating acacia seeds – and also in technology, e.g. magnetically propelled colloidal particles that can be utilised to increase and control the stiffness in shock absorbers.
When such a self-rotating object approaches a surface, two further factors contribute to the pressure exerted on the surface in addition to the normal reflection: On the one hand, the motion of the object itself results in increased pressure, known as active swim pressure. On the other hand, the rotation of the particles themselves causes the object to move tangentially on the surface, which reduces the swim pressure.
Professor Löwen: “In analytical terms, it resembles the ideal gas law but with an activity-dependent, effectively higher temperature. It is surprising that there is an equation of state in a non-equilibrium at all; this means that the pressure depends on just a few parameters, but not on the wall properties.”
Professor Dr Lorenzo Caprini from Sapienza University is lead author of the study: “The tangential motion at the walls results in a directed particle flow at the edges, which can be calculated quantitatively. Interestingly, this represents an analogy to so-called topological insulators, which can only carry current on the surface of a structure.
At HHU, the theoretical predictions were confirmed in an experiment involving chiral mini-robots. Doctoral researcher Marco Musacchio was largely responsible for conducting this experiment: “Our experimental setup replicates the rotating systems to optimum effect and allows a quantitative comparison with the theory, which is remarkably successful.”
Co-author Professor Dr Benno Liebchen from Darmstadt adds: “Our findings not only represent the fundamental basis for non-equilibrium thermodynamics of chiral microswimmers. They may also serve as a useful foundation for future applications, for example the creation of flow fields, which are only effective at the system edges and thus enable the efficient transport of e.g. drugs.”
L. Caprini, M. Musacchio, U. M. B. Marconi, B. Liebchen, H. Löwen; Active thermodynamics of inertial chiral active gases: equation of state and edge currents; PNAS 123 (29) e2532158123, 17 July 2026
Proceedings of the National Academy of Sciences
Active thermodynamics of inertial chiral active gases: equation of state and edge currents
17-Jul-2026