New insights into broken symmetries

The Lorentz reciprocal theorem can link the behavior of a microswimmer with a complex shape (left) to that of a non-motile object of the same geometry with different surface properties (right).

The Lorentz reciprocal theorem can now be applied to fluids with broken symmetries. Scientists at the Max Planck Institute for Dynamics and Self-Organization (MPI-DS) in Göttingen have found a way to accommodate this classical theorem also in fluids with odd viscosities. Their discovery opens a new way to explore systems with broken symmetries.

Symmetries are fundamental to physics. Generally, a physical process is considered symmetric if it appears identical when viewed as a mirror image or when time is reversed. In hydrodynamics, a consequence is that the flow around a moving object is the same as if the direction of motion is reversed. The invariance of the flow streamlines causes the force on the object to remain unchanged under motion reversal. This is described by the Lorentz reciprocal theorem, which enables researchers to solve even complex fluid dynamics problems.

However, some systems show broken symmetries resulting in a so-called odd viscosity of the fluid. Here, the Lorentz theorem so far could not be applied. The scientists Yuto Hosaka, Ramin Golestanian, and Andrej Vilfan from Department of Living Matter Physics at the MPI-DS now gained new insights into such systems with broken symmetries. “We found a way to extend the theorem to odd viscosity without violation”, reports Yuto Hosaka, first author of the study. “Our generalization allows for a wide range of analytical calculations for fluids, including self-propelled microorganisms in living systems”, he continues.

Novel applications of the Lorentz reciprocal theorem

The researchers used their new insights to analyze the behavior of different microswimmers. Thanks to their generalization, the Lorentz theorem can link the behavior of moving microswimmers to non-motile objects of the same geometry in odd viscosities. Yet, this application is not limited to the hydrodynamics of microswimmers: “Since related reciprocal theorems exist in a wide range of physical fields, our work gives researchers a new tool to explore the systems with broken symmetries”, concludes Hosaka.


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Max-Planck-Institut für Dynamik und Selbstorganisation

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