From a microscopic inertial active matter model to the Schrödinger equation

Michael te Vrugt, Tobias Frohoff-Hülsmann, Eyal Heifetz, Uwe Thiele*, Raphael Wittkowski*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


Active field theories, such as the paradigmatic model known as ‘active model B+’, are simple yet very powerful tools for describing phenomena such as motility-induced phase separation. No comparable theory has been derived yet for the underdamped case. In this work, we introduce active model I+, an extension of active model B+ to particles with inertia. The governing equations of active model I+ are systematically derived from the microscopic Langevin equations. We show that, for underdamped active particles, thermodynamic and mechanical definitions of the velocity field no longer coincide and that the density-dependent swimming speed plays the role of an effective viscosity. Moreover, active model I+ contains an analog of the Schrödinger equation in Madelung form as a limiting case, allowing one to find analoga of the quantum-mechanical tunnel effect and of fuzzy dark matter in active fluids. We investigate the active tunnel effect analytically and via numerical continuation.

Original languageEnglish
Article number1302
JournalNature Communications
Issue number1
StatePublished - Dec 2023


FundersFunder number
Deutsche Forschungsgemeinschaft433682494 – SFB 1459
Studienstiftung des Deutschen Volkes


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