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Résumé |
Active systems are made up of elementary agents that are able to self-propel, ``microscopically'' driving the
system out-of-equilibrium by doing so. The collective behaviors of such systems---of
which collective motion or motility induced phase separation (MIPS) are emblematic instances---are best
described by field theories. At this large-scale long-time level, the irreversibility of these
systems can be subtle to capture.
To fill this gap, we introduce a functional generalization of the exterior derivative, which is itself a
generalization of the curl operator to (finite) dimensions higher than 3. This functional exterior
derivative allows to associate to a field dynamics a functional ``vorticity'', or ``cycle affinity''. As soon as this
vorticity vanishes time-reversal symmetry (TRS) is effectively restored at the
macroscopic scale, which implies that all the machinery of equilibrium statistical field theory can be applied
to determine e.g. the phase diagram.
Whenever the dynamics of the system is macroscopically irreversible, the functional vorticity is not identically
zero and provides a valuable insight on the way TRS is broken, from the shape of the
stationary probability current to the counterparts of this current in the physical space. For instance, in the
case of the active model B -- that is a minimal field theoretic description of MIPS -- the
latter correspond to the permanent excitation of anisotropic, propagating modes that are localized at the
liquid-gas interface.
Furthermore, for stochastic vector fields over a one-dimensional space, we exhibit a basis of functional
vorticities, the elements of which can be seen as independent sources of irreversibility on
which the entropy production rate can be decomposed. In addition, these basis elements allow classifying the
potential out-of-equilibrium behaviours the field theory can display.
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