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Classifying nonequilibrium steady states via invariant manifolds


Lucht,  Jens
Group Non-equilibrium soft matter, Department of Dynamics of Complex Fluids, Max Planck Institute for Dynamics and Self-Organization, Max Planck Society;

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Lucht, J. (2018). Classifying nonequilibrium steady states via invariant manifolds. Bachelor Thesis, Georg-August-Universität, Göttingen.

Cite as: https://hdl.handle.net/21.11116/0000-0001-260C-8
Nonequilibrium steady states (NESS) give rise to nontrivial cyclic probability
that breach detailed balance (DB), and thus it is not clear how to define a potential analog
to the equilibrium case. In this thesis we argue that possibly there is a formal way to
define such a NESS potential for systems describable by a Fokker-Planck equation. DB in
NESS can be restored [1] by mapping the phase space into a parameterized family of non-
intersecting cycles containing the invariant manifolds of the corresponding deterministic,
dynamical system. Transition rates between neighboring cycles are obtained from the
microscopic dynamics, i.e., from the drift and diffusive currents. Since
fluxes between
cycles obey DB, we can integrate over the set of cycles. We present some evidence that
this gives us a nonequilibrium potential which reaches minimum solely for NESS.
The main goal of this thesis is to put forward a tentative theory for deriving a generalized potential function whose extrema identify the NESS. We will present results of a first
numerical test based on two well-known dynamical systems: the van-der-Pol oscillator and
the Brusselator. Our results, although not conclusive, are encouraging.