instead of $P_{mix}$. Moreover, maximizing the KSE appears today as a method to describe out of equilibrium complex systems [@monthus2011non], to find natural behaviors [@burda2009localization] or to define optimal diffusion coefficients in diffusion networks. This general observation however provides a possible rationale for selection of stationary states in out-of-equilibrium physics: it seems reasonable that in a physical system with two simultaneous equiprobable possible dynamics, the final stationary state will be closer to the stationary state corresponding to the fastest dynamics (smallest mixing time). Through the link found in this letter, this state will correspond to a state of maximal KSE. If this is true, this would provide a more satisfying rule for selecting stationary states in complex systems such as climate than the maximization of the entropy production, as already suggested in [@mihelich2014maximum].

[**Acknowledgments**]{} Martin Mihelich thanks IDEEX Paris-Saclay for financial support. Quentin Kral was supported by the French National Research Agency (ANR) through contract ANR-2010 BLAN-0505-01 (EXOZODI).


---
abstract: 'Using a scenario of a hybridized mixture of localized bipolarons and conduction electrons, we demonstrate for the latter the simultaneous appearance of a pseudogap and of strong incoherent contributions to their quasi-particle spectrum which arise from phonon shake-off effects. This can be traced back to temporarily fluctuating local lattice deformations, giving rise to a double-peak structure in the pair distribution function, which should be a key feature in testing the origin of these incoherent contributions, recently seen in angle resolved photoemission spectroscopy ($ARPES$).'
address:
- |
    $^{(a)}$ Centre de Recherches sur les Très Basses Températures, Laboratoire Associé á l’Université Joseph Fourier,\
    Centre National de la Recherche Scientifique, BP 166, 38042, Grenoble Cédex 9, France
- |
    $^{(b)}$ Dipartimento di Scienze Fisiche “E.R. Caianie