**7. Conclusions**

In this paper, we have presented a method for reduction of detailed dynamics to less detailed dynamics called Dynamic MaxEnt. The key feature of the method is that conjugate variables are promoted to independent variables and as such they can relax to a quasi-equilibrium in a different way than state variables. While relaxation of the state variables generates the entropy on the lower level of description, relaxation of conjugate variables ensures that the vector field on the higher level becomes tangent to the quasi-equilibrium manifold.

First, in Section 2, the usual MaxEnt is recalled, which gives state variables on the higher (detailed) level as functions of state variables on the lower (less detailed) level. The DynMaxEnt method is then presented in Section 3 including the infinite chain of higher order DynMaxEnt corrections, and it is compared to asymptotic expansions in Section 3.3.3. Then, the method is used on the reduction of dynamics of complex fluids equipped with conformation tensor and Reynolds stress to the Navier–Stokes equations, reduction of hyperbolic heat conduction to the Fourier law, where we again compare the result to the formal asymptotic methods, and reduction of electromagnetohydrodynamics to magnetohydrodynamics. Finally, motivation for the DynMaxEnt method by contact geometry is shown in Section 5.

In summary, this paper contains a relatively straightforward method for reduction from dynamics on a detailed level of description to dynamics on a less detailed level of description.

**Author Contributions:** M.G. stands behind the main ideas for DynMaxEnt, V.K. and M.P. formulated the energetic DynMaxEnt and wrote most of the text. P.V. formulated the application of DynMaxEnt on electromagnetic field.

**Funding:** This work was supported by the Czech Science Foundation, project No. 17-15498Y, and by Charles University Research program No. UNCE/SCI/023. This research has been supported partially by the Natural Sciences and Engineering Research Council of Canada, Grants 3100319 and 3100735.

**Acknowledgments:** We are grateful to Petr Pelech and Ilya Peshkov for discussing the DynMaxEnt method.

**Conflicts of Interest:** The authors declare no conflict of interest.
