*3.4. Summary of the Dynamic MaxEnt Method*

A few remarks should be pointed out:


where **w** is among the state variables. On the other hand, if one only measures **w** in the relaxed state, where it is proportional to the gradient of temperature, one does not see the effect of Galilean boosts anymore, and one can safely work on the Fourier level, where **w** is no longer among the state variables. In summary, the behavior of physical quantities with respect to time reversal and space-time transformations crucially depends on the chosen level of description, and this level-dependent behavior is compatible with the multiscale point of view of physical systems.

The comparison to the asymptotic expansion methods provided further insight into the Dynamic MaxEnt method. For example, the asymptotic analysis provides corrections to the solution at each order, whereas Dynamic MaxEnt provides relations for closures/fluxes. In the particular examples above, it may seem that both methods yield the same set of leading order evolution equations but note that this is only when one can explicitly solve the asymptotic problems in each order. As noted above, typically asymptotic methods do not change the number of equations as opposed to Dynamic MaxEnt, where the number is given by the projection. Asymptotic methods also typically do not change parity with respect to time reversal, whereas DynMaxEnt does [28]. Finally, note that Dynamic MaxEnt does not rely per se on the presence of a small parameter but rather on the existence of a projection which is similar but not the same.

In summary, Dynamic MaxEnt does not have (and perhaps should not have) an aim to find other than the leading order evolution equations on a specified level. From all the observations and discussion above, we propose that the most appropriate choice for this aim is to find and keep the static MaxEnt values of direct variables exactly while identifying the conjugate state variables in such a way that evolution (dynamics) stays exactly on this MaxEnt value of direct variables; entropy is modified in such a way as to correspond to conjugate variables and the first correction of the direct variables. It seems that, if the zero-th order asymptotic solution coincides with the static MaxEnt value of the direct state variable **x**˜(**y**), the asymptotic expansion method yields the same leading order behaviour as the Dynamic MaxEnt (although without the thermodynamic structure and new phenomena linked to the change of entropy accompanying the transition of scales). Interestingly, we found in the studied example that even higher order asymptotic solutions seem to be related to the Dynamic MaxEnt method, although it is yet unclear how exactly and will be subjected to further research.
