*3.2. Higher Order DynMaxEnt*

From the first order DynMaxEnt, we have acquired independent relations among direct and conjugate variables, **x**˜(**y**) and **x**˜ †(**y**, **y**†), but at the same time direct and conjugate variables are linked via the potential ↑Φ. This entails that these two relations cannot be independent and some of the relations have to be violated.

There are two possible remedies of this situation both being iterative: (i) to correct the upper entropy so that the MaxEnt value is indeed a conjugate to the identified relation **x**˜ †(**y**, **y**†) on the lower level; however, this change in entropy means that MaxEnt has to be recalculated including everything that follows as well and the situation repeats itself; (ii) to correct the MaxEnt value of the direct variable so that the direct and conjugate values are now in line with the upper entropy evaluated on the lower level; however, this again modifies the dependent evolution equations on the lower level, in turn the value of upper conjugate variables and the situation repeats.

Although the former correction, the correction of the upper entropy, might seem natural, it turns out that it cannot be a general approach as the corrected entropy would fail to meet the requirement of being an even function with respect to time reversal when the reduced state variables change parity during the transition to a lower level, see Appendix A.1. We shall now develop the latter approach in detail (forming a possibly infinite chain of corrections) and compare it with asymptotic methods for upscaling evolution equations.

To repair the link between **x** and **x**†, which remained broken after the first order DynMaxEnt reduction, an additional correction of the state variables **x** is required,

$$\frac{\partial^{\uparrow}\Phi}{\partial\mathbf{x}}\Big|\_{\mathbf{x}^{(2)}} = \ddot{\mathbf{x}}^{\dagger}(\mathbf{y},\mathbf{y}^{\dagger}).\tag{9}$$

The solution to this equation gives the second iteration of the value of the state variable, **x**˜(2)(**y**, **y**†).

However, is the consistency condition (7) satisfied at **x**˜(2)? Not in general. Therefore, the condition can be regarded as an equation for the second-order iteration of **x**†, **x**˜ †(2). This iterative chain can be summarized as

$$\mathbf{y} \stackrel{\text{MaxEnt}}{\rightarrow} \tilde{\mathbf{x}}(\mathbf{y}) \stackrel{\text{Equation (7)}}{\rightarrow} \tilde{\mathbf{x}}^{\dagger}(\mathbf{y}, \mathbf{y}^{\dagger}) \stackrel{\text{Equation(9)}}{\rightarrow} \tilde{\mathbf{x}}^{(2)}(\mathbf{y}, \mathbf{y}^{\dagger}) \stackrel{\text{Equation(7)}}{\rightarrow} \tilde{\mathbf{x}}^{\dagger(2)}(\mathbf{y}, \mathbf{y}^{\dagger}) \rightarrow \dots,\tag{10}$$

which can continue indefinitely. We have thus found an infinite chain of DynMaxEnt reduction, which leads to evolution equations for the reduced state variables **y** in a closed form. Note that the lower potential corresponds to the chosen step *k* in the infinite chain of DynMaxEnt as <sup>↓</sup>Φ(*k*)(**y**) = <sup>↑</sup>Φ(**x**˜(*k*)(**y**)).

It is now apparent that any correction in this infinite chain of DynMaxEnt reduction leads to another correction as any of the adaptations, be it in direct or conjugate variables or even entropy, yield new disparity. Where should one end the iteration and what is the best choice of a method generating evolution equations on the lower level of description while retaining the thermodynamic structure and knowledge of the equations?

First, it should be mentioned that the aim of dynamic MaxEnt is to arrive at reasonable evolution equations for a specified level of description. We conjecture that the static MaxEnt value of direct variables **x**˜(**y**) is the best choice as it corresponds to the most probable, least biased relation [20] between direct lower and upper state variables. The conjugate variables then follow from the requirement of the dynamics being such that it does not drive the system away from the MaxEnt values of the direct variables, i.e., **x**˜ †(**y**, **y**†). Finally, the lower potential is chosen as the first correction <sup>↓</sup>Φ(2)(**y**) = ↑Φ **x**˜(2)(**y**) . The reason is twofold: (i) a correction of the lower entropy neither affects the static MaxEnt value nor does it explicitly change the evolution equations that are given in terms of direct and conjugate variables (it affects them indirectly via a resulting change of the relation of conjugate to direct variables); (ii) the corrected entropy is more tightly linked to the evolution equations rather than to the static MaxEnt which we aim to extend. Additionally, as we shall see below, in the special case of projection corresponding to relaxation of fast variables, the ↓*S* entropy is simply the upper entropy but where all the effects of fast variables are neglected while the correction ↓*S* (2) includes additional (typically non-local) effects resulting from this transition between levels.

In short, direct state variables are set-up in such a way that their most probable value (with given information about the system comprised in entropy) is always kept on the lower level (although it might evolve with the evolution of the lower state variables). The choice of conjugate variables entails persistence of the direct variables exactly on their most probable values. Lower entropy corresponds to the set-up of conjugate rather than direct variables, which contain some non-trivial effects due to the transition of scales.
