*3.1. First Order DynMaxEnt Reduction*

Here, we propose an extension to the static version of MaxEnt (This extension is based on our previous work in [14]; however, we clarify, extend and elaborate on this method in this article.), which provides in addition to the lower entropy <sup>↓</sup>*S*(**y**) and the inverse projection *π*−1(**y**) the dynamics on the lower level, i.e., reduced evolution equations.

Assuming the evolution on the more microscopic (higher) level can be expressed as

$$\dot{\mathbf{x}}^i = V^i(\mathbf{x}, \mathbf{x}^\dagger),\tag{3}$$

where **x**† are conjugate variables which can be eventually identified with derivatives of energy (being the choice in this paper or e.g., [17–19]), entropy or another thermodynamic potential. Let us denote the yet unspecified functional by <sup>↑</sup>Φ, i.e., *x*† *<sup>i</sup>* = <sup>↑</sup>Φ*x<sup>i</sup>* . Components of the right-hand side of the evolution equations *<sup>V</sup><sup>i</sup>* can be interpreted as elements of a vector field on the manifold of state variables <sup>M</sup>, **<sup>x</sup>** ∈ M. With the inverse relation (we shall use simply **<sup>x</sup>**˜(**y**) instead of *<sup>π</sup>*−1(**y**) hereafter) at hand, which enslaves the microscale state variables **x** in terms of the macroscale (or reduced) variables **y**, we may evaluate the more detailed evolution equations on the MaxEnt manifold as

$$\hat{\boldsymbol{\mathfrak{x}}}^{i} = \frac{\partial \hat{\boldsymbol{\mathfrak{x}}}^{i}}{\partial y^{a}} \boldsymbol{\mathfrak{y}}^{a} = \boldsymbol{\mathcal{V}}^{i} \left( \left. \tilde{\mathbf{x}}(\mathbf{y}), \mathbf{x}^{\dagger} = \mathbf{y}^{\dagger} \cdot \frac{\partial \boldsymbol{\pi}}{\partial \mathbf{x}} \right|\_{\mathbf{x}(\mathbf{y})} \right), \tag{4}$$

noting that

$$\frac{\partial^{\top}\Phi(\bar{\mathbf{x}}(\pi(\mathbf{x})) \big|}{\partial\mathbf{x}^{i}}\Big|\_{\delta(\mathbf{y})} = \frac{\partial^{\top}\Phi}{\partial\mathbf{y}} \cdot \frac{\partial\pi}{\partial\mathbf{x}}\Big|\_{\delta(\mathbf{y})}\tag{5}$$

for ↓Φ(**y**) = ↑Φ(**x**˜(**y**)). By projection *π*, we obtain the reduced equations

$$\mathbf{y}^{a} = \frac{\partial \pi^{a}}{\partial \mathbf{x}^{i}} \Big|\_{\mathbf{x}(\mathbf{y})} \dot{\mathbf{x}}^{i} = \frac{\partial \pi^{a}}{\partial \mathbf{x}^{i}} \Big|\_{\mathbf{x}(\mathbf{y})} V^{i} \left( \ddot{\mathbf{x}}(\mathbf{y}), \mathbf{x}^{\dagger} = \mathbf{y}^{\dagger} \cdot \frac{\partial \pi}{\partial \mathbf{x}} \Big|\_{\mathbf{x}(\mathbf{y})} \right), \tag{6}$$

where it should be noted that the reduced equations have to live on the lower level. To this end, a relation **x**˜ †(**y**, **y**†) has to be identified. By comparing these last two evolution equations, it follows that

$$\frac{\partial \tilde{\mathbf{x}}^i}{\partial y^a} \frac{\partial \pi^a}{\partial \mathbf{x}^j} \Big|\_{\mathbf{x}(\mathbf{y})} V^j \left( \tilde{\mathbf{x}}(\mathbf{y}), \mathbf{x}^\dagger = \mathbf{y}^\dagger \cdot \frac{\partial \pi}{\partial \mathbf{x}} \Big|\_{\mathbf{x}(\mathbf{y})} \right) = V^i(\tilde{\mathbf{x}}(\mathbf{y}), \mathbf{x}^\dagger(\mathbf{y}, \mathbf{y}^\dagger)), \tag{7}$$

which is a consistency condition entailing relations among conjugate variables **x**˜ †(**y**, **y**†). Without this condition, the evolution (4) would leave the manifold of MaxEnt states **x**˜(**y**), the vector field would be sticking out of the MaxEnt manifold <sup>M</sup>˜ . Condition (7) is actually an equation for **<sup>x</sup>**†, the solution of which will be denoted by **x**˜ †(**y**, **y**†). After substitution into Equation (6), we obtain the reduced evolution equations for **y**, which is the result of the first order DynMaxEnt reduction.

The first order DynMaxEnt reduction can be summarized as the sequence

$$\mathbf{y} \stackrel{\text{MaxEnt}}{\rightarrow} \check{\mathbf{x}}(\mathbf{y}) \stackrel{\text{Equation (7)}}{\rightarrow} \check{\mathbf{x}}^{\dagger}(\mathbf{y}, \mathbf{y}^{\dagger}),\tag{8}$$

which ends up in a closed system of equations for the reduced state variables **y**. The lower level potential follows from the projection as ↓Φ(**y**) = ↑Φ(**x**˜(**y**)).

Note, however, that the link tying **x** and **x**† was broken because **x**˜ † was determined by solving Equation (7) instead of differentiating the potential ↑Φ with respect to **x** at **x**˜(**y**). This leads to the higher order DynMaxEnt reduction.
