**2. Static MaxEnt**

As a thorough understanding of (static) maximum entropy (MaxEnt) method is required, we shall recapitulate its key steps, what it means and what it provides.

Let us denote the state variables on the more microscopic level as **x** ∈ M, M being the manifold (often vector space) of the state variables, with conjugate variables **x**<sup>∗</sup> via entropy <sup>↑</sup> *S*. Furthermore, let us assume that there is a projection that relates two sets of variables **x** ∈ M, **y** ∈ N via *π*(**x**) = **y** where the latter corresponds to the more macroscopic level of description. The static MaxEnt provides an inverse mapping **x**˜(**y**) such that **x**˜(**y**) is the point of preimage *π*−1(**y**) with the highest entropy. This determines a manifold in <sup>M</sup>, the MaxEnt manifold <sup>M</sup>˜ .

The inverse mapping can be achieved by maximization of entropy while keeping the constraint that *π*(**x**) = **y**, i.e., by the method of Lagrange multipliers. More geometrically, it can be also done via two consecutive Legendre transformations which together correspond to maximisation with a constraint **y** = *π*(**x**) [14,16]. In particular, as a first step, one obtains a relation **x**˜(**y**∗) from solving

$$\mathbf{0} = \partial\_{\mathbf{x}} \underbrace{\left(-{}^{\uparrow}S(\mathbf{x}) + \langle \mathbf{y}^{\*}, \pi(\mathbf{x}) \rangle\right)}\_{\uparrow\downarrow} \tag{1}$$

while the lower conjugate entropy is ↓*S* ∗ (**y**∗) = ↑*φ*(**x**˜(**y**∗), **y**∗). Note that this (actually generalized) Legendre transformation can not be inverted.

Next, Legendre transformation of the lower conjugate entropy provides lower entropy and a relation **y**˜∗(**y**):

$$\mathbf{0} = \partial\_{\mathbf{y}^\*} \underbrace{\left( -^{\downarrow} \boldsymbol{S}^\*(\mathbf{y}^\*) + \langle \mathbf{y}, \mathbf{y}^\* \rangle \right)}\_{\circ\_{\Phi}^\*} \tag{2}$$

which solution yields **y**˜∗(**y**) and <sup>↓</sup>*S*(**y**) = <sup>↓</sup>*φ*∗(**y**, **y**˜∗(**y**)). Finally, the "inverse" mapping to the projection **y** = *π*(**x**) is **x** = **x**˜(**y**˜∗(**y**)) := *π*−1(**y**). Note that it is now evident that *π*−<sup>1</sup> is not a one-to-one mapping but rather a mapping identifying the most probable (with respect to the lower entropy) set of values of the microscale variable **x** that correspond to a given macroscale state **y**. We shall refer to this method as MaxEnt and see Figure 1 for a summary of the method.

**Figure 1.** A summary of static MaxEnt highlighting relations between state variables on the higher level and the lower level of description and their conjugates. MaxEnt provides lower entropy ↓*S*(**y**) and a relation **x** = *π*−1(**y**) from composition of **x**˜(**y**∗) and **y**˜∗(**y**). LT denotes a relation via Legendre transformation, *π* stands for a projection from the microscale to the macroscale variables and by an arrow we depict a mapping (written above or below the arrow) that relates the variables in the connected nodes.
