3.3.1. First Order DynMaxEnt

From relation (12), it follows that entropy on the higher level reads

$$\mathbf{r}^{\uparrow}\mathbf{s} = \mathbf{s} \left(\mathbf{c} - \frac{\mathbf{p}^{2}}{2m} - V(\mathbf{r})\right),$$

*s*(•) being the inverse function to *ε*(•). Entropy attains maximum (for a given energy and position) at **p**˜ = 0, which is the MaxEnt value of momentum.

The consistency condition (7) becomes in this particular case

$$
\begin{pmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} \mathbf{p}^{\dagger} \\ -\mathbf{r}^{\dagger} - \frac{1}{\tau} \frac{\mathbf{p}^{\dagger}}{\mathbf{r}^{\dagger}} \\ \frac{1}{\tau} \left(\frac{\mathbf{p}^{\dagger}}{\mathbf{r}^{\dagger}}\right)^{2} \end{pmatrix} = \begin{pmatrix} \mathbf{p}^{\dagger} \\ -\mathbf{r}^{\dagger} - \frac{1}{\tau} \frac{\mathbf{p}^{\dagger}}{\mathbf{r}^{\dagger}} \\ \frac{1}{\tau} \left(\frac{\mathbf{p}^{\dagger}}{\mathbf{r}^{\dagger}}\right)^{2} \end{pmatrix},\tag{14}
$$

which can be simplified to

$$0 = -\mathbf{r}^\dagger - \frac{1}{\pi} \frac{\mathbf{p}^\dagger}{s^\dagger}.\tag{15}$$

The solution to this equation is the first iteration of the conjugate momentum, **<sup>p</sup>**˜ † <sup>=</sup> <sup>−</sup>*τs*†**r**†. Plugging this back into the evolution equations leads to

$$\dot{\mathbf{r}} \quad = \ -\tau \mathbf{s}^{\dagger} \mathbf{r}^{\dagger} \,\tag{16a}$$

$$\dot{\mathbf{s}} \quad = \quad \pi(\mathbf{r}^\dagger)^2. \tag{16b}$$

These are the reduced equations obtained by the first order DynMaxEnt procedure. The † symbols denote derivative of a yet unspecified energy on the lower level of description ↓*e*(**r**,*s*). Note that the energy is again conserved and that entropy is produced. To make the reduced evolution equations closed, we should identify the lower energy ↓*e* as the MaxEnt value of the higher-level energy,

$$\mathbf{r}^{\perp}\mathbf{e} = {}^{\uparrow}\mathbf{e}(\mathbf{r}, \vec{p}, \mathbf{s}) = V(\mathbf{r}) + \varepsilon(\mathbf{s}).\tag{17}$$

Equations (16) then gain an explicit form manifesting that the particle tends to the minimum of the potential *V* while producing entropy.
