Appendix A.1.1. General Case

Let us now discuss the possibility to correct the higher level entropy instead of the infinite chain presented above. As a sequence of correction will be produced, we introduce a more detailed notation. Static MaxEnt starts with <sup>↑</sup>*S*(**x**), which we shall now denote as <sup>↑</sup>*S*0. Static MaxEnt provides <sup>↓</sup>*S*0(**y**) and **x**˜ *ME* <sup>0</sup> (**y**) and the reduction of the upper evolution equations to the lower MaxEnt manifold yields **x**˜ <sup>∗</sup> <sup>0</sup> (**y**) (where we rewrote the lower conjugate variables in terms of direct ones **y**<sup>∗</sup> = *∂***<sup>y</sup>** ↓*S*(**y**)).

As generally **x**˜ *ME* <sup>0</sup> (**y**) and **x**˜ <sup>∗</sup> <sup>0</sup> (**y**) are independent and hence likely are not conjugate to each other via ↑*S* evaluated at the lower MaxEnt manifold, we correct the upper entropy so that they are conjugate variables. Let us denote the conjugate counterpart to **x**˜ ∗ <sup>0</sup> (**y**) via <sup>↑</sup>*S* ∗ <sup>0</sup> as **x**˜*evo* <sup>0</sup> (**y**) <sup>=</sup> **<sup>x</sup>**˜ *ME* <sup>0</sup> (**y**). Hence, we find ↑*S* ∗ <sup>1</sup> (or equivalently <sup>↑</sup>*S*<sup>1</sup> via Legendre transformation) such that

$$
\partial\_{\mathbf{x}^\*}\,^\uparrow \mathcal{S}\_1^\*|\_{\mathbf{x}^\* = \mathfrak{x}\_0^\*(\mathbf{y})} = \mathfrak{x}\_0^{ME}(\mathbf{y})\_{\prime\prime}
$$

which may be rewritten as

$$\partial\_{\mathbf{x}^\*} \left( {}^\uparrow S\_1^\* - {}^\uparrow S\_0^\* \right) \Big|\_{\mathbf{x}^\* = \mathfrak{x}\_0^\*(\mathbf{y})} = \mathfrak{x}\_0^{ME}(\mathbf{y}) - \mathfrak{x}\_0^{evo}(\mathbf{y}). \tag{A1}$$

It might be difficult in practice to find such a correction to the upper entropy in general. However, we provide a worked out example below.

The change in the upper entropy ↑*S*<sup>1</sup> entails a change in static MaxEnt value and in everything that follows: **x**˜ *ME* <sup>1</sup> (**y**), **x**˜ <sup>∗</sup> <sup>0</sup> (**y**), <sup>↓</sup>*S*1(**y**) = <sup>↑</sup>*S*1(**x**˜ *ME* <sup>1</sup> (**y**)), **<sup>x</sup>**˜*evo* <sup>1</sup> (**y**). Hence, a sequence of correction is produced and if they are "small" they may provide a converging sequence.

Appendix A.1.2. The Correction of the Upper Energy for the Damped Particle

Let us illustrate the outlined difficulty of the correction of the upper energy on the simple example of a damped particle. This method corrects upper energy so that the direct and conjugate variables are indeed conjugate. The natural choice

$$\, ^\uparrow E\_0 = \frac{(\mathbf{p} + \mathbf{p}\_0^{\text{rev}})^2}{2m} + V(\mathbf{r}) + \varepsilon(\mathbf{s}) = \frac{(\mathbf{p} - m\frac{\tau}{\zeta}V\_\mathbf{r}(\mathbf{r}))^2}{2m} + V(\mathbf{r}) + \varepsilon(\mathbf{s})\tag{A2}$$

entails

$$\mathbf{p}^{\dagger} = \partial\_{\mathbf{p}}{}^{\uparrow} E\_0 = \frac{1}{m} (\mathbf{p} + \mathbf{p}\_0^{\mathrm{evo}})$$

and hence indeed the conjugate value to **p** = 0 is **p**† = <sup>1</sup> *<sup>m</sup>* **<sup>p</sup>***evo* <sup>0</sup> . Similarly,

$$\mathbf{r}^{\dagger} = \partial\_{\mathbf{r}}{}^{\uparrow}E\_{1} = \frac{1}{m}(\mathbf{p} - m\frac{\pi}{\zeta}V\_{\mathbf{r}})(-m\frac{\pi}{\zeta}V\_{\mathbf{r}\mathbf{r}}) + V\_{\mathbf{r}\mathbf{-}}$$

Static MaxEnt of ↑*E*<sup>1</sup> gives (minimum energy for given **r**,*s*)

$$\mathbf{p}\_1^{ME} = m \frac{\mathbf{r}}{\zeta} V\_\mathbf{r}$$

with a conjugate value **p**† = 0 and also **r**† = *V***r**. Note that lower energy is the same as in the previous step <sup>↑</sup>*E*1| **p***ME* 1 = <sup>↑</sup>*E*0| **p***ME* 0 . Finally, the evolution equation yields dependent relations

$$
\partial\_t \mathbf{p}\_1^{ME} = m \frac{\tau}{\zeta} V\_{\text{rr}} \partial\_t \mathbf{r} = m \frac{\tau}{\zeta} V\_{\text{rr}} \mathbf{p}\_1^{\dagger} = -\mathbf{r}^{\dagger} - \frac{\zeta}{\tau} \mathbf{p}\_{1'}^{\dagger}.
$$

requiring that

$$\mathbf{p}\_1^\dagger = -\frac{\tau}{\tilde{\zeta}} \mathbf{r}^\dagger \frac{1}{1 + m \frac{\tau^2}{\tilde{\zeta}^2} V\_{\mathbf{rr}}} \approx \mathbf{p}\_0^\dagger - m \frac{\tau^2}{\tilde{\zeta}^2} V\_{\mathbf{rr}} \mathbf{p}\_0^\dagger.$$

Note that this correction method seems to be working well, cf. (25). Namely, both correction methods of dynamic MaxEnt yield an asymptotic type series solution for small *τ* and, although it does not match the formal asymptotic expansion solution, some similarities can be seen.

However, a second inspection of the proposed upper energy (A2) reveals that energy is no longer an even function with respect to time reversal. One can show that, under the assumption of energy being an analytic function of state variables, the requirement that the conjugate value to **p***ME* is non-zero requires a presence of a linear term of **p** in the corrected energy ↑*E*1. In turn, energy cannot

be even with respect to time reversal when **p** is odd. Additionally, note that a natural term appearing in the correction is of the form (**<sup>p</sup>** <sup>−</sup> **<sup>p</sup>***ME* <sup>0</sup> + **<sup>p</sup>***evo* <sup>0</sup> )2, thus again failing to satisfy the requirement of being even with respect to time reversal once there is a change of parity during the transition (as it is here: the odd parity variable **p** evolves in such a way that it is enslaved to **r**† on the lower level, which is a variable with even parity).
