*3.3. Prototype Example—Damped Particle*

Let us now demonstrate the DynMaxEnt reduction on a prototype example—damped particle in a potential field. The state variables represent position, momentum and entropy of the particle, **x** = (**r**, **p**,*s*), and momentum is considered as the fast variable, which is to be reduced, i.e., *y* = (**r**,*s*). The projection to the lower (less detailed) level is thus *π* : (**r**, **p**,*s*) → (**r**,*s*).

The more detailed evolution equations are ([14], Ch 5.3.1)

$$
\dot{\mathbf{r}}\_{\cdot} = \mathbf{p}\_{\cdot}^{\dagger},\tag{11a}
$$

$$\dot{\mathbf{p}}\_{\parallel} = \begin{array}{c} -\mathbf{r}^{\dagger} - \frac{1}{\pi} \frac{\mathbf{p}^{\dagger}}{s^{\dagger}}, \end{array} \tag{11b}$$

$$\dot{s}\_{\text{tot}} = \frac{1}{\tau} \left( \frac{\mathbf{p}^{\dagger}}{s^{\dagger}} \right)^{2},\tag{11c}$$

where *x*† can be interpreted as conjugates with respect to energy

$$\mathbf{r}^{\uparrow}\boldsymbol{\varepsilon} = \frac{\mathbf{p}^{2}}{2m} + V(\mathbf{r}) + \boldsymbol{\varepsilon}(\mathbf{s}),\tag{12}$$

consisting of kinetic energy, potential energy and internal energy. The evolution equations represent Hamilton canonical equations (for **r** and **p**) equipped with friction in **p** and entropy production, c.f. [21]. Total energy is conserved by the evolution equations, *e*˙ = **r**†**r**˙ + **p**†**p**˙ + *s*†*s*˙ = 0.
