3.3.2. Second Order DynMaxEnt

By solving equation (14), we have actually broken the link between **<sup>p</sup>** and **<sup>p</sup>**†, namely **<sup>p</sup>**† <sup>=</sup> *<sup>∂</sup>*↑*e*/*∂***p**. To recover that link, we have to find a new value of **p** that corresponds to the obtained *p*˜†,

$$\frac{\partial \mathbf{p}^{(2)}}{\partial t} = \frac{\partial \uparrow\_{\mathcal{E}}}{\partial \mathbf{p}}\Big|\_{\mathbf{p}^{(2)}} = -\tau \frac{\partial \uparrow\_{\mathcal{E}}}{\partial \mathbf{s}} \frac{\partial \uparrow\_{\mathcal{E}}}{\partial \mathbf{r}} = -\tau \frac{\partial \varepsilon}{\partial \mathbf{s}} \frac{\partial V}{\partial \mathbf{r}}.\tag{18}$$

Plugging this new value of **p** into energy ↑*e* gives a new energy on the lower level,

$$\varepsilon^{\downarrow} \varepsilon^{(2)} = \frac{m}{2} \left( \tau \frac{\partial \varepsilon}{\partial s} \frac{\partial V}{\partial \mathbf{r}} \right)^2 + V(\mathbf{r}) + \varepsilon(s), \tag{19}$$

which can be seen as a weakly nonlocal correction of the MaxEnt lower energy ↓*e*.

Once the link between **p** and **p**† through derivative of energy (Legendre transformation) has been recovered by updating **p**˜ to **p**˜ (2), condition (14) no longer holds. To satisfy the condition, we have to find a new value of the conjugate momentum by solving (assuming that *τ∂ε*/*∂s* = *ζ* = const for simplicity)

$$\begin{split} \dot{\mathbf{p}}^{(2)} &= \ -m\zeta \frac{\partial^2 V}{\partial \mathbf{r}^2} \dot{\mathbf{r}} = \ -m\zeta \frac{\partial^2 V}{\partial \mathbf{r}^2} \dot{\mathbf{p}}^{\dagger(2)} \\ &= \ -\mathbf{r}^\dagger - \frac{1}{\zeta} \dot{\mathbf{p}}^{\dagger(2)} = -\frac{\partial V}{\partial \mathbf{r}} - \frac{1}{\zeta} \dot{\mathbf{p}}^{\dagger(2)}. \end{split} \tag{20}$$

The solution to this equation is

$$\tilde{\mathbf{p}}^{\dagger(2)} = \frac{-\tilde{\zeta}\frac{\partial V}{\partial \mathbf{r}}}{1 - m\tilde{\zeta}^2 \frac{\partial^2 V}{\partial \mathbf{r}^2}} \sim -\tilde{\zeta}\frac{\partial V}{\partial \mathbf{r}} - m\tilde{\zeta}^3 \frac{\partial V}{\partial \mathbf{r}^2} + O(\zeta)^5. \tag{21}$$

The latter contribution is obviously of the second order in the relaxation time *τ* while the former contribution is identical to the **p**˜ † found above. This second-order correction of conjugate momentum can be plugged back into the equations for **r**˙ and *s*˙ to obtain

$$\dot{\mathbf{r}} = \frac{-\zeta \frac{\partial V}{\partial \mathbf{r}}}{1 - m\_5^{r} \frac{\partial^2 V}{\partial \mathbf{r}^2}} \,' \tag{22a}$$

$$\dot{s}^{\prime} = \frac{\frac{\mathcal{T}}{\mathcal{T}} \quad \frac{\left(\frac{\partial V}{\partial \mathbf{r}}\right)^2}{\mathcal{S}^{\dagger}} \frac{1}{\left(1 - m\_{\circ}^{\prime 2} \frac{\partial^2 V}{\partial \mathbf{r}^2}\right)^2} \,\prime \,\tag{22b}$$

which is the second-order DynMaxEnt reduction. It is instructive to compare asymptotic expansion of the evolution equations for small *τ*/*ζ* with DynMaxEnt.

Note that the singularity suggested by this second order DynMaxEnt is not physically relevant but rather an invitation to a higher order as Equation (20) reveals—independence of **p**†(2) and *∂***r***V* = 0. In addition, if *τ* 1, i.e., *ζ* 1 is small, DynMaxEnt generates a converging sequence of evolution laws where there is no blow-up. Hence, when the parameter *ζ* or *τ* is not small, the "corrections" stemming from higher order DynMaxEnt might be large and significant. Hence, as we shall further explore below, the higher order corrections seem to be mainly relevant when a small parameter is present.

3.3.3. Damped Particle by Asymptotic Expansions

A hallmark example of the usage of asymptotic expansions in statistical physics is the Chapman–Enskog expansion in kinetic theory [2,22].

In general, the two approaches, asymptotic expansion vs. Dynamic MaxEnt, are very different concepts of obtaining description of a system on a coarser level. The asymptotic analysis (such as the method of multiple scales, homogenisation, regular and singular perturbation methods or boundary layer analysis) does not explicitly assess the lower level of description or what exactly does the asymptotic approximation represents, but, from intuition and the dimensionless form of the evolution equations, one can use the presence of a small parameter to relate two levels of description (more precisely, two different spatial or temporal scales). The number of state variables and evolution equations are typically not changed (although in Chapman–Enskog analysis, evolution equations for several moments of the distribution function somewhat naturally appear) and a suitable choice of the form of expansion series allows a nested problem formulation on the coarser level by sequentially solving each asymptotic order. Note, however, that there is not a unique way to include the small parameter in spatial scaling, etc.

Dynamic MaxEnt, on the other hand, uses projections to the lower level of description (e.g., by relaxation of fast variables). In particular, the number of evolution equations (state variables) can be very different on the two levels in contrast to asymptotic analysis.

Let us compare the asymptotic method of solution with the Dynamic MaxEnt method on the simple problem of a damped particle where the dimensionless parameter 1/*τ* measures the ratio of the reversible and irreversible evolution. Typically, the timescale of relaxation of dissipative processes is much shorter than the remaining evolution, hence *τ* 1. The solution to the problem can be searched in the form of asymptotic expansions

$$\mathbf{p} = \mathbf{p}\_0 + \tau \mathbf{p}\_1 + \tau^2 \mathbf{p}\_2 + O(\tau^3), \ \mathbf{r} = \mathbf{r}\_0 + \tau \mathbf{r}\_1 + O(\tau^2), \ s = s\_0 + \tau s\_1 + O(\tau^2) \tag{23}$$

and the conjugate **r**† variable has the following expansion

$$\mathbf{r}^{\dagger} = V\_{\mathbf{r}}(\mathbf{r}\_0 + \tau \mathbf{r}\_1 + \tau^2 \mathbf{r}\_2) = V\_{\mathbf{r}}(\mathbf{r}\_0) + \tau \mathbf{r}\_1 V\_{\mathbf{r}}(\mathbf{r}\_0) + O(\tau^2).$$

Noting that **p**† = **p**/*m*, the leading order solution is

$$
\tau^{-1}: \quad \mathbf{p}\_0 = 0,
$$

while the first subleading order gives

$$\tau^0: \mathbf{p}\_1 = -m\mathbf{s}^\dagger V\_\mathbf{r}(\mathbf{r}\_0), \quad \dot{\mathbf{r}}\_0 = \frac{\mathbf{p}\_0}{m} = 0.1$$

Therefore, **p**<sup>1</sup> is independent of time and the second subleading order reads

$$\mathbf{p}\_2 = -m\mathbf{s}^\dagger V\_\mathbf{r}(\mathbf{r}\_0)\mathbf{r}\_1 - m\mathbf{s}^\dagger \dot{\mathbf{p}}\_1 = -m\mathbf{s}^\dagger V\_\mathbf{r}(\mathbf{r}\_0)\mathbf{r}\_1, \quad \dot{\mathbf{r}}\_1 = \frac{\mathbf{p}\_1}{m} = -\mathbf{s}^\dagger V\_\mathbf{r}(\mathbf{r}\_0), \tag{24}$$

while, finally, **r**˙2 = **p**2/*m*. In summary, the reduced evolution up to the second order can be characterised by the following set of equations:

$$\dot{\mathbf{r}}\_0 = 0, \quad \dot{\mathbf{r}}\_1 = -\mathbf{s}^\dagger V\_\mathbf{r}(\mathbf{r}\_0), \quad \dot{\mathbf{r}}\_2 = -\mathbf{s}^\dagger V\_\mathbf{r}(\mathbf{r}\_0) \mathbf{r}\_1. \tag{25}$$

From the first subleading order, one could estimate that perhaps **<sup>p</sup>**<sup>0</sup> <sup>=</sup> 0, **<sup>p</sup>**<sup>1</sup> <sup>=</sup> <sup>−</sup>*ms*†**r**† and thus **<sup>r</sup>**˙ <sup>=</sup> <sup>−</sup>*τms*†**r**†. We can now straightforwardly verify this by using the asymptotic expansions (23).

Hence, indeed, we have **<sup>p</sup>** <sup>=</sup> <sup>−</sup>*τms*†**r**† while the remaining equations can be written in the following way:

$$\dot{\mathbf{r}} = -\tau \mathbf{s}^{\dagger} \mathbf{r}^{\dagger},\tag{26a}$$

$$
\dot{\mathbf{s}} = \tau \left(\mathbf{r}^\dagger\right)^2,\tag{26b}
$$

which means that the particle tends to the minimum of the potential *V*(**r**). These equations are in line with Equations (16), obtained by DynMaxEnt. Additionally, the leading order solution in the asymptotic method corresponds to MaxEnt **x**˜(**y**) value (as in the Chapman–Enskog solution to the Boltzmann equation where the leading order solution corresponding to vanishing collision term is the local Maxwell distribution).

Note that we proposed the DynMaxEnt method to be the first order in state variables (both direct and conjugate) accompanied by the second order lower energy <sup>↓</sup>*e*(2). In particular, if we now compare the lower evolution equations from the asymptotic (26) and dynamic MaxEnt (16), we observe that they are identical with the exception stemming from the correction of the lower energy (conjugates in the lower evolution equations are always with respect to the lower energy). Furthermore, the comparison of the asymptotic solution **p**, Equation (24), and of the dynamic MaxEnt result reveals that: (i) the structure of dynamic MaxEnt iteration resembles asymptotic expansion for a small parameter *τ*; (ii) the leading order solutions are the same, however, the subleading terms differ.
