# Dynamic Maximum Entropy Reduction

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## Abstract

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## 1. Introduction

## 2. Static MaxEnt

## 3. Dynamic MaxEnt

#### 3.1. First Order DynMaxEnt Reduction

#### 3.2. Higher Order DynMaxEnt

#### 3.3. Prototype Example—Damped Particle

#### 3.3.1. First Order DynMaxEnt

#### 3.3.2. Second Order DynMaxEnt

#### 3.3.3. Damped Particle by Asymptotic Expansions

#### 3.3.4. Relation to GENERIC

#### 3.4. Summary of the Dynamic MaxEnt Method

- MaxEnt with constraints corresponding to a chosen projection linking the two levels of description is equivalent to MinEne with the same constraints. This follows from the same argument as used in ([27], Ch 5.1) only with the addition of constraints which, crucially, describe the same manifold in the phase space of $(S,E,\mathbf{x})$.
- It is important to consider direct and conjugate variables to be independent until reaching the Legendre manifold where they are related via appropriate entropy. This stands out during the MaxEnt procedure as noted above and in the contact-geometric formulation in Section 5.
- Static MaxEnt should be seen as not providing a relation among conjugate variables but rather only among the direct ones. This can be appreciated in a special case of projection that corresponds to a relaxation of fast variables in the system, hence removing some of the state variables $\mathbf{x}$ when a transition to a less microscopic description is carried out. For this point of view, it is essential to consider direct and conjugate variables as independent.
- If direct and conjugate variables are not considered independent during MaxEnt and hence MaxEnt would provide MaxEnt values for both reduced and conjugate variables, the dynamics on MaxEnt manifold would be (typically) such that its vector field would be “sticking out”, i.e., driving the dynamics out of the MaxEnt manifold. The correction in finding ${\tilde{\mathbf{x}}}^{*}(\mathbf{y},{\mathbf{y}}^{*})$ is exactly such that the MaxEnt manifold $\tilde{\mathcal{M}}$ in direct variables is never left by the dynamics.
- The distinction of direct and conjugate variables has several crucial benefits. First, it guarantees the thermodynamic structure of the evolution equations (potentials, reversible and irreversible parts of equations, CR-GENERIC or possibly GENERIC structure). Secondly, this distinction of state variables enables to always sustain the MaxEnt, i.e., the most probable, value of the direct variables on the lower description (as discussed above). This cannot be achieved in an asymptotic framework/approach, where such knowledge is not at hand and the solution is searched in the form of asymptotic corrections to a leading order solution.
- Parity of the state variables with respect to time reversal typically changes during the DynMaxEnt reduction. For instance, the momentum, which was initially an odd variable, becomes proportional to the gradient of potential, which is even, or the conformation tensor (which is initially even) becomes proportional to the shear rate, which is odd. Similarly, behavior with respect to space-time transformations (e.g., Galilean boost) also changes. For instance, momentum of a particle, the value of which of course depends on the choice of inertial laboratory frame, becomes independent of Galilean boosts after the reduction (being proportional to gradient of the potential). This is compatible with the multiscale point of view of the studied systems. For instance, if one is able to measure the conjugate entropy flux $\mathbf{w}$ (see Section 4.3) directly, one can also see that Galilean boost affects it, and one should stay on the level of description where $\mathbf{w}$ is among the state variables. On the other hand, if one only measures $\mathbf{w}$ in the relaxed state, where it is proportional to the gradient of temperature, one does not see the effect of Galilean boosts anymore, and one can safely work on the Fourier level, where $\mathbf{w}$ is no longer among the state variables. In summary, the behavior of physical quantities with respect to time reversal and space-time transformations crucially depends on the chosen level of description, and this level-dependent behavior is compatible with the multiscale point of view of physical systems.

## 4. Applications in Continuum Thermodynamics

#### 4.1. Suspension of Hookean Dumbbells

#### 4.1.1. Non-Equilibrium Thermodynamics of Conformation Tensor

#### 4.1.2. DynMaxEnt to Hydrodynamics

#### 4.2. Reynolds Stress

#### 4.2.1. Non-Equilibrium Thermodynamics of Reynolds Stress

#### 4.2.2. DynMaxEnt to Hydrodynamics

#### 4.3. Hyperbolic Heat Conduction

#### 4.3.1. Non-Equilibrium Thermodynamics of Heat

#### 4.3.2. DynMaxEnt to Fourier Heat Conduction

#### 4.4. Magnetohydrodynamics

#### 4.4.1. Non-Equilibrium Thermodynamics of Charged Mixtures

_{left}is the free charge density.

#### 4.4.2. DynMaxEnt to the Displacement Field—Passage to MHD

## 5. Contact Geometry

## 6. Contact GENERIC

**Gibbs manifold**${\mathcal{M}}^{\left(G\right)}$ that is the image of the mapping

**Gibbs–Legendre manifold**${\mathcal{M}}^{\left(GL\right)}$ (in the shorthand notation GL manifold) that is the image of the mapping

#### 6.1. Relation to DynMaxEnt

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MaxEnt | Maximum Entropy Principle |

DynMaxEnt | Dynamic Maximum Entropy Principle |

GENERIC | General Equation for Non-Equilibrium Reversible-Irreversible Coupling |

FM | Fluid Mechanics |

MHD | Magnetohydrodynamics |

CIT | Classical Irreversible Thermodynamics |

## Appendix A.

#### Appendix A.1. Correction of Upper Entropy ^{↑} S

#### Appendix A.1.1. General Case

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

## Appendix B. Fluid Mechanics

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**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 ${}^{\downarrow}S\left(\mathbf{y}\right)$ and a relation $\mathbf{x}={\pi}^{-1}\left(\mathbf{y}\right)$ from composition of $\tilde{\mathbf{x}}\left({\mathbf{y}}^{*}\right)$ and $\tilde{{\mathbf{y}}^{*}}\left(\mathbf{y}\right)$. LT denotes a relation via Legendre transformation, $\pi $ 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.

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Klika, V.; Pavelka, M.; Vágner, P.; Grmela, M.
Dynamic Maximum Entropy Reduction. *Entropy* **2019**, *21*, 715.
https://doi.org/10.3390/e21070715

**AMA Style**

Klika V, Pavelka M, Vágner P, Grmela M.
Dynamic Maximum Entropy Reduction. *Entropy*. 2019; 21(7):715.
https://doi.org/10.3390/e21070715

**Chicago/Turabian Style**

Klika, Václav, Michal Pavelka, Petr Vágner, and Miroslav Grmela.
2019. "Dynamic Maximum Entropy Reduction" *Entropy* 21, no. 7: 715.
https://doi.org/10.3390/e21070715