# Kullback–Leibler Divergence of a Freely Cooling Granular Gas

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Free Cooling Evolution of Velocity Cumulants

#### 2.1. Boltzmann Equation and HCS

#### 2.2. Sonine Expansion Formalism

#### 2.3. Truncated Sonine Approximation

#### 2.4. Comparison with MD Simulations

## 3. KLD as a Lyapunov Functional

#### 3.1. Boltzmann’s H-Functional

#### 3.2. KLD

#### 3.3. MD Simulations

#### 3.3.1. Maxwellian Distribution as a Reference (${\varphi}_{\mathrm{ref}}={\varphi}_{\mathrm{M}}$)

#### 3.3.2. HCS Distribution as a Reference (${\varphi}_{\mathrm{ref}}={\varphi}_{\mathrm{H}}$)

#### 3.3.3. Relative Entropy of ${\varphi}_{\mathrm{H}}$ with Respect to ${\varphi}_{\mathrm{M}}$

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DSMC | Direct simulation Monte Carlo |

HCS | Homogenous cooling state |

KLD | Kullback–Leibler divergence |

MD | Molecular dynamics |

VDF | Velocity distribution function |

## Appendix A. Simulation and Numerical Details

## Appendix B. Initial Conditions

**Table A1.**Values of the fourth and sixth cumulants for the initial distributions $\delta $, M, I, $\mathsf{\Gamma}$, and S (see text).

$\mathit{\delta}$ | M | I | $\mathbf{\Gamma}$ | S | |
---|---|---|---|---|---|

${a}_{2}\left(0\right)$ | $\begin{array}{cc}-0.500& (d=2)\\ -0.400& (d=3)\end{array}$ | 0 | $\begin{array}{cc}0.151& (d=2)\\ 0.111& (d=3)\end{array}$ | $\begin{array}{cc}0.580& (d=2)\\ 0.580& (d=3)\end{array}$ | $\begin{array}{cc}0.885& (d=2)\\ 0.792& (d=3)\end{array}$ |

${a}_{3}\left(0\right)$ | $\begin{array}{cc}-0.667& (d=2)\\ -0.457& (d=3)\end{array}$ | 0 | $\begin{array}{cc}-0.080& (d=2)\\ -0.046& (d=3)\end{array}$ | $\begin{array}{cc}-0.062& (d=2)\\ -0.149& (d=3)\end{array}$ | $\begin{array}{cc}-4.733& (d=2)\\ -2.219& (d=3)\end{array}$ |

## Appendix C. Formal Expression for ${\mathbf{\partial}}_{\mathit{s}}{\mathcal{D}}_{\mathbf{KL}}\mathbf{(}\mathit{\varphi}\mathbf{\parallel}{\mathit{\varphi}}_{\mathbf{ref}}\mathbf{)}$

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**Figure 1.**Plot of (

**a**) the HCS fourth cumulant ${a}_{2}^{\mathrm{H}}$ and (

**b**) the HCS sixth cumulant ${a}_{3}^{\mathrm{H}}$ versus the coefficient of restitution $\alpha $. Symbols represent simulation results: MD (this work) for disks (∘) and spheres (∆), and DSMC [18,19,29] for disks (×) and spheres (□). The lines are the theoretical predictions ${a}_{2}^{\mathrm{H},b}$ (see Equation (20)) and ${a}_{3}^{\mathrm{H},ab}$ (see Equations (22) and (24)). The insets magnify the region $0.6\le \alpha \le 1$. The error bars in the simulation data are smaller than the size of the symbols.

**Figure 2.**Evolution of the fourth cumulant ${a}_{2}\left(s\right)$ as a function of the average number of collisions per particle for (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results, while the lines correspond to the theoretical prediction (26). The values of the coefficient of restitution are (from top to bottom) $\alpha =0.1$ (□), $0.4$ (×), 1 (∘), and $0.87$ (∆). The error bars in the simulation data are smaller than the size of the symbols.

**Figure 3.**Evolution of the sixth cumulant ${a}_{3}\left(s\right)$ as a function of the average number of collisions per particle for (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results, while the lines correspond to the theoretical prediction (29). The values of the coefficient of restitution are (from bottom to top on the right side) $\alpha =0.1$ (□), $0.4$ (×), $0.87$ (∆), and 1 (∘). The error bars in the simulation data are smaller than the size of the symbols, except in the stationary regime for $\alpha =0.1$.

**Figure 4.**Evolution of the fourth cumulant ${a}_{2}\left(s\right)$ for a coefficient of restitution $\alpha =0.1$ as a function of the average number of collisions per particle for (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results, while the lines correspond to the theoretical prediction (26). Five different initial conditions are considered (see Appendix B): $\delta $ (⋄), M (∘), $\mathsf{\Gamma}$ (×), I (□), and S (∆). The error bars in the simulation data are smaller than the size of the symbols, except in the early stage for the initial condition S.

**Figure 5.**Evolution of the sixth cumulant ${a}_{3}\left(s\right)$ for a coefficient of restitution $\alpha =0.1$ as a function of the average number of collisions per particle for (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results, while the lines correspond to the theoretical prediction (29). Five different initial conditions are considered (see Appendix B): $\delta $ (⋄), M (∘), $\mathsf{\Gamma}$ (×), I (□), and S (∆). The error bars in the simulation data are smaller than the size of the symbols, except in the early stage for the initial condition S.

**Figure 6.**Evolution of ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{M}})$ (in logarithmic scale) as a function of the average number of collisions per particle for (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results, while the lines correspond to the theoretical approximation (39) (the thin dashed lines for the first stage of the evolution mean that it was necessary to take the real part). The values of the coefficient of restitution are (from top to bottom on the right side) $\alpha =0.1$ (□), $0.4$ (×), $0.87$ (∆), and 1 (∘). The error bars in the simulation data are smaller than the size of the symbols, except when ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{M}})\lesssim {10}^{-4}$ for $\alpha =1$.

**Figure 7.**Evolution of ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{M}})$ (in logarithmic scale) for a coefficient of restitution $\alpha =0.1$ as a function of the average number of collisions per particle for hard (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results. Five different initial conditions are considered (see Appendix B): $\delta $ (⋄), M (∘), $\mathsf{\Gamma}$ (×), I (□), and S (∆). The error bars are smaller than the size of the symbols, except when ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{M}})\lesssim {10}^{-4}$ for the initial condition M.

**Figure 8.**Evolution of ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{H}})$ (in logarithmic scale) as a function of the average number of collisions per particle for (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results, while the lines correspond to the theoretical prediction (40) (the thin dashed lines for the first stage of the evolution meaning that it was necessary to take the real part). The values of the coefficient of restitution are $\alpha =0.1$ (□), $0.4$ (×), $0.87$ (∆), and 1 (∘) The error bars in the simulation data are smaller than the size of the symbols, except when ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{M}})\lesssim {10}^{-4}$.

**Figure 9.**Evolution of ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{H}})$ (in logarithmic scale) for a coefficient of restitution $\alpha =0.1$ as a function of the average number of collisions per particle for hard (

**a**) disks and (

**b**) spheres. Symbols represent MD simulation results. Five different initial conditions are considered (see Appendix B): $\delta $ (⋄), M (∘), $\mathsf{\Gamma}$ (×), I (□), and S (∆). The error bars are smaller than the size of the symbols, except when ${\mathcal{D}}_{\mathrm{KL}}(\varphi \parallel {\varphi}_{\mathrm{M}})\lesssim {10}^{-4}$.

**Figure 10.**Plot of ${\mathcal{D}}_{\mathrm{KL}}({\varphi}_{\mathrm{H}}\parallel {\varphi}_{M})$ as a function of the coefficient of restitution $\alpha $ for disks (– –, ∘) and spheres (—, ∆). Symbols represent MD simulation results, while the lines correspond to the theoretical prediction provided by Equation (39) with ${a}_{2}\left(s\right)\to {a}_{2}^{\mathrm{H}}$ and ${a}_{3}\left(s\right)\to {a}_{3}^{\mathrm{H}}$. The inset magnifies the region $0.6\le \alpha \le 1$. The error bars in the simulation data are smaller than the size of the symbols.

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Megías, A.; Santos, A. Kullback–Leibler Divergence of a Freely Cooling Granular Gas. *Entropy* **2020**, *22*, 1308.
https://doi.org/10.3390/e22111308

**AMA Style**

Megías A, Santos A. Kullback–Leibler Divergence of a Freely Cooling Granular Gas. *Entropy*. 2020; 22(11):1308.
https://doi.org/10.3390/e22111308

**Chicago/Turabian Style**

Megías, Alberto, and Andrés Santos. 2020. "Kullback–Leibler Divergence of a Freely Cooling Granular Gas" *Entropy* 22, no. 11: 1308.
https://doi.org/10.3390/e22111308