# Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres

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## Abstract

**:**

## 1. Introduction

## 2. A Collisional Model for the Effective Two-Dimensional Dynamics

## 3. Hydrodynamic Equations

## 4. Stability Analysis

## 5. Back to the Origins: Boltzmann–Enskog Kinetic Equation

## 6. Homogeneous Approximation: The Evolution of the Temperature

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the quasi-two-dimensional system described in the main text. The two parallel walls are vibrating, and the interest is on the dynamics observed when looking from above or below.

**Figure 2.**Relaxation of the granular temperature of the quasi-two-dimensional gas in a system with $\alpha =0.8$, $h=1.5\sigma $, and three-dimensional density $n=0.02{\sigma}^{-3}$. The walls are vibrating in a sawtooth way with a velocity ${v}_{b}$. Solid lines are the results from molecular dynamics (MD) simulations, while the dashed lines are the theoretical predictions obtained as indicated in the main text. (

**a**) On the left hand side, a system cooling towards its steady granular temperature; (

**b**) On the right hand side, the system started from a granular temperature smaller than the stationary one.

**Figure 3.**(

**a**) Dimensionless Euler transport coefficient ${\zeta}_{1}$ as a function of the dimensionless characteristic speed ${\Delta}^{\ast}$ for the effective two-dimensional granular gas. The coefficient of normal restitution is $\alpha =0.85$. The dots indicate the value of the transport coefficients at the steady state, whose characteristic speed is ${\Delta}_{st}^{\ast}$; (

**b**) the same for the adimensionalized shear viscosity $\overline{\eta}$.

**Figure 4.**(

**a**) Adimensionalized (thermal) heat conductivity $\overline{\kappa}$ as a function of the dimensionless characteristic speed ${\Delta}^{\ast}$ for the effective two-dimensional granular gas. The coefficient of normal restitution is $\alpha =0.85$. The dots indicate the value of the transport coefficients at the steady state, whose characteristic reduced speed is ${\Delta}_{st}^{\ast}$; (

**b**) the same for the adimensionalized diffusive heat conductivity $\overline{\mu}$.

**Figure 5.**Time evolution of the dimensionless perturbations of the hydrodynamic fields, $\rho \equiv (n-{n}_{H})/{n}_{h}$, $\mathit{\omega}\equiv \mathit{u}/{v}_{0}$, $\theta \equiv (T-{T}_{H})/{T}_{H}$, as predicted by the linearized hydrodynamic equations. The tildes indicate Fourier transforms, and the time s is a dimensionless scale defined from the original one by means of the thermal velocity ${v}_{0H}$ and the mean free path, Equation (18). The wavenumber is $k=0.2$, while ${k}_{\perp}=0.91$. The initial temperature of the system is larger than its stationary value, so the system is monotonically cooling in time. The values of the initial perturbations of the fields are on the order of ${10}^{-3}$.

**Figure 6.**Snapshot of the positions of the particles in a system of $N=2000$ smooth inelastic hard disks. The parameters defining the system are $n=0.06{\sigma}^{-2}$, $\alpha =0.6$, and $\Delta =2\times {10}^{-4}{v}_{0}\left(0\right)/\sqrt{2}$. At the time shown, the average accumulated number of collisions per particle was around 40.

**Figure 7.**Density profile between the two plates. The symbols are simulation results, and the dashed line is the theoretical prediction given in the main text. The number of particles used in the simulations is $N=500$, the separation of the plates is $h=1.9\sigma $, and the average density is given by $N{\sigma}^{2}/A=0.285$. The plotted dimensionless density is ${n}_{2}=n\sigma A$ and $\overline{z}\equiv (2z-\sigma )/2\sigma $.

**Figure 8.**Decay of the vertical, ${T}_{z}$, and horizontal, ${T}_{xy}$, temperatures towards their equal steady values in a homogenous system. Time is measured in the dimensionless units indicated in the label, where ${v}_{0}\equiv \sqrt{2T\left(0\right)/m}$. The solid lines are MD simulation results, while the dashed lines are the theoretical predictions given by Equations (39) and (40). Moreover, the two upper lines correspond to the vertical temperature, and the two lower lines to the horizontal temperature.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Brey, J.J.; Buzón, V.; García de Soria, M.I.; Maynar, P. Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres. *Entropy* **2017**, *19*, 68.
https://doi.org/10.3390/e19020068

**AMA Style**

Brey JJ, Buzón V, García de Soria MI, Maynar P. Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres. *Entropy*. 2017; 19(2):68.
https://doi.org/10.3390/e19020068

**Chicago/Turabian Style**

Brey, J. Javier, Vicente Buzón, Maria Isabel García de Soria, and Pablo Maynar. 2017. "Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres" *Entropy* 19, no. 2: 68.
https://doi.org/10.3390/e19020068