Kullback–Leibler Divergence of a Freely Cooling Granular Gas
Finding the proper entropy-like Lyapunov functional associated with the inelastic Boltzmann equation for an isolated freely cooling granular gas is a still unsolved challenge. The original <i>H</i>-theorem hypotheses do not fit here and the <i>H</i>-functional presents some a...
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MDPI AG
2020-11-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/11/1308 |
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| author | Alberto Megías Andrés Santos |
| author_facet | Alberto Megías Andrés Santos |
| author_sort | Alberto Megías |
| collection | DOAJ |
| description | Finding the proper entropy-like Lyapunov functional associated with the inelastic Boltzmann equation for an isolated freely cooling granular gas is a still unsolved challenge. The original <i>H</i>-theorem hypotheses do not fit here and the <i>H</i>-functional presents some additional measure problems that are solved by the Kullback–Leibler divergence (KLD) of a reference velocity distribution function from the actual distribution. The right choice of the reference distribution in the KLD is crucial for the latter to qualify or not as a Lyapunov functional, the asymptotic “homogeneous cooling state” (HCS) distribution being a potential candidate. Due to the lack of a formal proof far from the quasielastic limit, the aim of this work is to support this conjecture aided by molecular dynamics simulations of inelastic hard disks and spheres in a wide range of values for the coefficient of restitution (<inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>) and for different initial conditions. Our results reject the Maxwellian distribution as a possible reference, whereas they reinforce the HCS one. Moreover, the KLD is used to measure the amount of information lost on using the former rather than the latter, revealing a non-monotonic dependence with <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T14:47:43Z |
| format | Article |
| id | doaj.art-62e75e7a3e624b7faf8036245eb81069 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T14:47:43Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-62e75e7a3e624b7faf8036245eb810692023-11-20T21:15:11ZengMDPI AGEntropy1099-43002020-11-012211130810.3390/e22111308Kullback–Leibler Divergence of a Freely Cooling Granular GasAlberto Megías0Andrés Santos1Departamento de Física, Universidad de Extremadura, E-06006 Badajoz, SpainDepartamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06006 Badajoz, SpainFinding the proper entropy-like Lyapunov functional associated with the inelastic Boltzmann equation for an isolated freely cooling granular gas is a still unsolved challenge. The original <i>H</i>-theorem hypotheses do not fit here and the <i>H</i>-functional presents some additional measure problems that are solved by the Kullback–Leibler divergence (KLD) of a reference velocity distribution function from the actual distribution. The right choice of the reference distribution in the KLD is crucial for the latter to qualify or not as a Lyapunov functional, the asymptotic “homogeneous cooling state” (HCS) distribution being a potential candidate. Due to the lack of a formal proof far from the quasielastic limit, the aim of this work is to support this conjecture aided by molecular dynamics simulations of inelastic hard disks and spheres in a wide range of values for the coefficient of restitution (<inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>) and for different initial conditions. Our results reject the Maxwellian distribution as a possible reference, whereas they reinforce the HCS one. Moreover, the KLD is used to measure the amount of information lost on using the former rather than the latter, revealing a non-monotonic dependence with <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>.https://www.mdpi.com/1099-4300/22/11/1308Kullback–Leibler divergencegranular gaseskinetic theorymolecular dynamics |
| spellingShingle | Alberto Megías Andrés Santos Kullback–Leibler Divergence of a Freely Cooling Granular Gas Entropy Kullback–Leibler divergence granular gases kinetic theory molecular dynamics |
| title | Kullback–Leibler Divergence of a Freely Cooling Granular Gas |
| title_full | Kullback–Leibler Divergence of a Freely Cooling Granular Gas |
| title_fullStr | Kullback–Leibler Divergence of a Freely Cooling Granular Gas |
| title_full_unstemmed | Kullback–Leibler Divergence of a Freely Cooling Granular Gas |
| title_short | Kullback–Leibler Divergence of a Freely Cooling Granular Gas |
| title_sort | kullback leibler divergence of a freely cooling granular gas |
| topic | Kullback–Leibler divergence granular gases kinetic theory molecular dynamics |
| url | https://www.mdpi.com/1099-4300/22/11/1308 |
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