This article studies the kinetic dynamics of the rock-paper-scissors binary game in a measure setting given by a non local and non linear integrodifferential equation. After proving the wellposedness of the equation, we provide a precise description of the asymptotic behavior in large time. To do so we adopt a duality approach, which is well suited both as a first step to construct a measure solution by mean of semigroups and to obtain an explicit expression of the asymptotic measure. Even thought the equation is non linear, this measure depends linearly on the initial condition. This result is completed by a decay in total variation norm, which happens to be subgeometric due to the nonlinearity of the equation. This relies on an unusual use of a confining condition that is needed to apply a Harris-type theorem, taken from a recent paper [2] that also provides a way to compute explicitly the constants involved in the aforementioned decay in norm.
{"title":"Asymptotic behavior of a kinetic approach to the collective dynamics of a rock-paper-scissors binary game","authors":"Hugo Martin","doi":"10.3934/krm.2023022","DOIUrl":"https://doi.org/10.3934/krm.2023022","url":null,"abstract":"This article studies the kinetic dynamics of the rock-paper-scissors binary game in a measure setting given by a non local and non linear integrodifferential equation. After proving the wellposedness of the equation, we provide a precise description of the asymptotic behavior in large time. To do so we adopt a duality approach, which is well suited both as a first step to construct a measure solution by mean of semigroups and to obtain an explicit expression of the asymptotic measure. Even thought the equation is non linear, this measure depends linearly on the initial condition. This result is completed by a decay in total variation norm, which happens to be subgeometric due to the nonlinearity of the equation. This relies on an unusual use of a confining condition that is needed to apply a Harris-type theorem, taken from a recent paper [2] that also provides a way to compute explicitly the constants involved in the aforementioned decay in norm.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"35 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85245571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Landau equation and the Boltzmann equation are connected through the limit of grazing collisions. This has been proved rigorously for certain families of Boltzmann operators concentrating on grazing collisions. In this contribution, we study the collision kernels associated to the two-particle scattering via a finite range potential $Phi(x)$ in three dimensions. We then consider the limit of weak interaction given by $Phi_epsilon(x) = epsilon Phi(x)$. Here $epsilonrightarrow 0$ is the grazing parameter, and the rate of collisions is rescaled to obtain a non-trivial limit. The grazing collisions limit is of particular interest for potentials with a singularity of order $sgeq 0$ at the origin, i.e. $phi(x) sim |x|^{-s}$ as $|x|rightarrow 0$. For $sin [0,1]$, we prove the convergence to the Landau equation with diffusion coefficient given by the Born approximation, as predicted in the works of Landau and Balescu. On the other hand, for potentials with $s>1$ we obtain the non-cutoff Boltzmann equation in the limit. The Coulomb singularity $s=1$ appears as a threshold value with a logarithmic correction to the diffusive timescale, the so-called Coulomb logarithm.
{"title":"The grazing collisions limit from the linearized Boltzmann equation to the Landau equation for short-range potentials","authors":"Corentin Le Bihan, Raphael Winter","doi":"10.3934/krm.2023003","DOIUrl":"https://doi.org/10.3934/krm.2023003","url":null,"abstract":"The Landau equation and the Boltzmann equation are connected through the limit of grazing collisions. This has been proved rigorously for certain families of Boltzmann operators concentrating on grazing collisions. In this contribution, we study the collision kernels associated to the two-particle scattering via a finite range potential $Phi(x)$ in three dimensions. We then consider the limit of weak interaction given by $Phi_epsilon(x) = epsilon Phi(x)$. Here $epsilonrightarrow 0$ is the grazing parameter, and the rate of collisions is rescaled to obtain a non-trivial limit. The grazing collisions limit is of particular interest for potentials with a singularity of order $sgeq 0$ at the origin, i.e. $phi(x) sim |x|^{-s}$ as $|x|rightarrow 0$. For $sin [0,1]$, we prove the convergence to the Landau equation with diffusion coefficient given by the Born approximation, as predicted in the works of Landau and Balescu. On the other hand, for potentials with $s>1$ we obtain the non-cutoff Boltzmann equation in the limit. The Coulomb singularity $s=1$ appears as a threshold value with a logarithmic correction to the diffusive timescale, the so-called Coulomb logarithm.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"60 11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75721007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the following work, we consider the Boltzmann equation that models a polyatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section $mathcal{B}$, we prove that the linearized Boltzmann operator $mathcal{L}$ of this model is a Fredholm operator. For this, we write $mathcal{L}$ as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator $mathcal{K}$ is compact. The result is established after inspecting the kernel form of $mathcal{K}$ and proving it to be $L^2$ integrable over its domain using elementary arguments.
{"title":"Fredholm property of the linearized Boltzmann operator for a polyatomic single gas model","authors":"S. Brull, Marwa Shahine, P. Thieullen","doi":"10.3934/krm.2023021","DOIUrl":"https://doi.org/10.3934/krm.2023021","url":null,"abstract":"In the following work, we consider the Boltzmann equation that models a polyatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section $mathcal{B}$, we prove that the linearized Boltzmann operator $mathcal{L}$ of this model is a Fredholm operator. For this, we write $mathcal{L}$ as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator $mathcal{K}$ is compact. The result is established after inspecting the kernel form of $mathcal{K}$ and proving it to be $L^2$ integrable over its domain using elementary arguments.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"118 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87648117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we study the diffusion limit of the classical solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system with initial data near a global Maxwellian. We prove the convergence and establish the optimal convergence rate of the global strong solution to the VPFP system towards the solution to the drift-diffusion-Poisson system based on the spectral analysis with precise estimation on the initial layer.
{"title":"Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system","authors":"Mingying Zhong","doi":"10.3934/krm.2021041","DOIUrl":"https://doi.org/10.3934/krm.2021041","url":null,"abstract":"In the present paper, we study the diffusion limit of the classical solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system with initial data near a global Maxwellian. We prove the convergence and establish the optimal convergence rate of the global strong solution to the VPFP system towards the solution to the drift-diffusion-Poisson system based on the spectral analysis with precise estimation on the initial layer.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81900728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a bound on the entropy dissipation for the Boltzmann collision operator from below by a weighted $L^p$-Norm. The estimate holds for a wide range of potentials including soft potentials as well as very soft potentials. As an application, we study weak solutions to the spatially homogeneous Boltzmann equation and prove a weighted $L^1_t(L^p_v)$ estimate.
{"title":"Entropy dissipation estimates for the Boltzmann equation without cut-off","authors":"Jamil Chaker, L. Silvestre","doi":"10.3934/krm.2023006","DOIUrl":"https://doi.org/10.3934/krm.2023006","url":null,"abstract":"We prove a bound on the entropy dissipation for the Boltzmann collision operator from below by a weighted $L^p$-Norm. The estimate holds for a wide range of potentials including soft potentials as well as very soft potentials. As an application, we study weak solutions to the spatially homogeneous Boltzmann equation and prove a weighted $L^1_t(L^p_v)$ estimate.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"38 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82066010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the derivation of macroscopic traffic models out of optimal speed and follow-the-leader particle dynamics as hydrodynamic limits of non-local Povzner-type kinetic equations. As a first step, we show that optimal speed vehicle dynamics produce a first order macroscopic model with non-local flux. Next, we show that non-local follow-the-leader vehicle dynamics have a universal macroscopic counterpart in the second order Aw-Rascle-Zhang traffic model, at least when the non-locality of the interactions is sufficiently small. Finally, we show that the same qualitative result holds also for a general class of follow-the-leader dynamics based on the headway of the vehicles rather than on their speed. We also investigate the correspondence between the solutions to particle models and their macroscopic limits by means of numerical simulations.
{"title":"Macroscopic limits of non-local kinetic descriptions of vehicular traffic","authors":"F. A. Chiarello, A. Tosin","doi":"10.3934/krm.2022038","DOIUrl":"https://doi.org/10.3934/krm.2022038","url":null,"abstract":"We study the derivation of macroscopic traffic models out of optimal speed and follow-the-leader particle dynamics as hydrodynamic limits of non-local Povzner-type kinetic equations. As a first step, we show that optimal speed vehicle dynamics produce a first order macroscopic model with non-local flux. Next, we show that non-local follow-the-leader vehicle dynamics have a universal macroscopic counterpart in the second order Aw-Rascle-Zhang traffic model, at least when the non-locality of the interactions is sufficiently small. Finally, we show that the same qualitative result holds also for a general class of follow-the-leader dynamics based on the headway of the vehicles rather than on their speed. We also investigate the correspondence between the solutions to particle models and their macroscopic limits by means of numerical simulations.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"47 3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86256307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the long-time behaviour of a kinetic formulation of an Elo-type rating model for a large number of interacting players with variable strength. The model results in a non-linear mean-field Fokker-Planck equation and we show the existence of steady states via a Schauder fixed point argument. Our proof relies on the study of a related linear equation using hypocoercivity techniques.
{"title":"Steady states of an Elo-type rating model for players of varying strength","authors":"Bertram During, J. Evans, Marie-Therese Wolfram","doi":"10.3934/krm.2023020","DOIUrl":"https://doi.org/10.3934/krm.2023020","url":null,"abstract":"In this paper we study the long-time behaviour of a kinetic formulation of an Elo-type rating model for a large number of interacting players with variable strength. The model results in a non-linear mean-field Fokker-Planck equation and we show the existence of steady states via a Schauder fixed point argument. Our proof relies on the study of a related linear equation using hypocoercivity techniques.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"58 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80329976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Heinze, Jan-Frederik Pietschmann, M. Schmidtchen
We explore the dynamical behavior and energetic properties of a model of two species that interact nonlocally on finite graphs. The authors recently introduced the model in the context of nonquadratic Finslerian gradient flows on generalized graphs featuring nonlinear mobilities. In a continuous and local setting, this class of systems exhibits a wide variety of patterns, including mixing of the two species, partial engulfment, or phase separation. This work showcases how this rich behavior carries over to the graph structure. We present analytical and numerical evidence thereof.
{"title":"Nonlocal cross-interaction systems on graphs: Energy landscape and dynamics","authors":"G. Heinze, Jan-Frederik Pietschmann, M. Schmidtchen","doi":"10.3934/krm.2023008","DOIUrl":"https://doi.org/10.3934/krm.2023008","url":null,"abstract":"We explore the dynamical behavior and energetic properties of a model of two species that interact nonlocally on finite graphs. The authors recently introduced the model in the context of nonquadratic Finslerian gradient flows on generalized graphs featuring nonlinear mobilities. In a continuous and local setting, this class of systems exhibits a wide variety of patterns, including mixing of the two species, partial engulfment, or phase separation. This work showcases how this rich behavior carries over to the graph structure. We present analytical and numerical evidence thereof.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81861355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a particle system with uniform coupling between a macroscopic component and individual particles. The constraint for each particle is of full rank, which implies that each movement of the macroscopic component leads to a movement of all particles and vice versa. Skeletal muscle tissues share a similar property which motivates this work. We prove convergence of the mean-field limit, well-posedness and a stability estimate for the mean-field PDE.
{"title":"The mean-field limit for particle systems with uniform full-rank constraints","authors":"Steffen Plunder, B. Simeon","doi":"10.3934/krm.2023012","DOIUrl":"https://doi.org/10.3934/krm.2023012","url":null,"abstract":"We consider a particle system with uniform coupling between a macroscopic component and individual particles. The constraint for each particle is of full rank, which implies that each movement of the macroscopic component leads to a movement of all particles and vice versa. Skeletal muscle tissues share a similar property which motivates this work. We prove convergence of the mean-field limit, well-posedness and a stability estimate for the mean-field PDE.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78861448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A typical galaxy consists of a huge number of stars attracted to each other by gravity. For instance, the Milky Way has about $10^{11}$ stars. Thus it is typically modeled by the Vlasov-Poisson system. We prove an existence theorem for axisymmetric steady states of galaxies that may rotate rapidly. Such states are given in terms of a fairly general function $phi$ of the particle energy and angular momentum. The set $mathcal K$ of such states form a connected set in an appropriate function space. Along the set $mathcal K$, we prove under some conditions that either (a) the supports of the galaxies become unbounded or (b) both the rotation speeds and the densities somewhere within the galaxy become unbounded.
{"title":"Global continuation of a Vlasov model of rotating galaxies","authors":"W. Strauss, Yilun Wu","doi":"10.3934/krm.2023001","DOIUrl":"https://doi.org/10.3934/krm.2023001","url":null,"abstract":"A typical galaxy consists of a huge number of stars attracted to each other by gravity. For instance, the Milky Way has about $10^{11}$ stars. Thus it is typically modeled by the Vlasov-Poisson system. We prove an existence theorem for axisymmetric steady states of galaxies that may rotate rapidly. Such states are given in terms of a fairly general function $phi$ of the particle energy and angular momentum. The set $mathcal K$ of such states form a connected set in an appropriate function space. Along the set $mathcal K$, we prove under some conditions that either (a) the supports of the galaxies become unbounded or (b) both the rotation speeds and the densities somewhere within the galaxy become unbounded.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84763486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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