# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic and Related Models

April 2022 , Volume 15 , Issue 2

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2022, 15(2): 147-185 doi: 10.3934/krm.2021050 +[Abstract](347) +[HTML](111) +[PDF](622.76KB)
Abstract:

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces \begin{document}$L^{1}( \mathbb{R} _{+};\ x^{\alpha }dx)$\end{document} and \begin{document}$L^{1}( \mathbb{R} _{+};\ \left( 1+x\right) ^{\alpha }dx)$\end{document} for unbounded total fragmentation rates and continuous growth rates \begin{document}$r(.)$\end{document} such that \begin{document}$\int_{0}^{+\infty } \frac{1}{r(\tau )}d\tau = +\infty .\$\end{document} The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that \begin{document}$\alpha >\widehat{\alpha }$\end{document} for a suitable threshold \begin{document}$\widehat{ \alpha }\geq 1$\end{document} that depends on the moment space we consider. A systematic functional analytic construction is provided. Various examples of fragmentation kernels illustrating the theory are given and an open problem is mentioned.

2022, 15(2): 187-212 doi: 10.3934/krm.2021051 +[Abstract](296) +[HTML](96) +[PDF](519.47KB)
Abstract:

The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.

We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.

2022, 15(2): 213-237 doi: 10.3934/krm.2022005 +[Abstract](334) +[HTML](97) +[PDF](514.95KB)
Abstract:

In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e. it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation.

The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure.

2022, 15(2): 239-256 doi: 10.3934/krm.2022006 +[Abstract](276) +[HTML](74) +[PDF](646.29KB)
Abstract:

We consider kinetic vehicular traffic flow models of BGK type [24]. Considering different spatial and temporal scales, those models allow to derive a hierarchy of traffic models including a hydrodynamic description. In this paper, the kinetic BGK–model is extended by introducing a parametric stochastic variable to describe possible uncertainty in traffic. The interplay of uncertainty with the given model hierarchy is studied in detail. Theoretical results on consistent formulations of the stochastic differential equations on the hydrodynamic level are given. The effect of the possibly negative diffusion in the stochastic hydrodynamic model is studied and numerical simulations of uncertain traffic situations are presented.

2022, 15(2): 257-282 doi: 10.3934/krm.2022007 +[Abstract](204) +[HTML](68) +[PDF](729.76KB)
Abstract:

Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interactions of Rashba type are formally derived from a collisional Wigner equation. The collisions are modeled by a Bhatnagar–Gross–Krook-type operator describing the relaxation of the electron gas to a local equilibrium that is given by the quantum maximum entropy principle. Because of non-commutativity properties of the operators, the standard diffusion scaling cannot be used in this context, and a hydrodynamic time scaling is required. A Chapman–Enskog procedure leads, up to first order in the relaxation time, to a system of nonlocal quantum drift-diffusion equations for the charge density and spin vector densities. Local equations including the Bohm potential are obtained in the semiclassical expansion up to second order in the scaled Planck constant. The main novelty of this work is that all spin components are considered, while previous models only consider special spin directions.

2022, 15(2): 283-315 doi: 10.3934/krm.2022008 +[Abstract](315) +[HTML](72) +[PDF](1823.74KB)
Abstract:

By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular case that the collision invariant is of class \begin{document}$\mathscr{C}^{1}$\end{document}. This work extends that of Saint-Raymond and Wilkinson (Communications on Pure and Applied Mathematics (2018), 71(8), pp. 1494–1534), in which the authors characterise collision invariants only in the case of the so-called canonical physical scattering family. Indeed, our method extends to the case of non-canonical physical scattering, whose existence was reported in Wilkinson (Archive for Rational Mechanics and Analysis (2020), 235(3), pp. 2055–2083). Moreover, our new method improves upon the work in Saint-Raymond and Wilkinson as we place no symmetry hypotheses on the underlying non-spherical particles which make up the gas under consideration. The techniques established in this paper also yield a new proof of the result of Boltzmann for collision invariants of class \begin{document}$\mathscr{C}^{1}$\end{document} in the classical case of hard spheres.

2021 Impact Factor: 1.398
5 Year Impact Factor: 1.685
2021 CiteScore: 2.7