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1. Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

   https://doi.org/10.3934/krm.2022005  

   Shvydkoy, Roman ( January 2022 , Kinetic and Related Models)

   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.

    

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2. Epidemic modeling with heterogeneity and social diffusion

   https://doi.org/10.1007/s00285-022-01861-w  

   Berestycki, Henri ; Desjardins, Benoît ; Weitz, Joshua S. ; Oury, Jean-Marc ( April 2023 , Journal of Mathematical Biology)

   Abstract We propose and analyze a family of epidemiological models that extend the classic Susceptible-Infectious-Recovered/Removed (SIR)-like framework to account for dynamic heterogeneity in infection risk. The family of models takes the form of a system of reaction–diffusion equations given populations structured by heterogeneous susceptibility to infection. These models describe the evolution of population-level macroscopic quantities S ,  I ,  R as in the classical case coupled with a microscopic variable f , giving the distribution of individual behavior in terms of exposure to contagion in the population of susceptibles. The reaction terms represent the impact of sculpting the distribution of susceptibles by the infection process. The diffusion and drift terms that appear in a Fokker–Planck type equation represent the impact of behavior change both during and in the absence of an epidemic. We first study the mathematical foundations of this system of reaction–diffusion equations and prove a number of its properties. In particular, we show that the system will converge back to the unique equilibrium distribution after an epidemic outbreak. We then derive a simpler system by seeking self-similar solutions to the reaction–diffusion equations in the case of Gaussian profiles. Notably, these self-similar solutions lead to a system of ordinary differential equations including classic SIR-like compartments and a new feature: the average risk level in the remaining susceptible population. We show that the simplified system exhibits a rich dynamical structure during epidemics, including plateaus, shoulders, rebounds and oscillations. Finally, we offer perspectives and caveats on ways that this family of models can help interpret the non-canonical dynamics of emerging infectious diseases, including COVID-19. 

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3. A Transient Microsphere Model for Nonlinear Viscoelasticity in Dynamic Polymer Networks

   https://doi.org/10.1115/1.4052375  

   Lamont, Samuel ; Vernerey, Franck J. ( January 2022 , Journal of Applied Mechanics)

   Abstract Viscoelastic material behavior in polymer systems largely arises from dynamic topological rearrangement at the network level. In this paper, we present a physically motivated microsphere formulation for modeling the mechanics of transient polymer networks. By following the directional statistics of chain alignment and local chain stretch, the transient microsphere model (TMM) is fully anisotropic and micro-mechanically based. Network evolution is tracked throughout deformation using a Fokker–Planck equation that incorporates the effects of bond creation and deletion at rates that are sensitive to the chain-level environment. Using published data, we demonstrate the model to capture various material responses observed in physical polymers. 

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4. Understanding Fluid Dynamics from Langevin and Fokker–Planck Equations

   https://doi.org/10.3390/fluids5010040  

   Medved, Andrei ; Davis, Riley ; Vasquez, Paula A. ( March 2020 , Fluids)

   The Langevin equations (LE) and the Fokker–Planck (FP) equations are widely used to describe fluid behavior based on coarse-grained approximations of microstructure evolution. In this manuscript, we describe the relation between LE and FP as related to particle motion within a fluid. The manuscript introduces undergraduate students to two LEs, their corresponding FP equations, and their solutions and physical interpretation. 

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5. Data-Driven Computational Methods for Quasi-Stationary Distribution and Sensitivity Analysis

   https://doi.org/10.1007/s10884-022-10137-2  

   Li, Yao ; Yuan, Yaping ( January 2022 , Journal of Dynamics and Differential Equations)

   This paper studies computational methods for quasi-stationary distributions (QSDs). We first proposed a data-driven solver that solves Fokker–Planck equations for QSDs. Similar to the case of Fokker–Planck equations for invariant probability measures, we set up an optimization problem that minimizes the distance from a low-accuracy reference solution, under the constraint of satisfying the linear relation given by the discretized Fokker–Planck operator. Then we use coupling method to study the sensitivity of a QSD against either the change of boundary condition or the diffusion coefficient. The 1-Wasserstein distance between a QSD and the corresponding invariant probability measure can be quantitatively estimated. Some numerical results about both computation of QSDs and their sensitivity analysis are provided. 

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