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Large-time behavior in (hypo)coercive ODE-systems and kinetic models
Post-edited
Authors : Arnold, Anton (Author of the conference)
CIRM (Publisher )
35B20 - Perturbations (PDE)
35B40 - Asymptotic behavior of solutions of PDE
35H10 - Hypoelliptic equations
35K10 - General theory of second order parabolic equations
35Pxx - Spectral theory and eigenvalue problems for partial differential operators
47D06 - One-parameter semigroups and linear evolution equations
47D07 - Markov semigroups of linear operators and applications to diffusion processes
82C31 - Stochastic methods in time-dependent statistical mechanics (Fokker-Planck, Langevin, etc.)
35Q84 - Fokker-Planck equations
CIRM (Publisher )
Abstract :
In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive.
We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model.
The third part of the talk is concerned with the entropy method for (non)symmetric Fokker-Planck equations, which is a powerful tool to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For hypocoercive Fokker-Planck equations, i.e. degenerate parabolic equations (with drift terms that are linear in the spatial variable) we modify the classical entropy method by introducing an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. The obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with nonquadratic potential).
MSC Codes : We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive.
We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model.
The third part of the talk is concerned with the entropy method for (non)symmetric Fokker-Planck equations, which is a powerful tool to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For hypocoercive Fokker-Planck equations, i.e. degenerate parabolic equations (with drift terms that are linear in the spatial variable) we modify the classical entropy method by introducing an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. The obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with nonquadratic potential).
35B20 - Perturbations (PDE)
35B40 - Asymptotic behavior of solutions of PDE
35H10 - Hypoelliptic equations
35K10 - General theory of second order parabolic equations
35Pxx - Spectral theory and eigenvalue problems for partial differential operators
47D06 - One-parameter semigroups and linear evolution equations
47D07 - Markov semigroups of linear operators and applications to diffusion processes
82C31 - Stochastic methods in time-dependent statistical mechanics (Fokker-Planck, Langevin, etc.)
35Q84 - Fokker-Planck equations
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 26/09/2018 Conference Date : 18/09/2018 Subseries : Research talks arXiv category : Analysis of PDEs ; Mathematical Physics Mathematical Area(s) : PDE ; Mathematical Physics Format : MP4 (.mp4) - HD Video Time : 01:05:20 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2018-09-18_Arnold.mp4 
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Information on the Event
Event Title : Advances in computational statistical physics / Perspectives en physique statistique computationnelleEvent Organizers : Lelièvre, Tony ; Stoltz, Gabriel ; Pavliotis, Grigorios Dates : 17/09/2018 - 21/09/2018 Event Year : 2018 Event URL : https://conferences.cirm-math.fr/1866.html
Citation Data
DOI : 10.24350/CIRM.V.19445903Cite this video as: Arnold, Anton (2018). Large-time behavior in (hypo)coercive ODE-systems and kinetic models. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19445903 URI : http://dx.doi.org/10.24350/CIRM.V.19445903 |
See Also
- [Multi angle] Multiscale modeling for soft matter - Perspectives and challenges / Author of the conference Kremer, Kurt.
- [Multi angle] Semiclassical methods for Langevin dynamics at low temperature / Author of the conference Laurent, Michel.
- [Multi angle] From cluster algorithms to PDMP algorithms: a Monte Carlo story of symmetry exploitation / Author of the conference Michel, Manon.
- [Multi angle] Rare event simulation for molecular dynamics / Author of the conference Guyader, Arnaud.
Bibliography
- Achleitner, F., Arnold, A., & Carlen, E.A. (2016). On linear hypocoercive BGK models. In P.C. Gonçalves, & A.J. Soares (Eds.), From Particle Systems to Partial Differential Equations III (pp. 1-37). Cham: Springer - https://doi.org/10.1007/978-3-319-32144-8_1
- Achleitner, F., Arnold, A., & Carlen, E.A. (2018). On multi-dimensional hypocoercive BGK models. Kinetic & Related Models, 11(4), 953-1009 - http://dx.doi.org/10.3934/krm.2018038
- Achleitner, F., Arnold, A., & Signorello, B. (2018). On optimal decay estimates for ODEs and PDEs with modal decomposition.
- Achleitner, F., Arnold, A., & Stürzer, D. (2015). Large-time behavior in non-symmetric Fokker-Planck equations. Rivista di Matematica della Università di Parma. New Series, 6(1), 1-68 - http://rivista.math.unipr.it/wwwprotetto/2015-6-1/pdf/01.pdf
- Arnold, A., & Erb, J. (2014). Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift.
