{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/263044"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/263044","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Contributions to mixing and hypocoercivity in kinetic models","abstract":"The main results of my work contribute to the mathematical study of a stability mechanism common to both the Vlasov–Poisson equation and the Kuramoto equation. These kinetic models come from very different areas of physics: the Vlasov–Poisson equation models plasmas and the Kuramoto equation models synchronisation behaviour. The stability was first described by Landau in 1946 and is a subtle behaviour, because the damping only happens in a suitably weak sense. In fact, the models are not dissipative and cannot be stable in a strong topology. Instead, the so-called Landau damping happens through phase mixing. My contributions include a simplified linear analysis for the Vlasov–Poisson equation around the spatially homogeneous state. For the Kuramoto equation, I cover the linear analysis around general stationary states and show nonlinear stability results with algebraic and exponential decay. Moreover, I show how the mean-field estimate by Dobrushin can be improved around the incoherent state. In addition, I study how a kinetic system can reach a thermal equilibrium. This is modelled by adding a dissipative term, which by itself drives the system to a local equilibrium. In hypocoercivity theory, the complementary effect of the transport operator is used to show exponential decay to a global equilibrium. In particular, I show how a probabilistic treatment can complement the standard hypocoercivity theory, which constructs equivalent norms, and I discuss the necessity of the geometric control condition for the spatially degenerate kinetic Fokker–Planck equation. Finally, I study the possible discretisation of the velocity variable for kinetic equations. For the numerical stability, Hermite functions are a suitable choice, because their differentiation matrix is skew-symmetric. However, so far a fast expansion algorithm has been lacking and this is addressed in this work.","abstract_html":"The main results of my work contribute to the mathematical study of a stability mechanism common to both the Vlasov–Poisson equation and the Kuramoto equation. These kinetic models come from very different areas of physics: the Vlasov–Poisson equation models plasmas and the Kuramoto equation models synchronisation behaviour. The stability was first described by Landau in 1946 and is a subtle behaviour, because the damping only happens in a suitably weak sense. In fact, the models are not dissipative and cannot be stable in a strong topology. Instead, the so-called Landau damping happens through phase mixing. My contributions include a simplified linear analysis for the Vlasov–Poisson equation around the spatially homogeneous state. For the Kuramoto equation, I cover the linear analysis around general stationary states and show nonlinear stability results with algebraic and exponential decay. Moreover, I show how the mean-field estimate by Dobrushin can be improved around the incoherent state. In addition, I study how a kinetic system can reach a thermal equilibrium. This is modelled by adding a dissipative term, which by itself drives the system to a local equilibrium. In hypocoercivity theory, the complementary effect of the transport operator is used to show exponential decay to a global equilibrium. In particular, I show how a probabilistic treatment can complement the standard hypocoercivity theory, which constructs equivalent norms, and I discuss the necessity of the geometric control condition for the spatially degenerate kinetic Fokker–Planck equation. Finally, I study the possible discretisation of the velocity variable for kinetic equations. For the numerical stability, Hermite functions are a suitable choice, because their differentiation matrix is skew-symmetric. However, so far a fast expansion algorithm has been lacking and this is addressed in this work.","abstract_has_math":false,"creators":["Dietert, Helge"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Mouhot, Clement","Iserles, Arieh"],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-01-10","date_published":"2017-01-10","updated_at":"2026-07-22T22:23:54Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/d4164889-e8dd-4d10-aea8-90aca6b37e75/download","https://www.rioxx.net/licenses/all-rights-reserved/","https://www.repository.cam.ac.uk/bitstreams/7b068d2a-da01-4d40-9a14-5582900d298f/download"],"identifier_entries":[{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000220719202"],"render_values":[{"text":"0000-0002-2071-9202","href":"https://orcid.org/0000-0002-2071-9202","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.7765","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Mouhot, Clement","Iserles, Arieh"]},{"key":"dc:creator","label":"Author","values":["Dietert, Helge"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000220719202"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2017-01-10"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/263044"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/d4164889-e8dd-4d10-aea8-90aca6b37e75/download","https://www.rioxx.net/licenses/all-rights-reserved/","https://www.repository.cam.ac.uk/bitstreams/7b068d2a-da01-4d40-9a14-5582900d298f/download"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.7765"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/5ebda0e5-1892-4cd8-8046-4eb87209e9c4/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The main results of my work contribute to the mathematical study of a stability mechanism common to both the Vlasov–Poisson equation and the Kuramoto equation. 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This is modelled by adding a dissipative term, which by itself drives the system to a local equilibrium. In hypocoercivity theory, the complementary effect of the transport operator is used to show exponential decay to a global equilibrium. In particular, I show how a probabilistic treatment can complement the standard hypocoercivity theory, which constructs equivalent norms, and I discuss the necessity of the geometric control condition for the spatially degenerate kinetic Fokker–Planck equation. Finally, I study the possible discretisation of the velocity variable for kinetic equations. For the numerical stability, Hermite functions are a suitable choice, because their differentiation matrix is skew-symmetric. 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The stability was first described by Landau in 1946 and is a subtle behaviour, because the damping only happens in a suitably weak sense. In fact, the models are not dissipative and cannot be stable in a strong topology. Instead, the so-called Landau damping happens through phase mixing. My contributions include a simplified linear analysis for the Vlasov–Poisson equation around the spatially homogeneous state. For the Kuramoto equation, I cover the linear analysis around general stationary states and show nonlinear stability results with algebraic and exponential decay. Moreover, I show how the mean-field estimate by Dobrushin can be improved around the incoherent state. In addition, I study how a kinetic system can reach a thermal equilibrium. This is modelled by adding a dissipative term, which by itself drives the system to a local equilibrium. In hypocoercivity theory, the complementary effect of the transport operator is used to show exponential decay to a global equilibrium. In particular, I show how a probabilistic treatment can complement the standard hypocoercivity theory, which constructs equivalent norms, and I discuss the necessity of the geometric control condition for the spatially degenerate kinetic Fokker–Planck equation. Finally, I study the possible discretisation of the velocity variable for kinetic equations. For the numerical stability, Hermite functions are a suitable choice, because their differentiation matrix is skew-symmetric. 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