A rigorous derivation of the Hamiltonian structure for the Vlasov equation
We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie–Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise i...
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| Published in: | Forum of mathematics. Sigma Vol. 11 |
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05-09-2023
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| Abstract | We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie–Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to [MNP+20], which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison and Weinstein [MMW84] on providing a ‘statistical basis’ for the bracket structure of the Vlasov equation. |
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| AbstractList | We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie–Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to [MNP+20], which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison and Weinstein [MMW84] on providing a ‘statistical basis’ for the bracket structure of the Vlasov equation. We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie–Poisson type . In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to [MNP + 20], which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison and Weinstein [MMW84] on providing a ‘statistical basis’ for the bracket structure of the Vlasov equation. |
| ArticleNumber | e77 |
| Author | Rosenzweig, Matthew Staffilani, Gigliola Nahmod, Andrea R. Pavlović, Nataša Miller, Joseph K. |
| Author_xml | – sequence: 1 givenname: Joseph K. orcidid: 0000-0002-6916-5773 surname: Miller fullname: Miller, Joseph K. email: jkmiller@utexas.edu organization: 1Department of Mathematics, University of Texas at Austin, 2515 Speedway, Austin, 78712, United States of America; E-mail: jkmiller@utexas.edu – sequence: 2 givenname: Andrea R. surname: Nahmod fullname: Nahmod, Andrea R. email: nahmod@umass.edu organization: 2Department of Mathematics and Statistics, University of Massachusetts Amherst, 710 North Pleasant St, Amherst, 01003, United States of America; E-mail: nahmod@umass.edu – sequence: 3 givenname: Nataša surname: Pavlović fullname: Pavlović, Nataša email: natasa@math.utexas.edu organization: 3Department of Mathematics, University of Texas at Austin, 2515 Speedway, Austin, 78712, United States of America; E-mail: natasa@math.utexas.edu – sequence: 4 givenname: Matthew orcidid: 0000-0001-8842-9263 surname: Rosenzweig fullname: Rosenzweig, Matthew email: mrosenz2@andrew.cmu.edu organization: 4Department of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, 15213, United States of America; E-mail: mrosenz2@andrew.cmu.edu – sequence: 5 givenname: Gigliola surname: Staffilani fullname: Staffilani, Gigliola email: gigliola@math.mit.edu organization: 5Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, 02139, United States of America; E-mail: gigliola@math.mit.edu |
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| Snippet | We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system... |
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| SubjectTerms | 35Q70 35Q83 37K06 70S05 82C05 82C22 Algebra Analysis Bosons Brackets Derivation Hamiltonian functions Mathematical functions Ordinary differential equations Partial differential equations Physics Random variables Schrodinger equation Vlasov equations |
| Title | A rigorous derivation of the Hamiltonian structure for the Vlasov equation |
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