A rigorous derivation of the Hamiltonian structure for the Vlasov equation

• Published in: Forum of mathematics. Sigma Vol. 11
• Main Authors: Miller, Joseph K., Nahmod, Andrea R., Pavlović, Nataša, Rosenzweig, Matthew, Staffilani, Gigliola
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 05-09-2023
• Subjects: 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; 70S05; 82C05; 82C22; 37K06; 35Q83; 35Q70
• ISSN: 2050-5094; 2050-5094
• DOI: 10.1017/fms.2023.72

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.