Abstract
Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here, we present different approaches to Koopman's Hilbert-space formulation of classical mechanics in the context of Vlasov–Maxwell kinetic theory. The celebrated Koopman–von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is non-canonical and appears to overcome certain issues emerging in the canonical formalism. Furthermore, the canonical structure is restored for a variant of the Koopman–von Neumann construction that carries a different phase dynamics. Going back to van Hove's prequantum theory, the corresponding Koopman–van Hove equation provides an alternative Clebsch representation which is then coupled to the electromagnetic fields. Finally, the role of gauge transformations in the new context is discussed in detail.
Highlights
Ongoing discussions (Shenkel et al 2018) on the potential of quantum information and computation in plasma physics have recently led to exploiting Hilbert-space approaches in the numerical simulation of magnetized plasmas (Dodin & Startsev 2020; Engel, Smith & Parker 2019, 2021; Joseph 2020), of the Navier–Stokes equations (Gaitan 2020), and of arbitrary non-Hamiltonian systems of equations (Joseph 2020; Liu et al 2020)
The Koopman–von Neumann (KvN) and Koopman–van Hove equation’ (KvH) equations represent two valid approaches to developing a Hilbert-space formulation of classical mechanics on phase space. Both approaches lead to a generalized Clebsch representation for the particle distribution function (PDF) f
A specific choice of the complex phase factor allows this Clebsch representation for f to become equal to the Koopman prescription |Ψ |2. This choice requires the phase factor to become singular. In both formulations, the complex phase factor is generally involved in reproducing the classical dynamics
Summary
Ongoing discussions (Shenkel et al 2018) on the potential of quantum information and computation in plasma physics have recently led to exploiting Hilbert-space approaches in the numerical simulation of magnetized plasmas (Dodin & Startsev 2020; Engel, Smith & Parker 2019, 2021; Joseph 2020), of the Navier–Stokes equations (Gaitan 2020), and of arbitrary non-Hamiltonian systems of equations (Joseph 2020; Liu et al 2020). It is well known that the linearized Vlasov–Maxwell equation possesses Case–van Kampen eigenmodes and one might wish to understand how an expansion in eigenmodes, or some other complete set of eigenfunctions, converges to a solution of the nonlinear Vlasov–Maxwell equations This point of view was taken by Koopman and von Neumann in their development of ergodic theory.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
