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Register a new userTheory and applications of the Vlasov equation
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Added by
Francesco Pegoraro Dr.
Publication date
2015
fields
Physics
and research's language is
English
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Forty articles have been recently published in EPJD as contributions to the topical issue Theory and applications of the Vlasov equation. The aim of this topical issue was to provide a forum for the presentation of a broad variety of scientific results involving the Vlasov equation. In this editorial, after some introductory notes, a brief account is given of the main points addressed in these papers and of the perspectives they open.
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The low-frequency limit of Maxwell equations is considered in the Maxwell-Vlasov system. This limit produces a neutral Vlasov system that captures essential features of plasma dynamics, while neglecting radiation effects. Euler-Poincare reduction theory is used to show that the neutral Vlasov kinetic theory possesses a variational formulation in both Lagrangian and Eulerian coordinates. By construction, the model recovers all collisionless neutral models employed in plasma simulations. Then, comparisons between the neutral Vlasov system and hybrid kinetic-fluid models are presented in the linear regime.
Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here we present different approaches to Koopmans 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 noncanonical 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 Hoves 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.
Difficulties in founding microscopically the Vlasov equation for Coulomb-interacting particles are recalled for both the statistical approach (BBGKY hierarchy and Liouville equation on phase space) and the dynamical approach (single empirical measure on one-particle $(mathbf{r},mathbf{v})$-space). The role of particle trajectories (characteristics) in the analysis of the partial differential Vlasov--Poisson system is stressed. Starting from many-body dynamics, a direct derivation of both Debye shielding and collective behaviour is sketched.
The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $mathcal{O}(text{polylog}(N_v) t/delta)$ where $delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/delta$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.
We extend our analysis of divergence-free positive symmetric tensors (DPT) begun in a previous paper. On the one hand, we refine the statements and give more direct proofs. Next, we study the most singular DPTs, and use them to prove that the determinant is the only quantity that enjoys an improved integrability. Curiously, these singularities are intimately related to the Minkowskis Problem for convex bodys with prescribed Gaussian curvature. We then cover a list of models of mathematical physics that display a divergence-free symmetric tensor ; the most interesting one is probably that of nonlinear Maxwells equations in a relativistic frame. The case of the wave equation is the occasion to highlight the role of the positivity assumption. Last, but not least, we show that the Vlasov--Poisson equation for a plasma is eligible for our theory.
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