Abstract
In this work, we recast the collisional Vlasov–Maxwell and Vlasov–Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov–Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.
Highlights
The collisional Vlasov equation ∂f ∂t + v · ∇xf q (E m × B) ∇v f = C[f ]
We have considered novel stochastic formulations of the collisional Vlasov–Maxwell and Vlasov–Poisson equations, and we have identified new stochastic variational principles underlying these formulations
We have proposed a stochastic particle method for the Vlasov–Maxwell equations and proved the corresponding stochastic variational principle
Summary
We formulate a stochastic particle discretization for the collisional Vlasov–Maxwell equations and cast it in a form that allows the derivation of a variational principle. The main result of this section is theorem 5.1, in which a stochastic Lagrange–d’Alembert principle is proved for a class of the collisional Vlasov–Maxwell equations. It can be verified that an example decomposition (2.2) for M = 3 is given by the functions Note that these functions do not explicitly depend on f , in this case (1.1) is a linear Fokker–Planck equation. It can be verified by a straightforward calculation that an example decomposition (2.2) for M = 3 is given by the functions The collision operator (2.1) with Dij and Ki as in (2.9) can be expressed in an equivalent, more symmetric form, known as the Landau form of the Coulomb operator, or the Landau collision operator (e.g. [50])
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