Stochastic variational principles for the collisional Vlasov–Maxwell and Vlasov–Poisson equations

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.

Symplectic model reduction methods for the Vlasov equation

“…The results for the complex singular-value decomposition algorithm are nearly identical, and are therefore omitted for clarity fields satisfying the Maxwell equations. [24] Second, model reduction of collisional Vlasov equations stemming from metriplectic brackets [52] or stochastic action principles [53][54][55] would be an interesting and useful extension of the work presented here.…”

“…First, model reduction methods can be applied to the Vlasov equation coupled to a self‐consistent electric field satisfying the Poisson equation, or electromagnetic fields satisfying the Maxwell equations. [ 24 ] Second, model reduction of collisional Vlasov equations stemming from metriplectic brackets [ 52 ] or stochastic action principles [ 53–55 ] would be an interesting and useful extension of the work presented here.…”

Data-driven structure-preserving model reduction for stochastic Hamiltonian systems

“…A natural follow-up study would be the application of the presented model reduction techniques to semi-discretizations of other stochastic PDEs with underlying geometric structures, such as the stochastic Camassa-Holm equation ( [20], [79], [80]), the stochastic quasi-geostrophic equation ( [19], [52], [53], [54], [55]), the stochastic Korteweg-de Vries equation ( [46], [56], [75], [80]), or the stochastic Sine-Gordon equation ( [74], [145], [147], [155]) to name just a few. Particle discretizations of collisional Vlasov equations have been recently proved to have the structure of stochastic forced Hamiltonian systems ( [94], [146], [149]), therefore kinetic plasma theory is yet another area where structure-preserving model reduction techniques could be applied. Furthermore, it would be interesting to extend the methods discussed in our work to stochastic non-canonical systems.…”

“…Applications of such systems arise in many models in physics, chemistry, and biology. Particular examples include molecular dynamics (see, e.g., [18], [88], [98], [137]), dissipative particle dynamics (see, e.g., [131]), investigations of the dispersion of passive tracers in turbulent flows (see, e.g., [134], [144]), energy localization in thermal equilibrium (see, e.g., [130]), lattice dynamics in strongly anharmonic crystals (see, e.g., [71]), description of noise induced transport in stochastic ratchets (see, e.g., [100]), and collisional kinetic plasmas ( [91], [94], [139], [146]). While their stochastic flow is not symplectic in general, stochastic forced Hamiltonian systems have an underlying variational structure, that is, their solutions satisfy the stochastic Lagrange-d'Alembert principle (see [94]).…”

A stochastic Hamiltonian formulation applied to dissipative particle dynamics

“…The converse is also true providing the regularity of q and p [29]. In particular if h ij = h ij (q) are all independent of p, which is exactly the case for DPD, (q, p) is a solution of the SHF (1) if and only if it satisfies the stochastic Lagrange-d'Alembert principle (6) [30].…”