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

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.

Semi-martingale driven variational principles

Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler–Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

Two Methods for Analyzing Waves in Composites with Random Microstructure

ABSTRACTThe problem of calculating the mean wave in a composite with random microstructure is addressed. Exact characterizations of the problem can be given, in the form of stochastic variational principles. Substitution of simple configuration-dependent trial fields into these generates approximations which are, in a sense, ‘optimal’. It is necessary in practice to employ only trial fields which will generate, in the variational principle, no more statistical information than is actually available. Trial fields that require knowledge of two-point statistics generate equations that can also be obtained directly, through use of the QCA. The same fields can be substituted into an alternative variational principle to yield an approximation that makes use of three-point statistics – this approximation is less easy to obtain by direct reasoning. When not even two-point information is available, some more elementary approximation is needed. One such approximation, which is simple and direct in its application, is an extension to dynamics of a “self-consistent embedding” scheme which is widely used in static problems. This is also discussed, together with some illustrative results for a matrix containing inclusions and for a polycrystal.