ترغب بنشر مسار تعليمي؟ اضغط هنا

Stochastic Evolution of Augmented Born--Infeld Equations

157   0   0.0 ( 0 )
 نشر من قبل Darryl D. Holm
 تاريخ النشر 2017
  مجال البحث فيزياء
والبحث باللغة English
 تأليف Darryl D. Holm




اسأل ChatGPT حول البحث

This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born-Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations (SPDE) retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases; which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born-Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two Appendices treat the Hamiltonian structures underlying these results.



قيم البحث

اقرأ أيضاً

A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary differential equat ion. In this representation it can be shown that solitons and solutions with compact support only exist in systems with linear or quadratic dispersion, respectively. In particular, this article deals with so the called K(n,m) equations. It is shown that these equations can be classified according to a simple point transformation. As a result, all equations that allow for soliton solutions join the same equivalence class with the Korteweg-deVries equation being its representative.
We study standing-wave solutions of Born-Infeld electrodynamics, with nonzero electromagnetic field in a region between two parallel conducting plates. We consider the simplest case which occurs when the vector potential describing the electromagneti c field has only one nonzero component depending on time and on the coordinate perpendicular to the plates. The problem then reduces to solving the scalar Born-Infeld equation, a nonlinear partial differential equation in 1+1 dimensions. We apply two alternative methods to obtain standing-wave solutions to the Born-Infeld equation: an iterative method, and a ``minimal surface method. We also study standing wave solutions in a uniform constant magnetic field background.
We discuss Bogomolnyi equations for general gauge theories (depending on the two Maxwell invariants $F^{mu u} F_{mu u}$ and $tilde F^{mu u} F_{mu u}$) coupled to Higgs scalars. By analysing their supersymmetric extension, we explicitly show why t he resulting BPS structure is insensitive to the particular form of the gauge Lagrangian: Maxwell, Born-Infeld or more complicated non-polynomial Lagrangians all satisfy the same Bogomolnyi equations and bounds which are dictated by the underlying supersymmetry algebra.
117 - Darryl D. Holm 2017
In the pattern matching approach to imaging science, the process of emph{metamorphosis} in template matching with dynamical templates was introduced in cite{ty05b}. In cite{HoTrYo2009} the metamorphosis equations of cite{ty05b} were recast into the E uler-Poincare variational framework of cite{HoMaRa1998} and shown to contain the equations for a perfect complex fluid cite{Holm2002}. This result related the data structure underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids cite{GBHR2013}. In particular, it cast the concept of Lagrangian paths in imaging science as deterministically evolving curves in the space of diffeomorphisms acting on image data structure, expressed in Eulerian space. (In contrast, the landmarks in the standard LDDMM approach are Lagrangian.) For the sake of introducing an Eulerian uncertainty quantification approach in imaging science, we extend the method of metamorphosis to apply to image matching along emph{stochastically} evolving time dependent curves on the space of diffeomorphisms. The approach will be guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in cite{holm2015variational,CrFlHo2017}.
It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Field s, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or Dyson) models.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا