9 research outputs found
Strong solvability of regularized stochastic Landau-Lifshitz-Gilbert equation
We examine a stochastic Landau-Lifshitz-Gilbert equation based on an exchange
energy functional containing second-order derivatives of the unknown field.
Such regularizations are featured in advanced micromagnetic models recently
introduced in connection with nanoscale topological solitons. We show that, in
contrast to the classical stochastic Landau-Lifshitz-Gilbert equation based on
the Dirichlet energy alone, the regularized equation is solvable in the
stochastically strong sense. As a consequence it preserves the topology of the
initial data, almost surely
The Allen-Cahn Action functional in higher dimensions
The Allen-Cahn action functional is related to the probability of rare events
in the stochastically perturbed Allen-Cahn equation. Formal calculations
suggest a reduced action functional in the sharp interface limit. We prove in
two and three space dimensions the corresponding lower bound. One difficulty is
that diffuse interfaces may collapse in the limit. We therefore consider the
limit of diffuse surface area measures and introduce a generalized velocity and
generalized reduced action functional in a class of evolving measures. As a
corollary we obtain the Gamma convergence of the action functional in a class
of regularly evolving hypersurfaces.Comment: 33 pages, 4 figures; minor changes and addition
Computing the optimal path in stochastic dynamical systems
In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in high- dimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly simple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higher- dimensional spaces
Multiscale temporal integrators for fluctuating hydrodynamics
Following on our previous work [S. Delong and B. E. Griffith and E.
Vanden-Eijnden and A. Donev, Phys. Rev. E, 87(3):033302, 2013], we develop
temporal integrators for solving Langevin stochastic differential equations
that arise in fluctuating hydrodynamics. Our simple predictor-corrector schemes
add fluctuations to standard second-order deterministic solvers in a way that
maintains second-order weak accuracy for linearized fluctuating hydrodynamics.
We construct a general class of schemes and recommend two specific schemes: an
explicit midpoint method, and an implicit trapezoidal method. We also construct
predictor-corrector methods for integrating the overdamped limit of systems of
equations with a fast and slow variable in the limit of infinite separation of
the fast and slow timescales. We propose using random finite differences to
approximate some of the stochastic drift terms that arise because of the
kinetic multiplicative noise in the limiting dynamics. We illustrate our
integrators on two applications involving the development of giant
nonequilibrium concentration fluctuations in diffusively-mixing fluids. We
first study the development of giant fluctuations in recent experiments
performed in microgravity using an overdamped integrator. We then include the
effects of gravity, and find that we also need to include the effects of fluid
inertia, which affects the dynamics of the concentration fluctuations greatly
at small wavenumbers.Comment: Published with some errors (fixed here) as Phys. Rev. E, 90, 063312,
201