Random perturbation to the geodesic equation

Abstract

We study random “perturbation” to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm 11. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by 8n(n−1)8n(n−1) where nn is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion

Entropy Anomaly in Langevin–Kramers Dynamics with a Temperature Gradient, Matrix Drag, and Magnetic Field

“…We assume γ and σ are C 2 and Σ(t, q) = 2β −1 (t, q)γ(t, q), (11) where β is a C 2 function that is bounded above and below by positive constants. Physically, β is related to the time and position-dependent effective temperature by β −1 = k B T , where k B is the Boltzmann constant.…”

“…Langevin-Kramers equations model the motion of a noisy, damped, diffusing particle of non-zero mass, m. In the simplest case, the stochastic differential equation (SDE) has the form on the early literature and [4,5,6,7,8,9,10,11,12,13,14] for further mathematical results in this direction.…”

Perturbation of Conservation Laws and Averaging on Manifolds

“…These will considered in terms of stochastic dynamics. See [IO90,OT96] and [Li12] concerning collapsing of Riemannian manifolds. The third example is a model on the principal bundle.…”

“…See [HP08, BvR14, FL11, FD11, DKK04, LO12, Ruf15, E11, BF95, HP04, CG16, KM17] for a range of more recent related work. We also refer to the following books [KLO12, PS08, SHS02] Averaging of stochastic differential equations on manifolds has been studied in the following articles [Kif88], [Li08], [Li12], and [GGR16]. In these studies either one restricts to local coordinates, or has a set of convenient coordinates, or one works directly with local coordinates.…”

Homogenization of dissipative, noisy, Hamiltonian dynamics

“…Applications cover the overdamped limit of particle motion in a time-dependent electromagnetic field, on a manifold with time-dependent metric, and the dynamics of nuclear matter. of processes and convergence modes on manifolds [6,7,8,9,10,11,12,13]. History of the subject and a review of the early literature can be found in [14].…”