Homogenization of dissipative, noisy, Hamiltonian dynamics

Abstract

We study the dynamics of a class of Hamiltonian systems with dissipation, coupled to noise, in a singular (small mass) limit. We derive the homogenized equation for the position degrees of freedom in the limit, including the presence of a noise-induced drift term. We prove convergence to the solution of the homogenized equation in probability and, under stronger assumptions, in an L p -norm. 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.

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

“…We will need the following lemma bounding the spectrum of a matrix. See, for example, Appendix A in [47] for a proof.…”

A Homogenization Theorem for Langevin Systems with an Application to Hamiltonian Dynamics

“…(3.3)-Eq. (3.4) under less restrictive assumptions on the form of the Hamiltonian, than those made in [8]. We emphasize that the convergence statements are proven in the strong sense, see Section 3.2.…”

“…t ) i =(γ −1 ) ij (t, q ǫ t )(−∂ t ψ j (t, q ǫ t ) − ∂ q j V (t, q ǫ t ) + F j (t, x ǫ t ))dt (3.6) + (γ −1 ) ij (t, q ǫ t )σ jρ (t, x ǫ t )dW ρ t − (γ −1 ) ij (t, q ǫ t )∂ q j K(t, q ǫ t , z ǫ t )dt + (z ǫ t ) j ∂ q l (γ −1 ) ij (t, q ǫ t )∂ z l K(t, q ǫ t , z ǫ t )dt − d((γ −1 ) ij (t, q ǫ t )(u ǫ t ) j ) + (u ǫ t ) j ∂ t (γ −1 ) ij (t, q ǫ t )dt, where u ǫ t ≡ p ǫ t − ψ(t, q ǫ t ), z ǫ t ≡ u ǫ t √ ǫ, and γ ik (t, q) ≡ γ ik (t, q) + ∂ q k ψ i (t, q) − ∂ q i ψ k (t, q). (3.7)We define the components ofγ −1 such that(γ −1 ) ijγ jk = δ i k ,(3.8)and for any v i we define(γ −1 v) i = (γ −1 ) ij v j .Under the additional Assumptions 5-7 in[8], which include further restrictions on the form of the Hamiltonian, we were then able to show that…”

Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

“…In this section, we study homogenization for a general class of perturbed SDEs with state-dependent coefficients. Homogenization of differential equations has been extensively studied, from the seminal works of Kurtz [37], Papanicolaou [56] and Khasminksy [33] to the more recent works [58,57,28,27,5,4,9].…”