N/v-Limit for Langevin Dynamics in Continuum

Abstract

We construct an infinite particle/infinite volume Langevin dynamics on the space of configurations in R d having velocities as marks. The construction is done via a limiting procedure using N -particle dynamics in cubes (−λ, λ] d with periodic boundary conditions. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N -particle systems in (−λ, λ] d with periodic boundary conditions. After proving tightness of the laws of finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space (and their weak limit) fulfilling a uniform Ruelle bound. Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for general repulsive interaction potentials φ of Ruelle type (e.g. the Lennard-Jones potential) and all temperatures, densities and dimensions d ≥ 1.Date: October 26, 2018.

Construction, ergodicity and rate of convergence of N-particle Langevin dynamics with singular potentials

“…We remark that besides [6] also the article [18] is based on the construction result presented here. The existence of the so-called fiber lay-down dynamics analyzed there is derived in the same way.…”

“…The original motivation of this article was to provide the basis for the article [6], namely an N -particle dynamics of interacting particles on a d-dimensional torus which interact via a pair potential having a nonintegrable singularity in the origin:…”

“…only interested in equilibrium dynamics. (Concerning the use of equilibrium dynamics we mention that in [6], starting from equilibrium N -particle Langevin dynamics constructed in the present article, we construct infinite particle Langevin dynamics for a large class of singular pair interactions (including Lennard-Jones type potentials) having a grand canonical Gibbs measure as initial distribution, see also Example 3.4(i) below. Equilibrium infinite particle Langevin dynamics are for example of interest when investigating the equilibrium fluctuations for infinite particle systems, see [27].…”

“…The method for proving the weak mixing property also applies to the fiber lay-down dynamics from [18] and at present we are investigating its applicability also to the infinite particle dynamics from [6].…”

Gibbs states of continuum particle systems with unbounded spins: Existence and uniqueness

“…An advantage of the τ -topology is that it makes Γ(X) a Polish space, in contrast to the vague topology inherited from Γ( X) (which is generated by the maps (2.3) with g ∈ C 0 ( X)). For an example of the τ -consistent metric on Γ(X) see Section 2 of [7]. We then endow Γ(X) with the associated Borel σ-algebra B( Γ), also coinciding with the trace σ-algebra B(Γ( X)) ∩ Γ(X).…”