Year: 2011
Type: article
Abstract: We study the dynamics of a tagged particle in an environment of infinitely many Brownian particles in continuum. All the particles interact via the gradient of an interaction potential. Our strategy is to construct the infinite particle environment process at first and afterwards the process coupling the motion of the tagged particle and the motion of the environment. First we derive an integration by parts formula with respect to the standard gradient ∇Γon configuration spaces Γ for a general class of grand canonical Gibbs measures μ, corresponding to pair potentials ϕ and intensity measures σ = z exp (-ϕ)dx, 0 < z < ∞, having correlation functions fulfilling a Ruelle bound. Combining this with a second integration by parts formula with respect to the gradient ∇Γin direction γ ∈ Γ, by Dirichlet form techniques we can construct the environment process and the coupled process, respectively. Our results give the first mathematically rigorous and complete construction of the tagged particle process in continuum with interaction potential. In particular, we can treat interaction potentials which might have a singularity at the origin, nontrivial negative part and infinite range as e.g., the Lennard–Jones potential.
Authors: TORBEN FATTLER, MARTIN GROTHAUS
Institution: University of Kaiserslautern
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2005773809