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

Abstract

We construct N -particle Langevin dynamics in R d or in a cuboid region with periodic boundary for a wide class of N -particle potentials and initial distributions which are absolutely continuous w.r.t. Lebesgue measure. The potentials are in particular allowed to have singularities and discontinuous gradients (forces). An important point is to prove an L p -uniqueness of the associated non-symmetric, non-sectorial degenerate elliptic generator. Analyzing the associated functional analytic objects, we also give results on the long-time behaviour of the dynamics, when the invariant measure is finite: Firstly, we prove the weak mixing property whenever it makes sense (i.e. whenever { < ∞} is connected). Secondly, for a still quite large class of potentials we also give a rate of convergence of time averages to equilibrium when starting in the equilibrium distribution. In particular, all results apply to N -particle systems with pair interactions of Lennard-Jones type.

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

“…This is done by constructing an explicit Lyapunov function. The class of admissible potentials, defined below in Section 2, is comparable to those considered in [2,9], but the results are in line with those proven in [3] and hence stronger. In particular, although we do not treat potentials that are merely weakly differentiable on the set where U < 1, our convergence results allow for the analysis of numerical methods used to simulate molecular dynamics or sample from the density using Monte Carlo methods.…”

“…A more recent set of works develops hypercoercivity in the L 2 setting [6,7,10], which simplifies the analysis in many respects. This was then adapted to dynamics with singular potentials in [9], which extended and simplified the original approach in [2]. However, in both [2,9], the system must start in equilibrium.…”

“…As a result, the system (1.1) with a singular potential U is not covered by most results on second-order Langevin dynamics. Notable exceptions are the papers [2,9], where a mixture of martingale techniques and ideas from hypocoercivity are used to prove the existence of the process as well as exponential mixing, in the sense of exponential decorrelation, provided the system is started in the equilibrium measure given by .dp dp/ D 1 Z e 1 T H.p;q/ dq dp: (1.4)…”

Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

“…Concerning these statements we refer to Conrad et al, [17,18] in particular Conrad and Grothaus [17,Theorem 2.5] and Conrad. Concerning these statements we refer to Conrad et al, [17,18] in particular Conrad and Grothaus [17,Theorem 2.5] and Conrad.…”

“…is solved by P (x, ) for quasi any starting point (x, ) ∈ R d × S d−1 . Concerning these statements we refer to Conrad et al, [17,18] in particular Conrad and Grothaus [17,Theorem 2.5] and Conrad. [18,Lemma 2.2.8] Such results can be established via the powerful theory of (generalised) Dirichlet forms.…”