Weighted L 2-contractivity of Langevin dynamics with singular potentials

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al, we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2(dμ) and L 2(W* dμ), where μ denotes the invariant probability measure and W* is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min(γ, γ −1). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

Inferring temporal dynamics from cross-sectional data using Langevin dynamics

Cross-sectional studies are widely prevalent since they are more feasible to conduct compared with longitudinal studies. However, cross-sectional data lack the temporal information required to study the evolution of the underlying dynamics. This temporal information is essential to develop predictive computational models, which is the first step towards causal modelling. We propose a method for inferring computational models from cross-sectional data using Langevin dynamics. This method can be applied to any system where the data-points are influenced by equal forces and are in (local) equilibrium. The inferred model will be valid for the time span during which this set of forces remains unchanged. The result is a set of stochastic differential equations that capture the temporal dynamics, by assuming that groups of data-points are subject to the same free energy landscape and amount of noise. This is a ‘baseline’ method that initiates the development of computational models and can be iteratively enhanced through the inclusion of domain expert knowledge as demonstrated in our results. Our method shows significant predictive power when compared against two population-based longitudinal datasets. The proposed method can facilitate the use of cross-sectional datasets to obtain an initial estimate of the underlying dynamics of the respective systems.