Abstract: In this talk we study the existence and stability of solutions to a system of ODEs which can be seen as the discrete-in-space $L^2$ gradient flow of the double well potential $W$. We show first that for a special class of initial data, convergence to the (deterministic) forward-backward parabolic equation can be proved rigorously. Such discrete-in-space schemes can be seen as systems of interacting particles driven by the potential $W$. We add an interacton term and a perturbation by independent Brownian motions to their dynamics and study the arising system of coupled SDEs. We address the question how many particles are allowed to interact with each other such that the resulting interacting particle system converges to a well-posed stochastic PDE. We will then discuss the long-time behaviour of solutions to the limit SPDE. This talk combines results obtained in collaboration with Giovanni Bellettini, Anton Bovier and Matteo Novaga.