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We construct generalized grand-canonical- and canonical Gibbs measures for aHamiltonian system described in terms of a complex scalar field that is definedon a circle and satisfies a nonlinear Schr\"odinger equation with a focusingnonlinearity of order $p<6$. Key properties of these Gibbs measures, inparticular absence of "phase transitions" and regularity properties of fieldsamples, are established. We then study a time evolution of this system givenby the Hamiltonian evolution perturbed by a stochastic noise term that mimicseffects of coupling the system to a heat bath at some fixed temperature. Thenoise is of Ornstein-Uhlenbeck type for the Fourier modes of the field, withthe strength of the noise decaying to zero, as the frequency of the mode tendsto $\infty$. We prove exponential approach of the state of the system to agrand-canonical Gibbs measure at a temperature and "chemical potential"determined by the stochastic noise term.

Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schr\"odinger Equation with Noise