Models to describe the flow of polymeric liquids are based on a variety of different approaches. The ``phenomenological" approach assumes a certain type of relationship between the stresses and the history of the deformation, typically given in the form of integrals or systems of differential equations. ``Molecular" theories, on the other hand, begin from a simple model of the polymer molecules and lead to equations describing how they interact with the flow. Mathematically, this leads to a diffusion equation, which is a partial differential equation in some molecular configuration space. It describes how the statistical distribution of molecules evolves in time. In addition, there is an equation relating the stress tensor to the molecular distribution. The mathematical analysis of these molecular models has received little attention in the literature. In this paper, the simplest type of molecular theory, known as the dumbbell model, is considered. In this model, polymer molecules are visualized as two spherical beads, connected by a spring. The motion of the molecules is then determined by the equilibrium of three forces: the spring force, a Stokes drag on the beads exerted by the surrounding fluid, and a stochastic Brownian motion force. The stress due to the polymer is then regarded as due to the connector forces in the springs. The model provides the simplest vehicle to express the qualitative idea that polymer molecules are stretched by the surrounding flow, and that stresses result from the resistance of the polymer to such stretching. The result in our paper focusses on the simplest problem, which is when there is no flow, and we only need to be concerned with the evolution of the molecular distribution function on its own. This leads to a parabolic partial differential equation. In contrast to results which are readily available in the literature, however, this parabolic equation has unbounded coefficients. Clearly, one expects that the molecular distribution will evolve towards equilibrium, and the main result of the paper is that it does so at an exponential rate. The crucial step in proving this will be the compactness of the associated semigroup of operators.