We present a study of the coarse-Braining of polymers into bead-spring chains using statistical mechanics. The force-extension behavior is examined at different levels of coarse-Braining. A direct result of the springs being decoupled is that the force-extension behavior depends only on the number of flexibility lengths (e.g. persistence or Kuhn lengths) represented by each spring. This dimensionless parameter is found to govern the fluctuations around the mean extension, analogous to the conventional role of temperature. The use of an effective flexibility length to correct the model behavior is analyzed, and we have calculated bounds on the choices of this correction factor. The analytic nature of the statistical mechanical framework has also allowed for the calculation of asymptotic and universal behavior. The zero Weissenberg number theological behavior is examined using the retarded-motion expansion coefficients of bead-spring chains at different levels of coarse-Braining. The results show the trade-off between using too few or too many springs. The general framework to analyze the force-extension and theological behavior is applied to the worm-like chain, FENE, and Fraenkel models. We introduce a new method for coarse-Braining a polymer into a bead-spring chain called the Polymer Ensemble Transformation (PET) method. Application to the freely jointed chain polymer yields a set of spring force-laws called the Random Walk Spring (RWS) model. This new method illustrates why the previous spring force-laws cannot be used to finely discretize polymers and also provides new insight into how to rationally proceed in the coarse-Braining of polymers into bead-spring chains. (C) 2004 Elsevier B.V. All rights reserved.