We present a statistical mechanics theory of rubber-like elasticity in swollen and unswollen polymer networks characterized by explicitly non-Gaussian distribution functions (Laplace's, Cauchy's, and continuous Poisson's in the exponential limit). An important outcome is the derivation of new families of statistical and mechanical laws, including a discussion of energy functions of strain invariants, which are reasonably simple for a model comparison with available data on polymer networks. Accordingly, a theoretical–experimental approach based on LF-NMR was devised to identify the most likely end-to-end length distribution in an arbitrary network. When this strategy is applied to agar 1 %, alginate 1 %, and scleroglucan 2 % hydrogels, it turns out that the end-to-end distribution should be never regarded as Gaussian even if, as in agar and scleroglucan systems, the normal statistics is the best among those here regarded. Remarkably, Poisson's distribution is proved instead to be the most realistic for the alginate hydrogel.
