  We address the statistical mechanics of randomly and permanently crosslinked
networks. We develop a theoretical framework (vulcanization theory) which can
be used to systematically analyze the correlation between the statistical
properties of random networks and their histories of formation. Generalizing
the original idea of Deam and Edwards, we consider an instantaneous
crosslinking process, where all crosslinkers (modeled as Gaussian springs) are
introduced randomly at once in an equilibrium liquid state, referred to as the
preparation state. The probability that two functional sites are crosslinked by
a spring exponentially decreases with their distance squared. After formally
averaging over network connectivity, we obtained an effective theory with all
degrees of freedom replicated 1 + n times. Two thermodynamic ensembles, the
preparation ensemble and the measurement ensemble, naturally appear in this
theory. The former describes the thermodynamic fluctuations in the state of
preparation, while the latter describes the thermodynamic fluctuations in the
state of measurement. We classify various correlation functions and discuss
their physical significances. In particular, the memory correlation functions
characterize how the properties of networks depend on their history of
formation, and are the hallmark properties of all randomly crosslinked
materials. We clarify the essential difference between our approach and that of
Deam-Edwards, discuss the saddle-point order parameters and its physical
significance. Finally we also discuss the connection between saddle-point
approximation of vulcanization theory, and the classical theory of rubber
elasticity as well as the neo-classical theory of nematic elastomers.
