Statistical Properties of Networks of Flexible Chains

A Gaussian network is defined as a network of flexible chain segments, linked to each other and to a system of fixed points, in which each unbranching chain segment can take on a number of configurations which is a Gaussian function of the distance between its ends. Real molecular networks, such as those of rubber, can under certain circumstances be treated as Gaussian networks. The present paper carries out a systematic mathematical discussion of the statistical properties of Gaussian networks: the total number of possible configurations of the network as a function of the fixed‐point coordinates, the probability of finding a given element of the network in a given position, or of finding two elements of the network in given relative positions, and so on. All probability‐density functions appear as exponentials of quadratic forms, with constants explicity expressible in determinant form. An explicit reduction to a sum of squares is given for all quadratic forms occurring in the theory of coherent Gaussian networks, and an explicit general formula is found for integrals of the form $${\displaystyle {\int }_{\mathrm{-\infty }}^{\mathrm{+\infty }}}{\mathrm{dX}}_1{\displaystyle {\int }_{\mathrm{-\infty }}^{\mathrm{+\infty }}}{\mathrm{dX}}_2\mathrm{···}{\displaystyle {\int }_{\mathrm{-\infty }}^{\mathrm{+\infty }}}{\mathrm{dX}}_q\hspace{0.17em}\exp \left\{-{\displaystyle \sum _i^n}{\displaystyle \sum _j^n}{\gamma }_{\mathrm{ij}}{X}_i{X}_j\right\}.$$ There is described a mechanical analog of a Gaussian network, by consideration of which the statistical properties of the Gaussian network can be determined. The method is applied to the discussion of the statistical properties of a Gaussian network with the connectivity of a regular cubic lattice.