Brownian motion of elastically deformable bodies

Extensions of the Case and Ornstein-Uhlenbeck treatments of rigid-body Brownian motion to an elastically deformable body in d dimensions are given. The motion of Brownian deformations in a fluid is considered to be locally damped by a retarding friction. To describe the expected fluctuations, velocity and displacement space-time correlation functions of a body are obtained from the relaxationfunction theorem. A general equipartition theorem is obtained without postulating certain fluctuating sources. However, it is possible to exhibit those fluctuating sources which would yield our correlation functions, and thus a generalized Einstein relation between fluctuations and retarding dissipations is obtained, which is proved to be necessary and sufficient. Sum rules and both Green's and scattering functions are given for strings, elastically deformable rods, and viscoelastic gels. Curves for the ensemble-average-square displacements and their integrations over the entire length are given.