On Equipartition of Energy and Integrals of Generalized Langevin Equations with Generalized Rouse Kernel

We show that if the motion of a particle in a linear viscoelastic liquid is described by a Generalized Langevin Equation with generalized Rouse kernel, then the resulting velocity process satisfies equipartition of energy. In doing so, we present a closed formula for the improper integration along the positive line of the product of a rational polynomial function and of even powers of the sinc function. The only requirements on the rational function are sufficient decay at infinity, no purely real poles and only simple nonzero poles. In such, our results are applicable to a family of exponentially decaying kernels. The proof of the integral result follows from the residue theorem and equipartition of energy is a natural consequence thereof. Furthermore, we apply the integral result to obtain an explicit formulae for the covariance of the position process both in the general case and for the Rouse kernel. We also discuss a numerical algorithm based on residue calculus to evaluate the covariance for the Rouse kernel at arbitrary times.