  We study the time correlation functions of coupled linear Langevin dynamics
without and with inertia effects, both analytically and numerically. The model
equation represents the physical behavior of a harmonic oscillator in two or
three dimensions in the presence of friction, noise, and an external field with
both rotational and deformational components. This simple model plays pivotal
roles in understanding more complicated processes. The presented analytical
solution serves as a test of numerical integration schemes, its derivation is
presented in a fashion that allows to be repeated directly in a classroom.
While the results in the absence of fields (equilibrium) or confinement (free
particle) are omnipresent in the literature, we write down, apparently for the
first time, the full nonequilibrium results that may correspond, e.g., to a
Hookean dumbbell embedded in a macroscopically homogeneous shear or mixed flow
field. We demonstrate how the inertia results reduce to their noninertia
counterparts in the nontrivial limit of vanishing mass. While the results are
derived using basic integrations over Dirac delta distributions, we mention its
relationship with alternative approaches involving (i) Fourier transforms, that
seems advantageous only if the measured quantities also reside in Fourier
space, and (ii) a Fokker--Planck equation and the moments of the probability
distribution. The results, verified by numerical experiments, provide
additional means of measuring the performance of numerical methods for such
systems. It should be emphasized that this manuscript provides specific details
regarding the derivations of the time correlation functions as well as the
implementations of various numerical methods, so that it can serve as a
standalone piece as part of education in the framework of stochastic
differential equations and calculus.
