=m Eq. [eqn:force] can be written as, m{dt}=F(t) It was Langevin who proposed that the net force on the BP had two components, the friction force \(bv(t)\), and the random force \(xi(t)\)\(xi(t)\) given by the equation then named after him, m{dt}=-bv(t)+\xi(t) This is the Langevin equation of a Free Brownian Particle. It can be used to describe the motion of a particle that undergoes normal diffusion. This is not the case for a BHO. It is confined within a harmonic potential $V(x)={2}$ and is therefore not free to undergo normal diffusion. In order to correctly describe the behavior of a BHO, Eq. [eqn:Lang] must be modified. m{dt}=-bv(t)-kx(t)+\xi(t) where _k_ is the spring constant, by dividing Eq. [eqn:bho] by _m_, {dt}=-\beta v(t)-\omega^2_ox(t)+\xi(t) This is the Langevin Equation for a Brownian Harmonic Oscillator, where $\beta = {m}$ and $\omega_o ={m}}$. Here the random force is a Gaussian random process (white noise) with moments <\xi(t)> = 0 <\xi(t)\xi(t+\tau)> = 2\beta k_BT\delta(t-(t+\tau)) All of the information we require from the Langevin Eq. is contained in the correlation function. From Eq. [eqn:bho2] the velocity auto-correlation \(C_v\) function can be obtained from the Wiener-Khinchin theorem. The position auto-correlation \(C_x\) function must first be obtained. x(\omega)= {m(\omega^2_o+i\omega\beta-\omega^2)}, and the Spectral Density \(\phi_x(\omega)\) of the displacement, \phi_x(\omega)={m^2[(\omega^2_o-omega^2)^2+\omega^2\beta^2]} Since ϕξ(ω)=2βmkBT, C_x = {\pi m}\int_\infty ^\infty {(\omega^2_o-\omega^2)^2+\omega^2\beta^2}. The integral can be evaluated as a contour integral and the imaginary part of the integral disappears, because it will result to an odd function in the integrand. For $\omega^2_o > {4}$ and τ ≥ 0, C_x = {\pi} \int_\infty^\infty {(\omega^2_o-\omega^2)^2+\omega^2\beta^2} = \frac {e^{2}}{\beta\omega^2_o}(\cos\omega_1\tau + {2\omega_1}\sin\omega_1\tau), where \(\omega_1\) is the damped natural frequency; \omega^2_1 = \omega^2_o - {4}. C_x = {m\omega^2_o} {2}}{\beta\omega^2_o}(\cos\omega_1\tau + {2\omega_1}\sin\omega_1\tau). By differentiating the position-velocity cross-correlation function \(C_xv\), C_xv = <x(t){d\tau}x(t+\tau)> = {d\tau}<x(t)x(t+\tau)>, leads to C_xv = -{m\omega_1} e^{2}\sin\omega_1\tau. By stationary shifting the time axis from \(t\) to t − τ, <x(t+\tau){dt}x(t)> = -{d\tau}<x(t)x(t+\tau)> = {m\omega_1}e^{2}\sin\omega_1\tau Calculating for the \(C_v\) by stationarity, C_v = -{d\tau}<x(t)v(t+\tau)> = -{d\tau^2}<x(t)x(t+\tau)>. C_v = {m}e^{2}(\cos\omega_1\tau - {2\omega_1}\sin\omega_1\tau) The relationship between the velocity auto-correlation function and the diffusion coefficient \(D\) is given by the Green-Kubo relation. D = \int_0^\infty C_v dt