We start with the Langevin equation $$m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}} = -\Gamma \frac{\mathrm{d}x}{\mathrm{d}t} +\sqrt{2\Gamma k_{B}T} \eta(t). $$
Now, we know that at $t \gg m/\Gamma$, the velocity degree of freedom relaxes to Maxwell distribution of velocities. Now we neglect inertia term in a viscous medium, and we are left with only the equation of motion of position variable: $$\frac{\mathrm{d}x}{\mathrm{d}t} = \sqrt{\dfrac{2 k_{B}T}{\Gamma}} \eta(t).$$
However, if we calculate $\langle v^{2}(t) \rangle_{equ}$ from the above equation, we get it to be equal to $\frac{2 k_{B}T}{\Gamma}\delta(t)$, which is not $k_{B}T/m$. Now can someone explain this violation?
