Why should $\langle xf_r\rangle=0$ but $\langle\dot{x}f_r\rangle\ne 0$? All the $\langle\rangle$ in this question is the mean value theorem over a large number of experiments.
Consider a Brownian particle moving in a liquid with the viscosity $\mu$. The equation of motion in the $x$ axis is:
$$m\frac{d^2x}{dt^2}=-\mu\frac{dx}{dt}+f_r$$
where $f_r$ is a random force representing the collision with the small moleculars in the liquid.
Multiply the equation by $x$ we have
$$m\frac{d^2}{dt^2}\left(\frac{x^2}{2}\right)-m\dot{x}^2=-\mu\frac{d}{dt}\left(\frac{x^2}{2}\right)+xf_r$$
Taking the mean value and if we assume that: $(1)$ $\langle xf_r\rangle=0$ and $(2)$ $\langle\frac{m\dot{x}^2}{2}\rangle=\frac{kT}{2}$ we got:
$$m\frac{d^2}{dt^2}\langle x^2\rangle=-\mu\frac{d}{dt}\langle x^2\rangle+\frac{2kT}{\mu}$$
This is a simple equation for $\langle x^2\rangle$ and when $t$ is larege enough we have $\langle x^2\rangle=\frac{2kT}{\mu}t$.
If we multiply the equation of motion by $\dot{x}$ and take the mean value, we will see that $\langle\dot{x}f_r\rangle\ne 0$. In fact this is the mean work per time done by the random force to win the viscous force.
How can I explain intuitively and simply why $\langle xf_r\rangle=0$ but $\langle\dot{x}f_r\rangle\ne 0$? The random force seem to not be related to $x$ or $\dot{x}$...
 A: Here is the intuitive explanation:
When a particle is moving, it will "run into" things. Thus, the "random force" from impacting another particle is not completely random: it is in part correlated to the motion of the particle before the collision - the force of the impact is more likely to be in a direction opposite to the current motion than any other direction. And it is this partial correlation which means that the expectation of $\langle \dot{x}f_r\rangle$ will not be zero.
On the other hand, there is no correlation between the current position of the particle and the direction of the force of the next impact - which is why tjhe expectation of $\langle x f_r\rangle$ is zero.
A: In brownian motion (using the langevin equation) the average (net) displacement is zero $\langle x\rangle=0$ and since the random force $f_r$ (usually white noise) is uncorrelated to the displacement (and has zero mean as well) the joint average $\langle xf_r\rangle=0$ is zero as well.
The correlation of the velocity (frictional viscocity term depending on velocity) and the random force is described in the fluctuation-dissipation theorem as such (reference):
The random force (the fluctuation) is the source and driving force of the brownian particle, thus is also the source of the secondary term of the friction (friction, depending on velocity, the dissipation), thus these two terms must be correlated.
The expression $\langle\dot{x}f_r\rangle\ne 0$ defines exactly this correlation between these two facets of the same effect (fluctuation-dissipation).
