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Specifically, suppose there is a function ##A(x)## satisfying a Gaussian white noise $$\left < A(x)A(x') \right > =A_0^2\exp \left ( -\frac 1 {L^2}(x-x')^2\right )$$ where the average is taken over an ensemble.

Now I need to calculate the average ##\left < \frac {d A(x)}{dx} \frac {d A(x')}{ dx'} \right >##. I am not sure how to do this. My guess is ##\left < \frac {d A(x)}{dx} \frac {d A(x')}{ dx'} \right >=\frac {d^2}{dxdx'}\left < A(x)A(x') \right >##.

I have look it up in

*An Introduction to Thermal Physics*by Schroeder without results. I am not sure what kind of materials is related to this kind of calculation.

Thanks!