Trying to understand the Langevin Equation. In particular, this passage from a Wikipedia article has me confused (section: "Thermal Noise in an Electrical Resistor"):
$\frac{dU}{dt} =-\frac{U}{RC}+\eta \left( t\right),\;\;$ $\left\langle\eta \left( t\right) \eta \left( t^{\prime }\right)\right\rangle = \frac{2k_{B}T}{RC^{2}}\delta \left(t-t^{\prime }\right).$
This equation may be used to determine the correlation function
$\left\langle U\left(t\right) U\left(t^{\prime }\right) \right\rangle =\left( k_{B}T/C\right) \exp \left( -\left\vert t-t^{\prime }\right\vert /RC\right) \approx 2Rk_{B}T\delta \left( t-t^{\prime}\right),$
Here's my derivation of a rather different result:
$\displaystyle \frac{dU(t)}{dt} =-\frac{U(t)}{RC}+\eta \left( t\right)$
$\displaystyle U(t+t^\prime)\frac{dU(t)}{dt} =-\frac{U(t+t^\prime)U(t)} {RC}+U(t+t^\prime)\eta(t)$
Now concentrate on LHS:
$\displaystyle U(t+t^\prime)\frac{dU(t)}{dt} = \frac{d}{dt}(U(t+t^\prime)U(t))-U(t)\frac{d}{dt}U(t+t^\prime)= \frac{d}{dt}(U(t+t^\prime)U(t))-U(t)\frac{d}{dt^\prime}U(t+t^\prime)$ $\displaystyle=\frac{d}{dt}(U(t+t^\prime)U(t))-\frac{d}{dt^\prime}(U(t)U(t+t^\prime))$
now do ensemble averaging and assuming system is statistically stationary. Then the first term on the new LHS vanishes as does the second term on the RHS, leaving:
$\displaystyle -\frac{d}{dt^\prime}\langle U(t+t^\prime)U(t)\rangle = -\frac{\langle U(t+t^\prime)U(t)\rangle} {RC}$
which gives the obviously unphysical
$\displaystyle \langle U(0)U(t^\prime)\rangle=\langle U(0)U(0) \rangle \exp(+t^\prime / RC)$
Can anyone tell me where I went wrong and supply the derivation of the Wikipedia result?