Let us consider a two-terminal, dissipation-free “probe” circuit, playing the role of the harmonic oscillator in our analysis carried out above, connected to a resistive device (Figure \(\PageIndex{1}...Let us consider a two-terminal, dissipation-free “probe” circuit, playing the role of the harmonic oscillator in our analysis carried out above, connected to a resistive device (Figure \PageIndex1), playing the role of the probe circuit's environment. (The noise is generated by the thermal motion of numerous electrons, randomly moving inside the resistive device.) For this system, one convenient choice of the conjugate variables (the generalized coordinate and generalized force) is, respe…
Let’s take a look at this equation after Fourier transforming from x to q: \[\begin{aligned} P(x,t)&=\int\limits_{-\infty}^\infty\!\!{dq\over 2\pi}\>e^{iqx}\>{\hat P}(q,t)\\ {\hat P}(q,t)&=\in...Let’s take a look at this equation after Fourier transforming from x to q: P(x,t)=∞∫−∞dq2πeiqxˆP(q,t)ˆP(q,t)=∞∫−∞dxe−iqxP(x,t). Then as should be well known to you by now, we can replace the operator \pz\pzx with multiplication by iq, resulting in \pz\pztˆP(q,t)=−(Dq2+iqu)ˆP(q,t), with solution \[{\hat P…
