Consider the Langevin equation (1) of Problem 15.4. (a) Write F^' and v in terms of Fourier inte

Step 1: First, let's recall the Langevin equation from Problem 15.4. Although it's not explicitl

Consider the Langevin equation (1) of Problem 15.4. (a)...

Consider the Langevin equation (1) of Problem 15.4. (a) Write $F^{\prime}$ and $v$ in terms of Fourier integrals, and show how their Fourier cuefficients must be related to atisfy the Langevin equation. Expreas the epectral density of the velocity in terms of the spectral density of the force $F^{\prime}(t)$. (b) Use the Wiener-Khintchine relations to find the spectral density of the velocity $\boldsymbol{v}$ from the known correlation function ( $\mathbf{1 5} \cdot 10 \cdot 9$ ) of this quantity. (c) By combining the preceding results of this problem, fied an explicit expression for the spectral density of the force $F^{\prime}$ in terms of $\boldsymbol{\gamma}$. This constitutes enother derivation of Nyquist's theorem.

Consider the Langevin equation (1) of Problem 15.4.
(a) Write $F^{\prime}$ and $v$ in terms of Fourier integrals, and show how their Fourier cuefficients must be related to atisfy the Langevin equation. Expreas the epectral density of the velocity in terms of the spectral density of the force $F^{\prime}(t)$.
(b) Use the Wiener-Khintchine relations to find the spectral density of the velocity $\boldsymbol{v}$ from the known correlation function ( $\mathbf{1 5} \cdot 10 \cdot 9$ ) of this quantity.
(c) By combining the preceding results of this problem, fied an explicit expression for the spectral density of the force $F^{\prime}$ in terms of $\boldsymbol{\gamma}$. This constitutes enother derivation of Nyquist's theorem.

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