Question

Express $\bar{G}^{\mathbf{i}}$ in terms of the correlation function $K(s)=\left(F^{\prime}(l) F^{\prime}(\ell+e)\right)$ of the random force and use the result of Problem 15.8 to rederive in this way the fluctuation-dissipation theorem ( $\mathbf{1 5} \cdot \mathrm{S} \cdot 8$ ) relating the friction constant to $K(\mathrm{~s})$.

   Express $\bar{G}^{\mathbf{i}}$ in terms of the correlation function $K(s)=\left(F^{\prime}(l) F^{\prime}(\ell+e)\right)$ of the random force and use the result of Problem 15.8 to rederive in this way the fluctuation-dissipation theorem ( $\mathbf{1 5} \cdot \mathrm{S} \cdot 8$ ) relating the friction constant to $K(\mathrm{~s})$.

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Fundamentals of Statistical and Thermal Physics

Fundamentals of Statistical and Thermal Physics

Rief F. 1st Edition

Chapter 15, Problem 10

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Step 1

This is the average impulse response function, which describes how the velocity of a Brownian particle responds to an impulse force. In the context of a Langevin equation, it relates to how the velocity evolves after an initial impulse. Show more…

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Express $\bar{G}^{\mathbf{i}}$ in terms of the correlation function $K(s)=\left(F^{\prime}(l) F^{\prime}(\ell+e)\right)$ of the random force and use the result of Problem 15.8 to rederive in this way the fluctuation-dissipation theorem ( $\mathbf{1 5} \cdot \mathrm{S} \cdot 8$ ) relating the friction constant to $K(\mathrm{~s})$.

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