Question

Use the solution (2) of Problem 15.4 to calculate directly $\overline{v^2(l)}$ for any time $t \gg \tau^*$ without expllcitly breaking up the range of integration of the integral into discrete intervals as was done in Problem 15.5. Use the fact that $\tau^* \ll \boldsymbol{\gamma}^{-1}$ so that the correlation function $\left(F^{\prime}(0) F^{\prime}(s)\right)$ is appreciable only when $\boldsymbol{\gamma}_8 \ll 1$. Show that the result thus obtained agrees with that derived previously. Show also that this result yields immediately the general fluctuation-dissipation theorem ( $15 \cdot 8 \cdot 8$ ) if one makes use of the requirement thet $\frac{1}{2} m \vec{v}=\frac{1}{4} k T$ in the final equilibrium situation when $t \rightarrow \infty$.

   Use the solution (2) of Problem 15.4 to calculate directly $\overline{v^2(l)}$ for any time $t \gg \tau^*$ without expllcitly breaking up the range of integration of the integral into discrete intervals as was done in Problem 15.5. Use the fact that $\tau^* \ll \boldsymbol{\gamma}^{-1}$ so that the correlation function $\left(F^{\prime}(0) F^{\prime}(s)\right)$ is appreciable only when $\boldsymbol{\gamma}_8 \ll 1$. Show that the result thus obtained agrees with that derived previously. Show also that this result yields immediately the general fluctuation-dissipation theorem ( $15 \cdot 8 \cdot 8$ ) if one makes use of the requirement thet $\frac{1}{2} m \vec{v}=\frac{1}{4} k T$ in the final equilibrium situation when $t \rightarrow \infty$.
 
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Fundamentals of Statistical and Thermal Physics
Fundamentals of Statistical and Thermal Physics
Rief F. 1st Edition
Chapter 15, Problem 13 ↓

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Use the solution (2) of Problem 15.4 to calculate directly $\overline{v^2(l)}$ for any time $t \gg \tau^*$ without expllcitly breaking up the range of integration of the integral into discrete intervals as was done in Problem 15.5. Use the fact that $\tau^* \ll \boldsymbol{\gamma}^{-1}$ so that the correlation function $\left(F^{\prime}(0) F^{\prime}(s)\right)$ is appreciable only when $\boldsymbol{\gamma}_8 \ll 1$. Show that the result thus obtained agrees with that derived previously. Show also that this result yields immediately the general fluctuation-dissipation theorem ( $15 \cdot 8 \cdot 8$ ) if one makes use of the requirement thet $\frac{1}{2} m \vec{v}=\frac{1}{4} k T$ in the final equilibrium situation when $t \rightarrow \infty$.
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