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The central limit theorem in its general form (aee Problem 1.27) can be applled to Eq. (1) of Problem 15.5 to 5 nd for large $N$ the probability distribution of $\boldsymbol{Y}=\boldsymbol{\Sigma} y_k$ since this quantity is a sum of atatiatically independent variables. By combining this result with the valuc of $\overline{Y^2}$ found in Problem 15.8, find the probability $P\left(v, t v_0\right) d v$ for the velocity $v$ after any time intsrval $t$. Show that the result thus obtained agrees with the solution $(15-12.8)$ of the Fokker-Planck equation.

   The central limit theorem in its general form (aee Problem 1.27) can be applled to Eq. (1) of Problem 15.5 to 5 nd for large $N$ the probability distribution of $\boldsymbol{Y}=\boldsymbol{\Sigma} y_k$ since this quantity is a sum of atatiatically independent variables. By combining this result with the valuc of $\overline{Y^2}$ found in Problem 15.8, find the probability $P\left(v, t v_0\right) d v$ for the velocity $v$ after any time intsrval $t$. Show that the result thus obtained agrees with the solution $(15-12.8)$ of the Fokker-Planck equation.

Fundamentals of Statistical and Thermal Physics

Rief F. 1st Edition

Chapter 15, Problem 9

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Step 1

From Problem 15.5, Equation (1) likely describes the velocity after time t as a sum of independent random variables: $$v(t) = v_0 e^{-\gamma t} + \sum_{k=1}^{N} y_k$$ where $v_0$ is the initial velocity, $\gamma$ is a damping constant, and $y_k$ are independent Show more…

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The central limit theorem in its general form (aee Problem 1.27) can be applled to Eq. (1) of Problem 15.5 to 5 nd for large $N$ the probability distribution of $\boldsymbol{Y}=\boldsymbol{\Sigma} y_k$ since this quantity is a sum of atatiatically independent variables. By combining this result with the valuc of $\overline{Y^2}$ found in Problem 15.8, find the probability $P\left(v, t v_0\right) d v$ for the velocity $v$ after any time intsrval $t$. Show that the result thus obtained agrees with the solution $(15-12.8)$ of the Fokker-Planck equation.

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