Generalize the analysis of the Langevin theory of a harmonic oscillator, as given by equation (15.3.33), to the case of an oscillator starting at time \(t=0\) with the initial position \(x(0)\) and the initial velocity \(v(0)\). Derive, for this system, the quantities \(\left\langle x^{2}(t)\rightangle\) and \(\left\langle v^{2}(t)\rightangle\) and show that, in the limit \(\omega_{0} \rightarrow 0\), these expressions reproduce equations (15.3.29) and (15.3.31) while, in the limit \(M \rightarrow 0\), they lead to the relevant results of Section 15.4.

Data From Equation (15.3.29)

Data From Equation (15.3.31)

Data From Equation (15.3.33)

Transcribed Image Text:

(v (t)) = v (0) + { 3kT M