Question 33.7: Find the velocity autocorrelation for the Brownian motion pa......
Find the velocity autocorrelation for the Brownian motion particle governed by eqn 33.1 where the random force F(t) is as described in the previous example, i.e. with 〈F(t)F(t^{\prime })〉 = Aδ(t−t^{\prime }). Hence relate the constant A to the temperature T.
m\dot{v} = −αv + F(t), (33.1)
Equation 33.1 states that
m\dot{v} = −αv + F(t), (33.88)
and the Fourier transform of this equation is
\widetilde{v} (ω) =\frac{\widetilde{F} (ω) }{α−imω}. (33.89)
This implies that the Fourier transform of the velocity autocorrelation function is
\widetilde{C} _{vv} (ω) = \left\langle \left|v(ω)\right|^{2} \right\rangle \frac{A}{α^{2} + m^{2} ω^{2} }, (33.90)
using the result of eqn 33.87. The Wiener–Khinchin theorem states that
\left\langle |F(ω)|^{2} \right\rangle = A (33.87)
\widetilde{C} _{vv} (ω) = \int e^{−iωt} \left\langle v(0)v(t)\right\rangle dt, (33.91)
and hence
\widetilde{C} _{vv} (t) = \left\langle v(0)v(t)\right\rangle = \left\langle v^{2} \right\rangle e^{−αt/m} , (33.92)
in agreement with eqn 33.21 derived earlier using another method. Parseval’s theorem (eqn 33.84) implies that
\left\langle v(0)v(t)\right\rangle =\left\langle v(0)^{2} \right\rangle e^{−αt/m} . (33.21)
\left\langle v^{2} \right\rangle = \int_{-\infty }^{\infty }{} \frac{dω}{2π} \widetilde{C} _{vv} (ω) = \frac{A}{2mα}. (33.93)
Equipartition, which gives that \frac{1}{2}m〈v²〉 = \frac{1}{2}k_{B}T, leads immediately to
A =2αk_{B}T. (33.94)