Question 33.2: Derive an expression for the velocity correlation function ......
Derive an expression for the velocity correlation function 〈v(0)v(t)〉 for the Brownian motion problem.
Correlation functions are discussed in more detail in Section 33.6. The velocity correlation function 〈v(0)v(t)〉 is defined by
\underset{T\rightarrow \infty }{\lim }\frac{1}{T} \int_{-T/2}^{T/2}{dt^{\prime} v(t^{\prime} )v(t + t^{\prime} ),}
and describes how well, on average, the velocity at a certain time is correlated with the velocity at a later time.
The rate of change of v is given by
\dot{v} (t)= \frac{v(t + τ)−v(t) }{τ} (33.17)
in the limit in which τ → 0. Inserting this into eqn 33.1 and premuliplying by v(0) gives
m\dot{v} = −αv + F(t), (33.1)
\frac{v(0)v(t + τ)−v(0)v(t) }{τ} = -\frac{α}{m} v(0)v(t)+\frac{v(0)F(t) }{m}. (33.18)
Averaging this equation, and noting that 〈v(0)F(t)〉 = 0 because v and F are uncorrelated, yields
\frac{\left\langle v(0)v(t + τ)\right\rangle -\left\langle v(0)v(t)\right\rangle }{τ}= -\frac{α}{m}\left\langle v(0)v(t)\right\rangle , (33.19)
and taking the limit in which τ → 0 yields
\frac{d}{dt} \left\langle v(0)v(t)\right\rangle =-\frac{α}{m} \left\langle v(0)v(t)\right\rangle , (33.20)
and hence
\left\langle v(0)v(t)\right\rangle =\left\langle v(0)^{2} \right\rangle e^{−αt/m} . (33.21)