Question 33.1: Find the solution to the equation of motion (known as the La......

Note first that in the absence of the random force, eqn 33.1 becomes

m\dot{v} = −αv ,           (33.2)

which has solution

v(t)=v(0)\exp [−t/(mα^{−1} )],           (33.3)

so that any velocity component dies away with a time constant given by m/α. The random force F(t) is necessary to give a model in which the particle’s motion does not die away.

To solve eqn 33.1, write v = \dot{x} and premultiply both sides by x. This leads to

mx\ddot{x} = −αx\dot{x} + xF(t).          (33.4)

Now

\frac{d}{dt}(x\dot{x} )=x\ddot{x} +\dot{x}^{ 2} ,           (33.5)

and hence we have that

m \frac{d}{dt} (x\dot{x} )=m\dot{x}^{2} −αx\dot{x} + xF(t).          (33.6)

We now average this result over time. We note that x and F are uncorrelated, and hence 〈xF〉 = 〈x〉〈F〉 = 0. We can also use the equipartition theorem, which here states that

\frac{1}{2}m\left\langle \dot{x}^{2} \right\rangle = \frac{1}{2} k_{B} T.          (33.7)

Hence, using eqn 33.7 in eqn 33.6, we have

m\frac{d}{dt}\left\langle x\dot{x} \right\rangle = k_{B} T – α\left\langle x\dot{x} \right\rangle ,          (33.8)

or equivalently

\left(\frac{d}{dt}+\frac{α }{m} \right)\left\langle x\dot{x} \right\rangle = \frac{k_{B} T}{m},          (33.9)

which has a solution

\left\langle x\dot{x} \right\rangle = Ce^{−αt/m} + \frac{ k_{B} T}{ α}.          (33.10)

Putting the boundary condition that x = 0 when t = 0, one can find that the constant C = −k_{B}T/α, and hence

\left\langle x\dot{x} \right\rangle = \frac{k_{B} T}{α}(1−e^{−αt/m} ).           (33.11)

Using the identity

\frac{1}{2} \frac{d}{dt}\left\langle x^{2} \right\rangle = \left\langle x\dot{x} \right\rangle ,          (33.12)

we then have

\left\langle x^{2} \right\rangle = \frac{ 2k_{B} T }{α}\left[t− \frac{m}{ α} (e^{−αt/m} )\right] .          (33.13)

When t \ll m/α,

\left\langle x^{2} \right\rangle = \frac{ k_{B}T t^{2} }{m},            (33.14)

while for t \gg m/α,

\left\langle x^{2} \right\rangle = \frac{ 2k_{B}T t }{α}.           (33.15)

Writing{}^{1} 〈x²〉 = 2Dt, where D is the diffusion constant, yields D = k_{B}T/α.

{}^{1}See Appendix C.12.