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(8.17)
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As r is very small and (6.) differs from zero only for small 6., we can expand both p(x, t + r) and p(x - 6., t) in power series of rand 6., respectively: p(x, t
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(8.18) (8.19)
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Substituting these expansions in (8.17) and keeping only the lowest nonvanishing order after zeroth order, Einstein found that p obeyed the diffusion equation 8p _ D 8 2p _ 0 (8.20) 8t 2 8x 2 - , where (8.21) is the diffusion coefficient. So Brownian motion is the mechanism behind diffusion.
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8.2.2 Langevin's approach to Brownian motion
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Three years after Einstein's paper, Paul Langevin revisited the problem of Brownian motion, adopting a more direct approach [392}. Assuming viscous friction, he wrote down the following equation of motion for a particle of pollen of mass m:
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where "I is the coefficient of friction and F is the random fluctuating force that the particle experiences due to the impact of the thermally agitated molecules of the liquid. Multiplying (8.22) by x and noticing that
2 2 2 d x 1d x (dx)2 dt2 = 2" dt 2 dt '
(8.23)
we obtain (8.24) Then Langevin takes the average of (8.24) over a large number of particles. He assumes that (xF) vanishes because the random force is uncorrelated to x and fluctuates equally between positive and negative values. He also notices that in thermal equilibrium,7 (8.25)
7This follows from the equipartition theorem (see. e.g. [515]).
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where k is Boltzmann's constant and T is the absolute temperature. Thus, the average of (8.24) yields (8.26) dt mdt m Calling d(x 2 ) / dt X, we have a first-order nonhomogeneous ordinary differential equation for X. The general solution of this equation is an arbitrary linear combination of the particular solution with the solution of the associated homogeneous equation (the equation where the right-hand side of this equation is set to zero):
d( _ kT - x2) --+ C e -"ftlm dt 'Y '
~(X2) + .1.~(x2) = kT. 2
(8.27)
where C is an arbitrary constant. Now notice that for high-enough viscosity such that 'Y /m l/T, the exponential in the solution above will be negligible in any observable time interval. Integrating once more, Langevin then arrived at the following result: (8.28) This is the same result that one obtains starting with the diffusion equation (8.20), provided that D = kT / 'Y. So Langevin determined Einstein's diffusion coefficient in terms of the temperature, the coefficient of viscous friction, and Boltzmann's constant.
8.2.3 The modern form of Langevin's equation used in laser physics
You can often get more insight by deriving a well-known result in a different way. Let us reflect a bit more about Langevin's approach. What happens if we use the assumption of quite high viscosity b /m l/T) right from the beginning Then neglecting m ~x/ dt 2 compared with 'Y dx/ dt, instead of (8.22) we can write
dx dt =F,
(8.29)
where F == 'YF. Now let us try to obtain the properties of F without using the thermal equilibrium result (8.25) explicitly. The question we ask is: What properties must F have for this simple Langevin equation to be equivalent to the diffusion equation (8.20) deduced by Einstein The ensemble average of F can be given in terms of that of x by averaging (8.29):
(F(t))
dt (x).
(8.30)
We can calculate (x) if we determine the probability distribution p(x, t) that obeys Einstein's diffusion equation (8.20). Assuming that the particle starts at the origin at t = 0, you can show (Problem 8.2) that the solution of (8.20) is given by