Brownian Motion

In 1827, Robert Brown was studying pollen grains of the plant Clarkia pulchella suspended in water under a microscope when he observed minute particles, ejected by the pollen grains, executing a jittery motion. Let us examine this phenomenon, which is known as Brownian motion.

Consider a particle of mass $m$ that is suspended in a fluid. Let us investigate the motion of this particle parallel to the $x$-axis. The particle is subject to two types of force. First, a set of impulsive forces due to molecular bombardment. Second, a retarding force that is proportional to the particle's instantaneous speed through the surrounding fluid. Thus, the particle's equation of motion along the $x$-axis can be written

$$m\,\frac{d^2 x}{dt^2} = X(t)-\alpha\,\frac{dx}{dt},$$ (5.197)

where $X(t)$ is the impulsive force due to molecular bombardment, and $-\alpha\,dx/dt$ the retarding force. It follows that

$$m\,x\,\frac{d^2 x}{dt^2} = x\,X -\alpha\,x\,\frac{dx}{dt},$$ (5.198)

which can also be written

$$m\,\frac{d}{dt}\!\left(x\,\frac{dx}{dt}\right)-m\left(\frac{dx}{dt}\right)^2 = x\,X-\frac{\alpha}{2}\,\frac{d x^2}{dt}.$$ (5.199)

Taking the ensemble average of the previous equation, we obtain

$$m\,\frac{d}{dt}\!\left(\left\langle x\,\frac{dx}{dt}\right\rangle... ...angle x\,X\rangle-\frac{\alpha}{2}\,\frac{d \left\langle x^2\right\rangle}{dt}.$$ (5.200)

However,

$$\langle x\,X\rangle = 0,$$ (5.201)

because $x$ and $X$ are uncorrelated random variables whose mean values are zero. Furthermore,

$$\left\langle x\,\frac{dx}{dt}\right\rangle=0,$$ (5.202)

because $x$ and $dx/dt$ are also uncorrelated random variables whose mean values are zero. Finally,

$$\frac{1}{2}\,m\left\langle\left(\frac{dx}{dt}\right)^2 \right\rangle= \frac{1}{2}\,k_B\,T,$$ (5.203)

by the law of equipartition of energy, where $T$ is the temperature of the fluid. (See Section 5.3.4.) It follows from the previous four equations that

$$\frac{d \left\langle x^2\right\rangle}{dt}= \frac{2\,k_B\,T}{\alpha},$$ (5.204)

which can be integrated to give

$$\left\langle x^2\right\rangle = 2\,D\,t,$$ (5.205)

where

$$D =\frac{k_B\,T}{\alpha}.$$ (5.206)

It can be seen, by comparison with the analysis of Sections 5.1.5 and 5.1.7, that molecular bombardment causes a particle immersed in a fluid to execute a random walk along the $x$-axis with diffusivity $D$.

Suppose that the particle is a sphere of radius $a$. Furthermore, suppose that the retarding force acting on the particle is due to fluid viscosity. According to Stokes's law,

$$\alpha = 6\pi\,\eta\,a,$$ (5.207)

where $\eta$ is the viscosity of the fluid. It follows that

$$D = \frac{k_B\,T}{6\pi\,\eta\,a}.$$ (5.208)

This result, which was first obtained by Einstein in 1905, and was verified experimentally by Jean B. Perrin in 1910, served as the first convincing evidence of the existence of atoms and molecules. Note that the previous diffusivity scales as the inverse of the particle radius. Hence, only relatively small particles are likely to exhibit noticable Brownian motion.