2. Brownian Motion

1. Historical Background

2. Characteristic Scales Of Brownian Motion

3. Random Walk

4. Brownian Motion, Random Force And Friction: The Langevin Equation

5. Solving The Langevin Equation: Approximations And Orders Of Magnitude

6. Applications And Implications

2.1. Historical Background

• Continuum vs Atomism– Mach vs Boltzmann

• Einstein’s kinetic theory on Brownian Motion– atomism

• Fluctuation-dissipation theorem– Transport, Noises, …

2.2. Characteristic Scales Of Brownian Motion

Brownian motion:

Random motion of particles (size ~ m) hovering in gas or on surface of fluid.

26 3 20 3~ 10 10n m cm

23 211.38 10 300 5 10kT J Thermal energy of a gas particle at room temperature:

Density of gas:

Mass density of each hovering particle: 30.1 g cm

Mass of hovering particle: 340.1 10M 1310 g

In equilibrium: 21

2MV

1 2 26 12 10 5 10 10 /m s 2 2V

M

Speed:

Mass of the gas molecule: 2310 gSpeed:

310 /m s

L = mean free path of hovering particleS = l 2 = its cross section n = number density of gas

1nLS

24 210S cm 20 310n cm

12 88 20

110 ~ 10

10 10L cm l

12

210

Ls

V = mean free time

2.3. Random Walk

Brownian motion: R 0 2 tR

Major achievement of Einstein’s theory :Evaluate using kinetic theory.

Random walk of a drunk

1N N L R R n 2 1n

N R 0

221N N L R R n

2 2 21 2N N NL L R R n R 2 2

N L R

20 0R 2 2

1 LR 2 2 2 22 1 2L L R R

2 2N NLR

2Lt

2 2 2N NL L R n R

Average over behavior of many drunks

= Average over different starting points of one drunk

( Average over Ensemble )

( Average over Initial Conditions )

See Ex.2.2