Brownian Motion, Thermal Fluctuations and Diffusion

Instructor: Cao Xin
中文版：布朗运动
Back to: Soft Matter Physics Course Notes

Core Picture

Brownian motion is the random motion of a suspended particle caused by molecular thermal collisions. At short time it is random force plus damping; at long time it becomes diffusion.

Langevin Equation

Langevin=$\frac{dV}{dt}=-\Gamma V+L(t)$

relaxation=$\tau=\Gamma^{-1}=\frac{M}{6\pi\eta R}$

overdamped=$\frac{dX}{dt}=\Gamma^{-1}\tilde L(t)$

For t much larger than tau, velocity thermalizes quickly and position becomes a Wiener process.

equipart=$\frac{1}{2}M\langle V^2\rangle=\frac{1}{2}k_B T$

FDT=$\gamma=\frac{k_B T}{M}\Gamma$

MSD=$\langle x^2\rangle=2Dt$

Stokes-Einstein=$D=\frac{k_B T}{6\pi\eta R}$

Fokker-Planck Equation

A stochastic differential equation can be rewritten as an evolution equation for probability distribution.

SDE=$\frac{d\xi}{dt}=a(\xi)+bL(t)$

FP=$\partial_tP=-\partial_\xi[aP]+\gamma\partial_\xi^2[b^2P]$

diffusion=$\partial_tP=D\partial_X^2P$

solution=$P(X,t)=(4\pi Dt)^{-1/2}e^{-X^2/(4Dt)}$

The Fokker-Planck equation for velocity requires Maxwell distribution as the stationary state, giving the fluctuation-dissipation relation.

velocity FP=$\partial_tP=\partial_V[(\Gamma V+\gamma\partial_V)P]$

Maxwell=$P_{eq}(V)\propto e^{-MV^2/(2k_BT)}$

Dimensional MSD

1D=$\langle\Delta x^2\rangle=2Dt$

2D=$\langle\Delta r^2\rangle=4Dt$

3D=$\langle\Delta r^2\rangle=6Dt$

Active Brownian Particles

Active particles consume energy and self-propel, so motion is no longer purely thermal.

passive=$v_0=0$

active=$v_0>0$

MIPS=motility-induced phase separation

Exam Cue

For random force, write Langevin. For distributions, write Fokker-Planck. For long-time position statistics, write MSD and D. The key thread is thermal noise plus damping gives diffusion.