Complete Soft Matter Physics

Brownian Motion

Brownian motion is random motion driven by thermal collisions with a surrounding fluid. Its central lesson is that microscopic thermal noise produces macroscopic stochastic trajectories.

Diffusion Equation

For concentration $c(\mathbf r,t)$, diffusion is modeled by

$$\frac{\partial c}{\partial t}=D\nabla^2 c.$$

The diffusion constant $D$ sets the spreading scale $\langle r^2\rangle \sim 2dDt$ in $d$ dimensions.

Stokes-Einstein Relation

For a spherical particle of radius $a$ in a fluid of viscosity $\eta$,

$$D=\frac{k_BT}{6\pi\eta a}.$$

This relation connects thermal motion to viscous drag.

Polymer Chain Statistics

A simple polymer can be modeled as a random walk of segments. For $N$ segments of length $b$, the typical end-to-end size scales as

$$R\sim b\sqrt{N}$$

for an ideal chain.

Osmotic Pressure

Solute particles separated by a semipermeable membrane create osmotic pressure. In the dilute limit it has the ideal form

$$\Pi = c k_B T,$$

where $c$ is the solute number density.

Liquid Crystals

Liquid crystals combine fluidity with orientational or partial positional order. The nematic phase is described by orientational alignment without full crystalline order.