% miko:type: complete
% miko:title: Complete Soft Matter Physics
% miko:summary: A route through Brownian motion, diffusion, polymers, osmotic pressure, and liquid crystals.
% miko:tags: softmatter
% miko:difficulty: undergrad
% miko:related: complete-statistical-mechanics, diffusion-equation
% miko:keywords: Brownian motion, polymer, colloid, liquid crystal

\documentclass{article}
\usepackage{amsmath}
\title{Complete Soft Matter Physics}
\begin{document}
\maketitle

\section{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.

\section{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.

\section{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.

\section{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.

\section{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.

\section{Liquid Crystals}

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

\end{document}
