---
title: Brownian Motion and the Langevin Equation
module: Fluctuations and Response
moduleNumber: 12
lessonNumber: 2
order: 1202
summary: >
  A pollen grain in water executes a random walk driven by molecular collisions.
  Einstein tied its diffusion constant to its mobility, $D=\mu_{\mathrm{mob}}k_BT$,
  turning a visible motion into a measurement of Avogadro's number. The Langevin
  equation splits the collisions into a systematic drag and a random force whose
  strength is fixed by the drag through $\langle\xi(t)\xi(t')\rangle=2\gamma
  k_BT\,\delta(t-t')$ — the first fluctuation–dissipation relation. The
  mean-square displacement grows ballistically at short times and linearly,
  $\langle r^2\rangle=2dDt$, at long times, and the Stokes–Einstein relation
  $D=k_BT/6\pi\eta a$ closes the loop to Perrin's experiments.
topics: [Fluctuations and Response]
sources:
  - book: Einstein
    ref: "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung (Ann. Phys. 17, 549, 1905)"
  - book: Reif
    ref: "Ch. 15 — Irreversible Processes and Fluctuations; §15.5–15.8"
  - book: Kardar (Statistical Physics of Particles)
    ref: "Ch. 9 — nonequilibrium; the Langevin equation and Brownian motion"
  - book: Pathria & Beale
    ref: "Ch. 13 — Fluctuations and Nonequilibrium Statistical Mechanics; §13.3–13.4"
draft: false
---

A grain of pollen suspended in water jitters ceaselessly, a motion Brown observed
under a microscope in 1827 and could not explain. The explanation, given by
Einstein in 1905, is that the grain is bombarded from all sides by water molecules
in thermal motion, and the imbalance between collisions on opposite faces at any
instant drives it on a random walk.[^einstein-1905] The argument connects a
visible, macroscopic motion to the invisible molecular one, and it does so
quantitatively: the diffusion of the grain measures the thermal energy per
molecule, hence Avogadro's number. This lesson builds the theory in two
complementary forms. Einstein's diffusion picture tracks the probability
distribution of the grain's position and yields the relation between diffusion and
mobility. Langevin's picture writes a stochastic equation of motion, splits the
molecular force into a drag and a random part, and fixes the random part by
demanding thermal equilibrium — the first instance of a fluctuation–dissipation
relation.

## The random walk and diffusion

Model the grain's displacement as a sum of independent random steps. In one
dimension, after $n$ steps of independent displacements $\ell_i$ with
$\langle\ell_i\rangle=0$ and $\langle\ell_i\ell_j\rangle=\langle\ell^2\rangle\,\delta_{ij}$,
the total displacement $x=\sum_i\ell_i$ has zero mean and variance

$$
\langle x^2\rangle=\sum_{i,j}\langle\ell_i\ell_j\rangle=n\,\langle\ell^2\rangle.
$$

The mean-square displacement grows linearly in the number of steps, hence linearly
in time if steps occur at a fixed rate. Writing $\langle x^2\rangle=2Dt$ defines
the **diffusion constant** $D$. The same content appears as a partial differential
equation for the probability density $P(x,t)$ of finding the grain at $x$: the
continuity of probability under a current proportional to the concentration
gradient, $J=-D\,\partial P/\partial x$, gives the **diffusion equation**

$$
\frac{\partial P}{\partial t}=D\,\frac{\partial^2 P}{\partial x^2},
$$

whose solution from a point release at the origin is a spreading Gaussian,

$$
P(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\!\left[-\frac{x^2}{4Dt}\right],
\qquad
\langle x^2\rangle=2Dt.
$$

In $d$ dimensions the independent Cartesian components add, giving
$\langle r^2\rangle=2dDt$. The linear growth of $\langle r^2\rangle$ in time is the
signature of diffusion, distinct from the $\langle r^2\rangle\propto t^2$ of
ballistic motion; it holds once many uncorrelated collisions have occurred.

$$
% caption: A Brownian trajectory is a random walk; successive independent displacements accumulate so that the mean-square distance from the start grows linearly in time, $\langle r^2\rangle=2dDt$, not as the square of time.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
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\draw[acc,thick]
 (0,0)--(0.4,0.5)--(0.2,1.1)--(0.9,1.3)--(0.7,0.7)--(1.4,0.9)--(1.2,1.6)
 --(1.9,1.8)--(2.4,1.3)--(2.2,2.0)--(2.9,2.2)--(3.4,1.7)--(3.2,2.4)
 --(3.9,2.7)--(4.4,2.2)--(4.2,3.0)--(4.9,3.1)--(5.3,2.6)--(5.1,3.3);
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\node[black!70,font=\scriptsize] at (3.4,0.7) {displacement grows as $t^{\frac{1}{2}}$};
\end{tikzpicture}
$$

## The Einstein relation between diffusion and mobility

Diffusion and drag are two faces of the same molecular collisions, and Einstein
tied them together with an equilibrium argument. Apply a weak steady force $F$ to
each suspended particle — gravity on a colloidal grain, for instance. The force
drives a steady drift at velocity $v_{\mathrm{drift}}=\mu_{\mathrm{mob}}F$, where
$\mu_{\mathrm{mob}}$ is the **mobility** (drift velocity per unit force). The drift
carries a particle current $J_{\mathrm{drift}}=n\,\mu_{\mathrm{mob}}F$ down the
force. Opposing it, the concentration gradient that builds up drives a diffusion
current $J_{\mathrm{diff}}=-D\,\partial n/\partial x$ back.

In equilibrium the two currents cancel and the concentration is barometric,
$n(x)\propto e^{-U(x)/k_BT}$ with $F=-\partial U/\partial x$. Then
$\partial n/\partial x=-nF/k_BT$, and the balance $J_{\mathrm{drift}}+J_{\mathrm{diff}}=0$
reads

$$
n\,\mu_{\mathrm{mob}}F=D\,\frac{nF}{k_BT}
\quad\Longrightarrow\quad
D=\mu_{\mathrm{mob}}\,k_BT.
$$

> **Theorem (Einstein relation).** The diffusion constant and the mobility of a
> particle in a thermal medium are proportional, with the temperature as the
> constant of proportionality:
> $$D=\mu_{\mathrm{mob}}\,k_BT.$$

The relation ties a fluctuation (diffusion, the spreading driven by random kicks)
to a dissipation (mobility, the response to a steady force). It is the first
appearance of the fluctuation–dissipation principle: the same collisions that
randomly push the particle also resist its steady drift, and the two effects are
locked together by $k_BT$. Nothing in the derivation depends on the mechanism of
the drag, only on the medium being at temperature $T$.

$$
% caption: A steady force drives a drift current down the concentration gradient while diffusion drives a current back up; in equilibrium the barometric profile balances the two, forcing $D=\mu_{\mathrm{mob}}k_BT$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(0,4.4) node[above,black]{height};
\draw[->,black] (0,0)--(4.8,0) node[right,black]{concentration $n$};
\draw[very thick] plot[domain=0:4.1,samples=60] ({3.6*exp(-0.55*\x)},\x);
\node[align=center] at (1.9,3.9) {barometric\\density};
\draw[acc,->,very thick] (2.8,2.2)--(2.8,1.4);
\node[acc,right,font=\scriptsize] at (2.9,1.8) {drift down};
\draw[acc,->,very thick] (2.8,2.6)--(2.8,3.4);
\node[acc,right,font=\scriptsize] at (2.9,3.0) {spreading up};
\end{tikzpicture}
$$

## The Langevin equation

Langevin recast the problem as a stochastic equation of motion. The water exerts
two forces on the grain: a systematic drag opposing its velocity, and a rapidly
varying random force from individual collisions. For a grain of mass $m$ and
velocity $v$,

$$
m\,\frac{\d v}{\d t}=-\gamma\,v+\xi(t),
$$

where $\gamma$ is the **friction coefficient** — related to the mobility by
$\mu_{\mathrm{mob}}=1/\gamma$, since a steady force $F$ balances the drag at
$v=F/\gamma$ — and $\xi(t)$ is the random force. The split is a separation of time
scales: the drag is the slow average of the collisions, and $\xi$ is the fast
fluctuating remainder with zero mean, $\langle\xi(t)\rangle=0$, so that
$\langle v\rangle$ obeys the deterministic decay $m\,\d\langle v\rangle/\d t=-\gamma\langle v\rangle$.

$$
% caption: The molecular bombardment on a Brownian grain separates into a systematic drag $-\gamma v$ opposing the motion and a rapidly varying random force $\xi(t)$ of zero mean; the two are the slow and fast parts of the same collisions.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[very thick,fill=black!6] (0,0) circle (0.9);
\node[black] at (0,0) {grain};
\draw[->,black,very thick] (0.9,0)--(2.5,0) node[right,black]{velocity $v$};
\draw[->,black,very thick] (-0.9,0)--(-2.3,0) node[left,black,align=center]{drag\\opposes $v$};
\foreach \a in {35,80,130,175,220,270,315}{
  \draw[->,acc] ({1.25*cos(\a)},{1.25*sin(\a)})--({0.95*cos(\a)},{0.95*sin(\a)});
}
\node[acc,font=\scriptsize] at (0,-1.75) {random kicks};
\end{tikzpicture}
$$

## Fixing the noise by the fluctuation–dissipation relation

The random force cannot have an arbitrary strength. If it were too weak, drag
would drain the grain's kinetic energy and it would fall still; if too strong, the
grain would heat without bound. Thermal equilibrium at temperature $T$ pins the
strength exactly. Model $\xi(t)$ as Gaussian white noise: uncorrelated from one
instant to the next,

$$
\langle\xi(t)\rangle=0,
\qquad
\langle\xi(t)\,\xi(t')\rangle=2B\,\delta(t-t'),
$$

with $B$ the noise strength to be determined. Solve the linear equation for the
velocity,

$$
v(t)=v(0)\,e^{-\gamma t/m}+\frac1m\int_0^t e^{-\gamma(t-t')/m}\,\xi(t')\,\d t',
$$

square it, and average using the correlation of $\xi$. At long times the initial
condition decays and the stationary variance is

$$
\langle v^2\rangle=\frac{1}{m^2}\int_0^\infty\!\!\int_0^\infty
e^{-\gamma(t_1+t_2)/m}\,2B\,\delta(t_1-t_2)\,\d t_1\,\d t_2
=\frac{B}{\gamma m}.
$$

Equipartition fixes the left side: a grain in equilibrium has
$\tfrac12 m\langle v^2\rangle=\tfrac12 k_BT$ per translational degree of freedom,
so $\langle v^2\rangle=k_BT/m$. Matching the two expressions,

$$
\frac{B}{\gamma m}=\frac{k_BT}{m}
\quad\Longrightarrow\quad
B=\gamma\,k_BT,
\qquad
\langle\xi(t)\,\xi(t')\rangle=2\gamma\,k_BT\,\delta(t-t').
$$

> **Theorem (Fluctuation–dissipation, Langevin form).** For the Langevin equation
> $m\dot v=-\gamma v+\xi$ to reach thermal equilibrium at temperature $T$, the
> random force must satisfy
> $$\langle\xi(t)\,\xi(t')\rangle=2\gamma\,k_BT\,\delta(t-t').$$
> The strength of the fluctuating force is set by the friction coefficient and the
> temperature.

The friction $\gamma$ appears twice: as the coefficient of the systematic drag
(dissipation) and as the strength of the random force (fluctuation). They are not
independent because both arise from the same molecular collisions. A medium that
resists motion strongly also kicks strongly, and the ratio is fixed by $k_BT$.
This is the microscopic content of the Einstein relation and the direct ancestor
of the general [fluctuation–dissipation theorem](/statistical-mechanics/fluctuations/linear-response-and-the-fluctuation-dissipation-theorem).

$$
% caption: The friction coefficient sets both the systematic drag and the strength of the random force; the two are locked by $\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t')$, so a strongly damping medium is a strongly fluctuating one.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
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\node[draw,thick,fill=black!6,align=center,minimum width=2.8cm,minimum height=1.1cm] (g) at (0,0) {friction};
\node[draw,black,thick,fill=black!6,align=center,minimum width=3.0cm,minimum height=1.1cm] (d) at (5.4,1.4) {drag: dissipation};
\node[draw,black,thick,fill=black!6,align=center,minimum width=3.0cm,minimum height=1.1cm] (f) at (5.4,-1.4) {noise: random force};
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\draw[<->,acc,thick] (d)--(f) node[midway,right,acc,font=\scriptsize,align=left]{locked by\\$k_BT$};
\end{tikzpicture}
$$

## Velocity correlation and the mean-square displacement

The same solution gives the velocity autocorrelation function. For $t\ge0$,

$$
\langle v(t)\,v(0)\rangle=\frac{k_BT}{m}\,e^{-\gamma t/m},
$$

an exponential decay on the momentum relaxation time $\tau=m/\gamma$. The grain
remembers its velocity for a time $\tau$, then the memory is erased by collisions.
The diffusion constant is the time integral of this correlation, a relation of the
[Green–Kubo type](/statistical-mechanics/fluctuations/linear-response-and-the-fluctuation-dissipation-theorem),

$$
D=\int_0^\infty\langle v(t)\,v(0)\rangle\,\d t
=\frac{k_BT}{m}\cdot\frac{m}{\gamma}
=\frac{k_BT}{\gamma}=\mu_{\mathrm{mob}}\,k_BT,
$$

recovering the Einstein relation from the dynamics. Integrating twice gives the
mean-square displacement at all times,

$$
\langle x^2(t)\rangle=\frac{2k_BT}{\gamma}
\left[t-\frac{m}{\gamma}\big(1-e^{-\gamma t/m}\big)\right],
$$

with two regimes separated by $\tau=m/\gamma$:

- **Ballistic**, $t\ll\tau$: expanding the exponential gives
  $\langle x^2\rangle\approx(k_BT/m)\,t^2$, the free-flight motion of a particle at
  the thermal speed $\sqrt{k_BT/m}$ before it has collided.
- **Diffusive**, $t\gg\tau$: the bracket is $\approx t$, so
  $\langle x^2\rangle\approx2Dt$ with $D=k_BT/\gamma$, the random walk.

$$
% caption: The mean-square displacement crosses over from ballistic growth $\langle x^2\rangle\propto t^2$ at times shorter than the momentum relaxation time $\tau=m/\gamma$ to diffusive growth $\langle x^2\rangle=2Dt$ at longer times.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(7.2,0) node[right,black]{$t$};
\draw[->,black] (0,0)--(0,4.3) node[above,black]{mean square $x$};
\draw[black,thick,densely dotted] plot[domain=0:2.7,samples=50] (\x,{0.214*\x*\x});
\node[black!70,font=\scriptsize] at (1.15,3.05) {ballistic};
\draw[acc,thick,dashed] plot[domain=1.4:6.9,samples=40] (\x,{0.6*\x-0.84});
\node[acc,font=\scriptsize] at (5.5,2.55) {random walk};
\draw[black,very thick] plot[domain=0:6.9,samples=140] (\x,{0.6*(\x-1.4*(1-exp(-\x/1.4)))});
\draw[black,dashed] (1.4,0)--(1.4,1.05);
\node[black!70,font=\scriptsize,below] at (1.4,0) {relaxation time};
\end{tikzpicture}
$$

## The Stokes–Einstein relation and Avogadro's number

The friction on a sphere of radius $a$ moving slowly through a fluid of viscosity
$\eta$ is given by Stokes' law, $\gamma=6\pi\eta a$. Substituting it into
$D=k_BT/\gamma$ gives the **Stokes–Einstein relation**,

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

Every quantity on the right except $k_B$ is measurable: the temperature, the
fluid's viscosity, and the grain's radius under the microscope. Every quantity on
the left is measurable by tracking the grain, since $\langle r^2\rangle=2dDt$
gives $D$ from the slope of the mean-square displacement against time. The relation
therefore delivers $k_B$, and with the gas constant $R=N_A k_B$ known from the
ideal-gas law it delivers Avogadro's number $N_A=R/k_B$.

> **Worked example.** Perrin tracked colloidal grains of radius
> $a\approx0.2\ \mathrm{\mu m}$ in water ($\eta\approx1.0\times10^{-3}\ \mathrm{Pa\,s}$)
> at $T\approx300\ \mathrm{K}$. The predicted diffusion constant is
> $$
> D=\frac{k_BT}{6\pi\eta a}
> =\frac{(1.38\times10^{-23})(300)}{6\pi(1.0\times10^{-3})(0.2\times10^{-6})}\ \mathrm{m^2\,s^{-1}}
> \approx1.1\times10^{-12}\ \mathrm{m^2\,s^{-1}}.
> $$
> In one second the root-mean-square displacement in the plane is
> $\sqrt{\langle r^2\rangle}=\sqrt{4Dt}\approx2\ \mathrm{\mu m}$, a distance
> resolvable under an optical microscope. Measuring $D$ this way and inverting the
> Stokes–Einstein relation gave $N_A\approx6\times10^{23}$, confirming the
> molecular hypothesis and earning Perrin the 1926 Nobel Prize.

Perrin's experiments settled the physical reality of molecules. A visible grain,
too large to be a molecule and too small to ignore the molecular buffeting,
translated the thermal energy $k_BT$ of the invisible medium into a measurable
diffusion, and the number that came out matched Avogadro's number obtained from
entirely independent methods.[^perrin]

## Summary

- A Brownian particle executes a random walk; independent steps make
  $\langle r^2\rangle$ grow linearly in time, $\langle r^2\rangle=2dDt$, and the
  probability density obeys the diffusion equation with a spreading-Gaussian
  solution.
- The **Einstein relation** $D=\mu_{\mathrm{mob}}k_BT$ follows from balancing drift
  and diffusion currents in a barometric equilibrium; it ties a fluctuation
  (diffusion) to a dissipation (mobility) through $k_BT$.
- The **Langevin equation** $m\dot v=-\gamma v+\xi$ splits the molecular force into
  drag and noise; equilibrium forces the fluctuation–dissipation relation
  $\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t')$, so the same $\gamma$
  sets both the drag and the noise strength.
- The velocity autocorrelation decays as $(k_BT/m)e^{-\gamma t/m}$; its time
  integral is $D=k_BT/\gamma$, and the mean-square displacement crosses over from
  ballistic $t^2$ at $t\ll m/\gamma$ to diffusive $2Dt$ at $t\gg m/\gamma$.
- The **Stokes–Einstein relation** $D=k_BT/6\pi\eta a$ made the grain's diffusion a
  measurement of $k_B$, hence of Avogadro's number; Perrin's confirmation
  established the molecular hypothesis.

[^einstein-1905]: **Einstein**, "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen," _Annalen der Physik_ **17**, 549 (1905) — the diffusion equation for the suspended particle, $\langle x^2\rangle=2Dt$, and the relation $D=k_BT/6\pi\eta a$. Translation and history at <https://www.physik.uni-augsburg.de/theo1/hanggi/History/BM-History.html>.
[^perrin]: **Reif**, _Fundamentals of Statistical and Thermal Physics_, §15.5–15.8, and **Pathria & Beale**, _Statistical Mechanics_ (4th ed.), §13.3–13.4 — the Langevin equation, the velocity autocorrelation, the mean-square displacement, and Perrin's determination of Avogadro's number.