6.5 The Einstein relation and fluctuation–dissipation

A Brownian particle does two things at once: it diffuses, characterised by $D$ , and it resists steady forcing, characterised by the friction coefficient $\gamma$ . These look like separate properties, but they are driven by the very same molecular collisions, and Einstein showed in 1905 that temperature ties them together.

Drift–diffusion balance

▶ Deriving the Einstein relation Derivation

Take a dilute suspension of Brownian particles in a weak external force $F$ pointing in the $-z$ direction (gravity, for concreteness). In steady state two currents oppose each other:

A drift current: each particle reaches terminal velocity $F/\gamma$ , giving $J_\text{drift} = c\,F/\gamma$ .
A diffusive counter-current down the concentration gradient the drift builds up, $J_\text{diff} = -D\,\partial c/\partial z$ .

In steady state the net current vanishes, $J_\text{drift} + J_\text{diff} = 0$ , so

\frac{1}{c}\frac{\partial c}{\partial z} \;=\; \frac{F}{D\,\gamma}.

But equilibrium already fixes the concentration profile: the particles sit in the potential of the force, so by the Boltzmann distribution $c(z) \propto e^{-Fz/k_B T}$ , giving $\partial\ln c/\partial z = -F/k_B T$ . Matching the two expressions for the logarithmic gradient,

\frac{F}{k_B T} \;=\; \frac{F}{D\,\gamma}.

The force cancels, leaving the Einstein relation:

D \;=\; \frac{k_B T}{\gamma}.

Diffusion and friction are reciprocal: a particle that is hard to push (large $\gamma$ ) is also slow to diffuse (small $D$ ), and raising the temperature speeds diffusion in direct proportion. For a sphere, substituting $\gamma = 6\pi\mu a$ from the previous lesson gives the Stokes–Einstein relation

D \;=\; \frac{k_B T}{6\pi\mu a}.

F (force)0.50

γ (friction)1.00

k_BT1.00

v_drift = F/γ0.500

l_eq = k_BT/F2.000

D = v_drift · l_eq = k_BT/γ1.000

External force F = 0.50 Friction γ = 1.00 Temperature k_BT = 1.00

At equilibrium, the drift current downward (F/γ per particle) is exactly cancelled by the diffusive current upward (D times the concentration gradient). The equilibrium profile is exponential with decay length k_BT/F (Boltzmann). Equating the two currents gives the **Einstein relation** D = k_BT/γ — fluctuations and dissipation arise from the same microscopic process.

The sedimenting drift and the spreading diffusion settle into a stationary exponential atmosphere of particles, its scale height set by the balance the Einstein relation expresses. The Stokes–Einstein relation underlies every method that reads molecular size from a diffusion measurement — dynamic light scattering and fluorescence correlation spectroscopy among them — by inverting $D$ for the hydrodynamic radius $a$ .

Fluctuation and dissipation

The Einstein relation is the first and simplest instance of the fluctuation–dissipation theorem. The friction coefficient $\gamma$ measures dissipation — how the system drains energy when driven. The diffusion coefficient $D$ measures fluctuation — the spontaneous random motion the system exhibits at rest. The theorem states these are not independent: both spring from the same molecular collisions, and $k_B T$ is the exchange rate between them. A system that dissipates strongly must also fluctuate strongly, and vice versa.

The principle reaches far beyond suspended particles. The thermal (Johnson–Nyquist) noise voltage across a resistor is tied to its resistance by the identical relation; the width of a spectral line is tied to the rate at which the emitter loses energy; and the irreducible noise floor of any sensitive detector is set by the dissipation in it. Wherever a system both responds to forcing and jitters on its own, the two are locked together by temperature.

⏳ The history — Einstein, Perrin, and the molecular reality of fluids

Einstein’s 1905 paper “On the motion of small particles suspended in a stationary liquid required by the molecular-kinetic theory of heat” was one of his three annus mirabilis papers. He argued that visible Brownian particles undergo a random walk driven by molecular collisions, that their mean-squared displacement grows linearly in time, and — most consequentially — that the diffusion coefficient is fixed by Boltzmann’s constant and the macroscopic friction, $D = RT/(6\pi\mu a N_A)$ , expressing the otherwise-hidden $k_B = R/N_A$ in terms of measurable quantities.

The prediction was directly testable. Jean Perrin spent 1908–1910 making the measurement: tracking individual mastic and gamboge grains under a microscope, recording their positions at fixed intervals, computing the mean-squared displacement, and inverting Einstein’s formula for Avogadro’s number. His value, $N_A \approx 7\times10^{23}$ , landed within twenty percent of the modern figure. After Perrin the molecular-kinetic theory of heat was no longer a hypothesis — the existence of atoms had been weighed on a microscope.