Brownian Motion, Langevin and Fokker–Planck: The Physics of a Sampler
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The Wiener process
Einstein’s 1905 account of Brownian motion explained the jitter of pollen grains as the accumulated effect of molecular collisions. Its mathematical idealisation is the Wiener process $W_t$: $W_0 = 0$, independent increments, $W_t - W_s \sim \mathcal N(0, t-s)$ for $t>s$, and continuous sample paths.
The defining scaling is $\mathbb E[W_t^2] = t$, so typical displacement grows like $\sqrt t$. Over an interval $\Delta t$ the increment has size $\sqrt{\Delta t}$, which dwarfs $\Delta t$ as $\Delta t\to 0$ — this is why Wiener paths are nowhere differentiable and why stochastic calculus needs its own rules rather than the ordinary chain rule.
Concretely, a 1 µm-diameter bead in water at 20 °C has $D = kT/(6\pi\eta a) \approx 0.43\ \mathrm{µm^2\,s^{-1}}$. Its root-mean-square displacement along one axis is $\sqrt{2Dt}$:
| elapsed time | RMS displacement (one axis) |
|---|---|
| 1 s | 0.93 µm |
| 10 s | 2.93 µm |
| 100 s | 9.27 µm |
A hundredfold increase in time buys only a tenfold increase in distance. Diffusion is a poor way to travel and an excellent way to explore.
Fick’s law and the diffusion equation
Particles flow from crowded regions to empty ones, at a rate proportional to the concentration gradient — Fick’s first law, $\mathbf J = -D\nabla\rho$, with $\mathbf J$ the flux and $\rho$ the density. Combine it with conservation, $\partial_t\rho + \nabla\cdot\mathbf J = 0$, and the diffusion equation follows:
Started from a point source in one dimension, the solution is a Gaussian \(\rho(x,t) = (4\pi D t)^{-1/2}\exp\!\left(-x^2/4Dt\right)\) of variance $2Dt$ — the same $\sqrt t$ spreading, now as a statement about the whole density rather than one path.
The Langevin equation
Add a force. A heavy particle in a potential $V$, in a medium with friction coefficient $\gamma$, in the overdamped limit where inertia is negligible, obeys
Two terms, two roles: the drift $-\nabla V/\gamma$ pulls downhill, the noise kicks in random directions with a strength set by temperature. Their ratio is fixed by the fluctuation–dissipation relation, not free to choose — the same collisions that cause friction cause the kicks. The Einstein relation $D = \mu kT$, with mobility $\mu = 1/\gamma$, is exactly that statement.
Fokker–Planck: what the density does
Track the density instead of the path. For $dx = f(x)\,dt + \sqrt{2D}\,dW$ the density evolves as
The first term transports probability along the drift; the second smooths it. Setting $f = -\nabla U$ and trying \(p_\infty \propto e^{-U/D}\) gives \(D\nabla p_\infty = -\nabla U\,p_\infty = f p_\infty\), so the current $J$ vanishes identically. The stationary state is not merely stationary but has zero flux — detailed balance. With $U = V$ and $D = kT/\gamma$ this reproduces the Boltzmann distribution \(p_\infty \propto e^{-V/kT}\): equilibrium statistical mechanics falls out of a stochastic differential equation.
The line to score-based generative models
Now run the argument backwards. You want to sample from \(p_{\text{data}}\). Pick \(U = -\log p_{\text{data}}\) and $D = \tfrac12$:
Discretised with step $\eta$, this is \(x_{k+1} = x_k + \tfrac{\eta}{2}\nabla_x\log p(x_k) + \sqrt{\eta}\,z_k\) with $z_k\sim\mathcal N(0,I)$ — unadjusted Langevin. The only thing needed is the score $\nabla_x\log p$, which is free of the intractable normaliser, and that is precisely what a score network is trained to estimate. Song & Ermon (2019) run this at a ladder of noise levels, annealing from large to small; the continuous-time generalisation and its reverse-time SDE are covered in score-based SDEs.
Recap
- Wiener process: independent Gaussian increments with \(\mathbb{E}[W_t^2]=t\), so displacement scales as \(\sqrt{t}\) and paths are nowhere differentiable.
- Fick's law \(J=-D\nabla\rho\) plus conservation gives \(\partial_t\rho = D\nabla^2\rho\), with point-source variance \(2Dt\).
- Langevin combines drift and noise, tied together by fluctuation–dissipation; the Einstein relation is \(D=\mu kT\).
- Fokker–Planck governs the density; for \(f=-\nabla U\) the zero-current stationary state is \(p_\infty\propto e^{-U/D}\).
- Setting \(U=-\log p_{\text{data}}\) makes Langevin a sampler for the data distribution driven only by the score — the basis of score-based generative models.
References
- Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 322(8):549–560, 1905.
- Risken, H. The Fokker–Planck Equation: Methods of Solution and Applications, 2nd ed. Springer, 1989.
- Welling, M., & Teh, Y. W. Bayesian Learning via Stochastic Gradient Langevin Dynamics. ICML 2011.
- Song, Y., & Ermon, S. Generative Modeling by Estimating Gradients of the Data Distribution. NeurIPS 2019.
- Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., & Poole, B. Score-Based Generative Modeling through Stochastic Differential Equations. ICLR 2021.