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.